# Variational Weakly Supervised Sentiment Analysis with
Posterior Regularization

Ziqian Zeng, Yangqiu Song
Department of CSE, HKUST, Hong Kong, China
{zzengae<EMAIL_ADDRESS>

###### Abstract

Sentiment analysis is an important task in natural language processing (NLP).
Most of existing state-of-the-art methods are under the supervised learning
paradigm. However, human annotations can be scarce. Thus, we should leverage
more weak supervision for sentiment analysis. In this paper, we propose a
posterior regularization framework for the variational approach to the weakly
supervised sentiment analysis to better control the posterior distribution of
the label assignment. The intuition behind the posterior regularization is
that if extracted opinion words from two documents are semantically similar,
the posterior distributions of two documents should be similar. Our
experimental results show that the posterior regularization can improve the
original variational approach to the weakly supervised sentiment analysis and
the performance is more stable with smaller prediction variance.

##  1 Introduction

Sentiment analysis is a task of identifying the sentiment polarity expressed
in textual data Liu (2012). Most state-of-the-art sentiment analysis methods
in the literature are supervised methods which require many labeled training
data. However, human annotations in the real world are scarce. While we assume
there is abundant annotated data to train more and more complex models, there
is still a need to consider weakly supervised methods that require less human
annotation.

One way to perform weakly supervised sentiment analysis is using a
predefinited lexicon Turney (2002); Taboada et al. (2011). A lexicon consists
of many opinion words. For each opinion word, its polarity (positive or
negative) and strength (the degree to which the opinion word is positive or
negative) are annotated by domain experts. lexicon-based weakly supervised
methods perform a dictionary lookup and assign a polarity according to all
opinion words extracted from a document. A good lexicon requires high
precision and high coverage, which needs a lot of human effort.

Another way to do weakly supervised sentiment analysis is using limited
keywords Meng et al. (2018); Zeng et al. (2019). Compared with lexicon-based
methods, user-provided keywords require less human effort. Among keyword-based
methods, there are two directions. First, Meng et al. (2018) leveraged limited
keywords to expand more keywords and generate pseudo-labeled data, and then
performed self-training on real unlabeled data for model refinement. Possible
improvements of this direction include investigating more advanced keywords
expansion techniques to generate better pseudo-labeled samples Miller et al.
(2012) and developing more advanced self-training algorithmsCoden et al.
(2014).

Second, the Variational Weakly Supervised (VWS) sentiment analysis Zeng et al.
(2019) used target-opinion word pairs as supervision signal. Its objective
function is to predict an opinion word given a target word. For example, in a
sentence “the room is big,” “room” is a target word and “big” is an opinion
word. By introducing a latent variable (the sentiment polarity), they can
learn a well-approximated posterior distribution via optimizing the evidence
lower bound. The posterior probability here is the probability of a possible
polarity (e.g., positive or negative) given a document, which is a typical
sentiment classifier.

A potential issue with VWS is that optimizing the objective function may not
guide the role of the latent variable to be sentiment polarity. For example,
when half of reviews mention “big room” and half of reviews mention “small
room,” the latent variable is possibly related to the size of rooms, but the
expected role of the latent variable is the sentiment polarity of rooms. Hence
how to control and regularize the posterior distribution is very important.
One indirect way to control the posterior distribution is clever
initialization Ganchev et al. (2010). Originally, VWS aims to predict the
sentiment polarity of each aspect for the multi-aspect sentiment analysis. So
it uses the overall sentiment polarity to pretrain the model so that the
posterior distribution has a good initialization. The overall polarity and
polarity of each aspect are highly correlated. Thus, the initialization is
highly likely to be similar to the true posterior distribution.

In this paper, we propose to use posterior regularization to regularize the
VWS approach for sentiment analysis. There are two types of side information
we can leverage to regularize the latent variable. First, calculating the
similarity between keywords and extracted opinion words from a document can
guide the model to decide which polarity that the document belongs to. Second,
calculating the similarity between extracted opinion words from two documents
can guide the model to decide whether two documents belong to the same
polarity. The first type of side information is very easy to leverage, and it
is reflected by our pretraining process. When a document is similar to
keywords associated with a specific polarity, then we enforce that the
posterior probability of a specific polarity should be large. In this case, in
the pretraining process, we assign a pseudo label to the document. The second
type of side information does not directly suggest which sentiment polarity
that a document should be assigned to. It enforces pairwise constraints to the
model. Our proposed posterior regularization leverage the second side
information to ensure that when two documents are similar (dissimilar), the
regularization enforces the posterior distribution of two documents to be
similar (dissimilar).

Our contributions are summarized as follows,

$\bullet$ We develop a posterior regularization framework for the variational
weakly supervised sentiment analysis.

$\bullet$ The experimental results show that the proposed regularization can
improve the VWS model, make the results more stable, and outperform other
weakly supervised baselines.

Our code is available at https://github.com/HKUST-KnowComp/VWS-PR.

##  2 Methodology

In this section, we first review the variational weakly supervised (VWS)
sentiment analysis method in Section 2.1. Then we introduce our posterior
regularization in Section 2.2.

###  2.1 VWS Sentiment Analysis

Before formally introducing the VWS framwork, we give a concrete example to
illustrate how VWS works. Let $\mathbf{x}$ be the representation of a document
$x$, e.g., bag of words or feature outputs of neural networks. Let $C$ be a
random variable, indicating the sentiment polarity of a document. The possible
value assignment of $C$ can be positive or negative, or rating from $1$ to
$10$. Suppose there is a document $x$ where we extract an opinion word
“terrific.” The objective function is to maximize the probability of opinion
word “terrific.” By introducing a latent variable $C$, the objective function
is split into two probabilities, corresponding to two classifiers, namely,
sentiment polarity classifier and opinion word classifier. The input of
sentiment classifier is the document representation $\mathbf{x}$, and it
produces a probability distribution of sentiment polarity, i.e.,
$p(C=positive|\mathbf{x})$ and $p(C=negative|\mathbf{x})$. The input of
opinion word classifier is extracted opinion words and the estimated sentiment
polarity distribution, and it produces a probability distribution of opinion
word given estimated sentiment polarity distribution, i.e.,
$p(\text{``terrific''}|C=positive)$ and $p(\text{``terrific''}|C=negative)$.

####  2.1.1 Sentiment Polarity Classifier

The sentiment polarity classifier aims to estimate a distribution
$q(C|\mathbf{x})$, where $C$ is a discrete random variable representing the
sentiment polarity of a document. Let $c$ denote a possible value of the
random variable $C$, representing a possible value of sentiment polarity,
e.g., positive or negative. The sentiment classifier estimates the probability
as

$q(C=c|\mathbf{x})=\frac{\exp\big{(}\mathbf{w}_{c}^{T}\mathbf{x}\big{)}}{\sum_{c^{\prime}}{\exp\big{(}\mathbf{w}_{c^{\prime}}^{T}\mathbf{x}\big{)}}}\;,$
(1) $\mathbf{x}=CNN(x)\;,$ (2)

where $\mathbf{w}_{c}$ is a trainable vector associated with a sentiment
polarity $c$, $x$ is a document, and $\mathbf{x}$ is the document
representation. The representation of a document $\mathbf{x}$ can be various.
We use Convolutional Neural Network (CNN) in the experiment.

####  2.1.2 Opinion Word Classifier

The opinion word classifier aims to estimate the probability of an opinion
word $w_{o}$ given a possible value of sentiment polarity $c$:

$p(w_{o}|c)=\frac{\exp\big{(}\varphi({w_{o}},c)\big{)}}{\sum_{{w_{o}^{\prime}}}{\exp\big{(}\varphi({w_{o}^{\prime}},c)\big{)}}}\;,$
(3)

where $\varphi(\cdot)$ is a scoring function taking opinion word $w_{o}$ and a
possible value of sentiment polarity $c$ as inputs. The nature of the scoring
function is about the frequency of occurrence. If an opinion word and a
possible value of sentiment polarity co-occur frequently, the score will be
high, otherwise, it will be low. Specifically, we define:

$\varphi({w_{o}},c)=\mathbf{a}_{c}^{T}\mathbf{w}_{o}\;,$ (4)

where $\mathbf{w}_{o}$ is the trainable word embedding of opinion word
$w_{o}$, $\mathbf{a}_{c}$ is a trainable vector associated with $c$. The
scoring function can be various, e.g., multilayer perceptron (MLP). Here we
only introduce the simplest case.

Given a possible value of sentiment polarity $c$, VWS aims to maximize the
probability of opinion words that frequently occurred with $c$. For example,
the opinion word “good” is usually occurred with sentiment polarity positive,
and the opinion word “terrible” is usually occurred with sentiment polarity
negative.

####  2.1.3 Training Objective

The objective function of VWS is to maximize the log-likelihood of an opinion
word $w_{o}$. After introducing a latent variable (i.e., the sentiment
polarity of a document) to the objective function, we can derive a variational
lower bound of the log-likelihood which can incorporate two classifiers. The
first one corresponds to the sentiment classifier. The second one corresponds
to the opinion word classifier. The variational lower bound of log-likelihood
is shown as follows:

$\displaystyle{\mathcal{L}}_{1}$ $\displaystyle=\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}\log p(w_{o})$ $\displaystyle=\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}\log\sum_{c}p(w_{o},c)$
$\displaystyle=\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}\log\sum_{c}q(c|\mathbf{x})\Big{[}\frac{p(w_{o},c)}{q(c|\mathbf{x})}\Big{]}$
$\displaystyle\geq\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}\sum_{c}q(c|\mathbf{x})\Big{[}\log\frac{p(w_{o},c)}{q(c|\mathbf{x})}\Big{]}$
$\displaystyle=\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}\operatorname{\mathbb{E}}_{q(C|\mathbf{x})}\big{[}\log
p(w_{o}|c)p(c)\big{]}$ $\displaystyle+\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}H(q(C|\mathbf{x})),$ (5)

where $X$ is the training set containing all documents, and
${\mathcal{P}}_{x}$ is the set of all opinion words extracted from a document
$x$, $H(\cdot)$ refers to the Shannon entropy, and $q(c|\mathbf{x})$ is short
for $q(C=c|\mathbf{x})$. By applying Jensen’s inequality, the log-likelihood
is lower-bounded by Eq. (5). The equality holds if and only if the KL-
divergence of two distributions, $q(C|\mathbf{x})$ and $p(C|w_{o})$, equals to
zero. Maximizing the evidence lower bound is equivalent to minimizing the KL-
divergence. Hence, VWS can learn a sentiment classifier that can produce a
similar distribution to the true posterior $p(C|w_{o})$. We assume that the
training set is perfectly balanced, which means the prior distribution of
sentiment polarity, i.e., $p(C)$, is a uniform distribution. Hence, $p(c)$ is
a constant, which can be ignored.

####  2.1.4 Approximation

The partition function in Eq. (3) requires the summation over all opinion
words in the vocabulary. Since the size of the opinion word vocabulary is
large, VWS uses the negative sampling technique Mikolov et al. (2013) to
approximate Eq. (3). Specifically, VWS approximates $p(w_{o}|c)$ in the
objective (3) with the following objective function:

$\log\sigma\big{(}\varphi({w_{o}},c)\big{)}+\sum_{w_{o}^{\prime}\in{\mathcal{N}}}\log\big{(}1-\sigma\big{(}\varphi({w_{o}^{\prime}},c)\big{)}\big{)}\;,$
(6)

where $w_{o}^{\prime}$ is a negative sample in opinion words vocabulary,
${\mathcal{N}}$ is the set of negative samples and $\sigma(\cdot)$ is the
sigmoid function. In order to ensure that the approximation part and the
entropy term are on the same scale Marcheggiani and Titov (2016), a hyper-
parameter $\alpha$ is added to the entropy term. The objective function
becomes:

$\displaystyle{\mathcal{L}}_{2}$ $\displaystyle=\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}\operatorname{\mathbb{E}}_{q(C|\mathbf{x})}\big{[}\log\sigma\big{(}\varphi({w_{o}},c)\big{)}$
$\displaystyle+\sum_{w_{o}^{\prime}\in{\mathcal{N}}}\log\big{(}1-\sigma\big{(}\varphi({w_{o}^{\prime}},c)\big{)}\big{)}+\log
p(c)\big{]}$ $\displaystyle+\sum_{x\in
X}\sum_{w_{o}\in{\mathcal{P}}_{x}}\alpha H(q(C|\mathbf{x})).$ (7)

###  2.2 Posterior Regularization

As pointed out by Ganchev et al. (2010), controlling the posterior
distribution is crucial for models that estimate posterior distribution by
maximizing the likelihood of the observed data via marginalizing over the
latent variables. We need side information to regularize the posterior
distribution. The side information we leveraged is that if the opinion words
extracted from two documents are similar semantically, then these two
documents probably are in the same class, and if the opinion words are
opposite semantically, then these two documents are probably not in the same
class. For example, if one document $x^{i}$ contains opinion words “great” and
“awesome,” another document $x^{j}$ contains opinion words “great” and
“excellent,” and another document $x^{k}$ contains opinion words “awful” and
“terrible,” it is highly possible that $x^{i}$ and $x^{j}$ belong to the same
class because their extracted opinion words are similar semantically, and
$x^{i}$ and $x^{k}$ do not belong to the same class because their extracted
opinion words are opposite semantically.

We formulate our posterior regularization as:

$\displaystyle{\mathcal{R}}(x^{i},x^{j})$
$\displaystyle=-d(x^{i},x^{j})\cdot{\mathcal{S}}({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j})),$
(8)

where $d(x^{i},x^{j})$ is short for
$d(q(C|\mathbf{x}^{i}),q(C|\mathbf{x}^{j}))$, meaning the distance of two
posterior distributions, and we use Euclidean distance metric;
${\mathcal{S}}(\cdot,\cdot)$ is a score function which measures the similarity
or dissimilarity between two sets of opinion words; ${\mathcal{O}}(x^{i})$
represents all opinion words extracted from a document $x^{i}$. We finally
maximize Eq. (8) in the objective function. When
${\mathcal{S}}({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))$ is positive
(suggesting similar), this regularization enforces the distance to be small,
and when ${\mathcal{S}}({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))$ is
negative (suggesting dissimilar), this regularization enforces the distance to
be large. When ${\mathcal{S}}({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))$ is
zero, it suggests comparison between opinion words cannot decide whether two
documents are similar or not.

Next, we will introduce the scoring function ${\mathcal{S}}(\cdot,\cdot)$.
Suppose a document $x^{i}$ contains a set of opinion words
${\mathcal{O}}(x^{i})=\\{w^{i}_{o_{1}},w^{i}_{o_{2}},\cdots,w^{i}_{o_{k}}\\}$
and a document $x^{j}$ contains a set of opinion words
${\mathcal{O}}(x^{j})=\\{w^{j}_{o_{1}},w^{j}_{o_{2}},\cdots,w^{j}_{o_{k}}\\}$.
We define an operation $cos({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))$ over
two sets of opinion words. It will return all cosine similarity values of all
valid opinion word pairs where one word must come from ${\mathcal{O}}(x^{i})$
and the other must come from ${\mathcal{O}}(x^{j})$. We represent opinion
words using embeddings, i.e., the embeddings in the opinion word classifier in
Eq. (4). If there are $k$ opinion words in each set,
$cos({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))$ will return
$\frac{k*(k-1)}{2}$ cosine similarity values. When we want to know whether two
documents are similar in opinion words, we pay attention to the maximum value,
i.e.,
$max\\_cos=max\big{(}cos({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))\big{)}$.
When we want to know whether two documents are dissimilar in opinion words, we
pay attention to the minimum value, i.e.,
$min\\_cos=min\big{(}cos({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))\big{)}$.
So we define:

${\mathcal{S}}(\cdot,\cdot)=\begin{cases}\hbox{\multirowsetup${max\\_cos}$,}&max\\_cos>\gamma_{1}\quad\hbox{\multirowsetup$\&$}\\\
&min\\_cos\geq\gamma_{2},\vspace{0.1in}\\\
\hbox{\multirowsetup${min\\_cos}$,}&max\\_cos\leq\gamma_{1}\quad\hbox{\multirowsetup$\&$}\\\
&min\\_cos<\gamma_{2},\vspace{0.1in}\\\
\hbox{\multirowsetup$\delta$,}&max\\_cos>\gamma_{1}\quad\hbox{\multirowsetup$\&$}\\\
&min\\_cos<\gamma_{2},\vspace{0.1in}\\\ 0,&otherwise,\end{cases}$ (9)

where ${\mathcal{S}}(\cdot,\cdot)$ is short for
${\mathcal{S}}(({\mathcal{O}}(x^{i}),{\mathcal{O}}(x^{j}))$ due to space
limit. The first condition means two documents have some semantically similar
opinion words ($max\\_cos>\gamma_{1}$) and have no semantically dissimilar
opinion words ($min\\_cos\geq\gamma_{2}$). The value returned by the function
score is $max\\_cos$. It should be a positive value. The second condition
means two documents have no semantically similar opinion words
($max\\_cos\leq\gamma_{1}$) and have some semantically dissimilar opinion
words ($min\\_cos<\gamma_{2}$). The value returned by function score is
$min\\_cos$. It should be a negative number. The third condition means two
documents have some semantically similar opinion words
($max\\_cos>\gamma_{1}$) and also have some semantically dissimilar opinion
words ($min\\_cos<\gamma_{2}$). This condition corresponds to a real-world
situation that when some customers want to express some negative sentiment,
they usually point out some positive aspects first, and then start with a
“but”, and emphasize some negative aspects. The opinion words sets extracted
from these type of documents have both negative and positive opinion words.
When we compare two of them, they will have some similar opinion words and
dissimilar opinion words. In this case, we tend to assume they are in the same
class. If the third condition is satisfied, it will return an non-negative
value $\delta\in[0,1]$; The final condition means two documents have no
semantically similar opinion words ($max\\_cos\leq\gamma_{1}$) and have no
semantically dissimilar opinion words ($min\\_cos\geq\gamma_{2}$). It will
return $0$.

The mechanism of the regularization is that if the posterior distributions
$q(C|\mathbf{x}^{i})$ and $q(C|\mathbf{x}^{j})$ are different from each other,
i.e., $d\big{(}q(C|\mathbf{x}^{i}),q(C|\mathbf{x}^{j})\big{)}$ is large, but
opinion words suggest that these two documents should be in the same cluster
i.e., $s(w^{i}_{d},w^{j}_{d})$ is large, then
$d\big{(}q(C|\mathbf{x}^{i}),q(C|\mathbf{x}^{j})\big{)}$ will be encouraged to
be small by applying the regularization. Oppositely, if the posterior
distributions $q(C|\mathbf{x}^{i})$ and $q(C|\mathbf{x}^{j})$ are similar,
i.e., $d\big{(}q(C|\mathbf{x}^{i}),q(C|\mathbf{x}^{j})\big{)}$ is small, but
opinion words suggest that these two documents should be in the different
cluster i.e., $s(w^{i}_{d},w^{j}_{d})$ is small, then
$d\big{(}q(C|\mathbf{x}^{i}),q(C|\mathbf{x}^{j})\big{)}$ will be encouraged to
be large by applying the regularization.

The final objective function with posterior regularization is as follows,

$\displaystyle{\mathcal{J}}$
$\displaystyle={\mathcal{L}}_{2}+\beta\sum_{x^{i}\in X}\sum_{x^{j}\in
X}R(x^{i},x^{j})\;,$ (10)

where $x^{i}$ and $x^{j}$ are documents in the training set $X$. The
constraints are defined in a $|X|\times|X|$ space. In practice, we train our
model batch by batch. So we only apply the constraints within a mini-batch.
There are at most $|X_{b}|\times|X_{b}|$ constraints in a mini-batch, where
$|X_{b}|$ is the number of samples in a mini-batch.

##  3 Experiments

In this section, we evaluate the empirical performance of our method on binary
sentiment classification tasks.

###  3.1 Datasets

Datasets | Yelp | IMDB | Amazon
---|---|---|---
Size of training set | 38,000 | 20,000 | 20,000
Size of development set | 3,800 | 2,000 | 2,000
Average doc length | 155.12 | 252.90 | 88.28
Opinion vocabulary size | 1,097 | 688 | 394
Table 1: Statistics of Yelp, IMDB, and Amazon dataset.

We use three corpora to evaluate the performance of our proposed method. All
corpora have two classes and perfectly balanced. For all methods, we use a
development set for hyper-parameter tuning. For all methods, we use the
training set as the test set since all methods do not use the ground truth in
the training set.

(1) Yelp Review: We use the Yelp reviews polarity dataset from Zhang et al.
(2015) and take its test set containing 38,000 documents as the corpus for
evaluation. For hyper-parameter tuning, we also extract 3,800 documents from
the original training set of Zhang et al. (2015) to serve as a development
set.

(2) IMDB Review: We use the IMDB reviews polarity dataset from Maas et al.
(2011) and randomly extract $20,000$ reviews from its original test set as the
corpus for evaluation. For hyper-parameter tuning, we also extract $2,000$
documents from the original training set of Maas et al. (2011) to serve as a
development set.

(3) Amazon Review: We use the Amazon reviews polarity dataset from Zhang et
al. (2015) and randomly extracted $20,000$ reviews from its original test set
as the corpus for evaluation. For hyper-parameter tuning, we also extract
$2,000$ documents from the original training set of Zhang et al. (2015) to
serve as a development set.

Table 1 provides the details of these datasets.

###  3.2 Compared Methods

Lexicon uses an opinion lexicon to assign sentiment polarity to a document
Read and Carroll (2009); Pablos et al. (2015). We combine two popular opinion
lexicons used by Hu and Liu (2004) and Wilson et al. (2005) to get a larger
lexicon. If an extracted opinion is in the positive (negative) lexicon, it
votes for positive (negative). When the opinion word is with a negation word
such as “no” and “not”, its polarity will be the opposite. Then, the polarity
of a document is determined by using majority voting among all extracted
opinion words. When the number of positive and negative words is equal, the
document will be randomly assigned a polarity.

WeSTClass Meng et al. (2018) first generates pseudo labels for documents which
contain user-provided keywords. Keywords are expanded to generate more pseudo
samples. It pretrains a CNN/LSTM model using pseudo samples as the training
set and then performs a self-training process. Here, we use CNN because it
empirically outperforms LSTM. The CNN architecture we used here is the same as
the one described in Meng et al. (2018).

Keyword Pretrain generates pseudo labels for documents in training set which
contain user-provided keywords. We pretrain a CNN model using pseudo samples
as the training set. In the CNN model, four different filter sizes
$\\{2,3,4,5\\}$ are applied, and a max-pooling layer is applied to each
convolutional layer, and each convolutional layer has $100$ filters.

VWS Zeng et al. (2019) uses target opinion word pairs as supervision signal.
It trains a sentiment polarity classifier and opinion word classifier
simultaneously via optimizing the variational lower bound. We use a CNN model
as the sentiment polarity classifier. And we pretrain it by generating pseudo
labels for documents which contain user-provided keywords in the training set.
The CNN architecture is the same as the one in Keyword Pretrain.

VWS-PR is VWS method with proposed posterior regularization.

###  3.3 Keywords and Opinion Word Extraction

Datasets | Positive | Negative
---|---|---
Yelp | terrific | amazing | awesome | horrible | worst | bad
IMDB | great | fantastic | awesome | awful | worst | bad
Amazon | great | fantastic | awesome | poor | worst | bad
Table 2: Keywords of Yelp, IMDB, and Amazon datasets.

We manually select three keywords for each class. The details of keywords of
three datasets are shown in table 2.

For opinion word extraction, we adopt four rules proposed by VWS Zeng et al.
(2019) in the implementation. All rules rely on dependency parser Chen and
Manning (2014). When a target word and an opinion word satisfy a dependency
relation, we will extract the opinion word. The details of dependency relation
and examples are provided in Table 3. When a pair of words satisfy one rule,
there are still some restrictions on head and tails words to be satisfied.
There is no restriction for Rule $1$. For Rule $2$, the head word should be an
adjective and the tail word should a noun. For Rule $3$, the head word should
be one of the following four words: “like,” “dislike,” “love,” and “hate.” For
Rule $4$, the head word should be one of the following word: “seem,” “look,”
“feel,” “smell,” and “taste.”

Rule | Dependency Relation | Example | Extracted Word
---|---|---|---
1 | adjectival modifier | they have delicious food | delicious
2 | nominal subject | the room is big | big
3 | direct object | i like it | like
4 | open clausal complement | i feel comfortable | comfortable
Table 3: Opinion words extraction rules.

###  3.4 Result Analysis

Methods Datasets | Yelp | IMDB | Amazon
---|---|---|---
| Mean | Std | Mean | Std | Mean | Std
Lexicon | 0.5982 | 0.0006 | 0.5998 | 0.0013 | 0.5754 | 0.0014
Keyword Pretraining | 0.7441 | 0.0060 | 0.7496 | 0.0050 | 0.6375 | 0.0171
WeSTClass Meng et al. (2018) | 0.8061 | 0.0105 | 0.7354 | 0.0096 | 0.7374 | 0.0082
VWS Zeng et al. (2019) | 0.8014 | 0.0179 | 0.7825 | 0.0045 | 0.7530 | 0.0025
VWS-PR | 0.8431 | 0.0013 | 0.8025 | 0.0041 | 0.7644 | 0.0020
Table 4: F1 scores of weakly supervised sentiment analysis methods on Yelp,
IMDB, and Amazon.

Table 4 shows that our method VWS-PR outperforms VWS by $\mathbf{4\%}$,
$\mathbf{2\%}$, and $\mathbf{1\%}$ on Yelp, IMDB, and Amazon datasets
respsectively. Compared with WeSTClass and VWS, our method is more stable,
i.e., smaller standard deviation, which shows that the regularization confine
the posterior distribution to a smaller space. The performance of lexicon
method is bad across three datasets. The main reason is that it does not
involve any learning process. Keyword pretraining method can outperform
lexicon method. But pseudo labels are not ground truths, hence the pseudo
training set contains noises. Also, user provided keywords are limited, so the
training samples with pseudo labels are restricted to some samples which
contain certain keywords. For example, reviews with an extreme polarity (only
expressing positive polarity or only express negative polarity) are likely to
be pseudo samples. But most of reviews express mixed polarities. This will
hinder the generalization ability. WeSTClass outperforms the keyword
pretraining method on Yelp and Amazon dataset due to keyword expansion and
self-training process. But in IMDB dataset, WeSTClass is slightly worse than
the keyword pretraining method. Possible reason would be keyword expansion
involve some harmful keywords and self-training procedure amplifies errors.
VWS outperforms WeSTClass on IMDB and Amazon datasets and is comparable to
Yelp dataset.

###  3.5 Hyper-parameters Sensitivity Analysis

(a) Varying $\beta$ with fixed $\gamma_{1}$ and $\gamma_{2}$.

(b) Varying $\gamma_{1}$ with fixed $\beta$ and $\gamma_{2}$.

(c) Varying $\gamma_{2}$ with fixed $\beta$ and $\gamma_{1}$.

Figure 1: Hyper-parameters sensitivity analysis.

We first show F1 scores on three datasets with varied $\beta$ in Figure 1(a).
It shows that optimal $\beta$ values of our model on three datasets are
different. When they achieve optimal $\beta$ value, the standard deviation is
much smaller than others. The regularization makes models more stable. Our
method on IMDB and Amazon is not very sensitive. The changes are within $2\%$.
Our method on Yelp is more sensitive. But we could still find a range, e.g.,
$0.1$ to $0.5$, where the changes are within $2\%$. When $\beta$ keeps
growing, the performance in Yelp deteriorates a lot. The main reason is
probably that the opinion word vocabulary size in Yelp is much larger than
other datasets, and hence it is likely that the vocabulary in Yelp contains
more noisy opinion words. When $\beta$ is large, the noises may harm the
performance.

We then show F1 scores on three datasets with varied $\gamma_{1}$ in Figure
1(b). The optimal $\gamma_{1}$ values of three datasets are the same, i.e.,
$0.7$. The trends are consistent on three datasets. F1 score first increases
and then decreases. When $\gamma_{1}$ is small, the constraints are easier to
satisfy and the performance is bad because it may involve more noisy
constraints. It probably enforces two samples to be similar but in reality,
they are not similar. When $\gamma_{1}$ is large, the performance is also bad
because it has fewer constraints that enforce two samples to be similar. It
could have made use of more constraints.

We show F1 scores on three datasets with varied $\gamma_{2}$ in Figure 1(c).
The optimal $\gamma_{2}$ values of three datasets are the same, i.e., $-0.1$.
The trends are consistent on three datasets. F1 score first increases and then
decreases. When $\gamma_{2}$ is small, the performance is bad because it has
fewer constraints that enforce two samples to be dissimilar. It could have
made use of more constraints. When $\gamma_{2}$ is large, the performance is
bad because it has more noisy constraints.

###  3.6 Error Analysis

Dataset | Document | Prediction | Ground Truth | Extracted Words
---|---|---|---|---
Yelp | It was good, but not at that price. | good | bad | good
| There are so many other good italian places |  |  |
| in the area for half the price. |  |  |
IMDB | I love gheorghe muresan, | bad | good | love
| so i automatically loved this movie. |  |  | good
| Everything else about it was so so. |  |  | annoying
| Billy crystal is a good actor, |  |  |
| even if he is annoying. |  |  |
Amazon | No wrist strap. Nice light, well made. | good | bad | nice
| But why would anyone design a |  |  |
| tactical style flashlight without |  |  |
| a place to attach a wrist strap. |  |  |
Table 5: Documents that are predicted incorrectly by VWS-PR.

We show some incorrectly predicted documents by VWS-PR on three datasets in
Table 5. For the first document, the customer emphasizes price a lot. Our
method cannot extract opinion words on snippets such as “not at that price”
and “for half the price.” For the second document, the reviewer loves this
movie because he/she loves the basketball player. The reviewer thinks that the
movie itself does not deserve a high score. Our method detects both positive
and negative polarities on this document, so it tends to predict as negative.
Because most mixed polarities are likely to be negative polarity. The
regularization enforces this pattern. This document obviously is different
from other documents with mixed opinion words. For the last document, our
method cannot extract opinion words on snippets such as “no wrist strap” and
“without a place to attach a wrist strap.” Our method is good at extracting
words on subjective expression such as “nice light,” but not on descriptive
expression such as “no wrist strap.” Our method fails because no other
knowledge source indicates that “no wrist strap” is negative.

###  3.7 Implementation Details

For WeSTClass and VWS, we used code released by Meng et al. (2018) and Zeng et
al. (2019) respectively, and followed their preprocessing steps and optimal
settings. For VWS and VWS-PR, we pretrain a CNN model using pseudo-labeled
samples. After that, the embeddings are untrainable. The rest of parameters
are trainable. For our method, the hyperparameter settings of VWS part is the
same as described in Zeng et al. (2019). We implemented our models using
TensorFlow Abadi et al. (2016). When tuning hyper-parameters of regularization
term, we perform grid search on $\gamma_{1}\in[0.5,0.6,\cdots,1.0]$ and
$\gamma_{2}\in[-0.5,-0.4,\cdots,0]$. After than, we fix $\gamma_{1}$ and
$\gamma 2$, then tune $\beta\in[0.1,0.2,\cdots,1.0]$. $\delta$ is fixed to
$1$.

##  4 Related Work

In this section, we review the related work on weakly supervised sentiment
analysis.

Using a lexicon is a typical way to perform weakly supervised sentiment
analysis. One line of works perform simple assignment, i.e., majority voting,
based on sentiment orientation scores of extracted opinion words. Some methods
Missen and Boughanem (2009); Tsytsarau et al. (2010) used sentiment
orientation scores in existing lexicons directly, and aggregated them within a
document to determine polarity. Some methods developed their own semantic
orientation estimation algorithm. For example, Turney (2002) first identified
phrases in the review and then estimated the semantic orientation of each
extracted phrases. The semantic orientation of a given phrase is calculated by
comparing its similarity to a positive reference word (“excellent”) with its
similarity to a negative reference word (“poor”). This method determined the
sentiment polarity based on the average semantic orientation of the phrases
extracted from the review. Kamps et al. (2004) used the minimum path distance
between a phrase and pivot words (“good” and “bad”) in WordNet to estimate the
semantic orientation of extracted phrases.

Another line of works involve learning process when using a lexicon. Li et al.
(2009); Zhou et al. (2014) proposed a constrained non-negative matrix tri-
factorization approach to sentiment analysis, and used a sentiment lexicon as
prior knowledge. In these models, a term-document matrix is approximated by
three factors that specify soft membership of terms and documents in one of
$k$ classes. All three factors are non-negative matrices. The first factor is
a matrix representing knowledge in the word space, i.e., each row represents
the posterior probability of a word belonging to the $k$ classes. The second
factor is a matrix providing a condensed view of the term-document matrix. The
third factor is a matrix representing knowledge in document space, i.e., each
row represents the posterior probability of a document belonging to the $k$
classes. Li et al. (2009) applied a regularization to encourage that the first
factor is close to prior knowledge. This regularization is different from ours
because it requires prior knowledge such as a predefined lexicon. A predefined
lexicon needs a lot of human effort. Zhou et al. (2014) applied a
regularization based on an intuition that if two documents are sufficiently
close to each other, they tend to share the same sentiment polarity. This
intuition of this regularization is similar to ours. But when they compare
document similarity, they use textual similarity (e.g., cosine similarity of
bag of words) rather than similarity on opinion words. The regularization is
applied under matrix factorization framework, it is not straightforward to fit
in neural network based models.

Using keywords is another way to perform weakly supervised sentiment analysis.
Meng et al. (2018) leveraged keywords to generate pseudo-labeled samples for
model pretraining, and then performed self-training on unlabeled data for
model refinement. The possible improvement of this direction would be
investigating more advanced keywords expansion techniques to generate better
pseudo-labeled samples and developing a more advanced self-training algorithm.
LOTClass Meng et al. (2020), a parallel work to ours, fine-tuned a masked
language model to generate relevant words that can replace label name such as
“good” and “bad,” and performed self-training on unlabeled data for model
refinement. Fine-tuning in LOTClass can be viewed as an advanced keyword
expansion process using language models. VWS Zeng et al. (2019) used target-
opinion word pairs as supervision signal. Its objective function is to predict
an opinion word given a target word. By introducing a latent variable (the
sentiment polarity), they can learn a well-approximated posterior distribution
via optimizing the evidence lower bound. The posterior probability here is the
probability of a possible polarity (e.g., positive or negative) given text
representation.

##  5 Conclusion

We propose a posterior regularization framework for the VWS sentiment analysis
to better control the posterior distribution. The intuition behind the
posterior regularization is that if extracted opinion words from two documents
are semantically similar (dissimilar), the posterior distribution of two
documents should be similar (dissimilar). Our experiments show that our
posterior regularization can improve VWS and the performance is more stable.

## Acknowledgments

We thank anonymous reviewers for their valuable comments. This paper was
supported by the NSFC Grant U20B2053 from China, the Early Career Scheme (ECS,
No. 26206717), the General Research Fund (GRF, No. 16211520), and the Research
Impact Fund (RIF, No. R6020-19) from the Research Grants Council (RGC) of Hong
Kong.

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Regge phenomenology and coherent photoproduction

of $\mathbf{J/\psi}$ in peripheral heavy ion collisions

László Jenkovszky a⋆, Vladyslav Libova⋄, and Magno V. T. Machadob♠

a Bogolyubov ITP, National Academy of Sciences of Ukraine, Kiev 03143 Ukraine

b HEP Phenomenology Group, CEP 91501-970, Porto Alegre, RS, Brazil

###### Abstract

By using models based on Regge phenomenology we analyse the coherent
photoproduction of charmonium in peripheral heavy-ion collisions at the Large
Hadron Collider (LHC). The centrality dependence is investigated and compared
to the experimental results for coherent $J/\psi$ production in lead-lead LHC
runs at the energies of 2.76 and 5.02 TeV. Theoretical uncertainties and
possible limitations of the formalism are also discussed.

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##  1 Introduction

High precision data on exclusive quarkonium production in semi-central and
peripheral heavy ion collisions became recently available [1, 2, 3, 4].
Systematically, event excess over the expected hadronic $J/\psi$ production at
very low transverse momentum, $p_{T}$, both at Large Hadron Collider (LHC) and
Relativistic Heavy Ion Collider (RHIC) energies has been interpreted as due to
coherent quarkonium photoproduction in hard collisions with nuclear overlap.
At RHIC, an excess in the $J/\psi$ yield at low-$p_{T}$ has been observed by
the STAR Collaboration in AuAu collsions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV
and UU collisions at $\sqrt{s_{\mathrm{NN}}}=193$ GeV at mid-rapidity [4].
ALICE collaboration has also measured an excess at very low-$p_{T}$ in the
yield of $J/\psi$ in PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV [1].
The excess observed in peripheral heavy ion collisions correspond to the
centrality classes of $30-50\%$, $50-70\%$ and $70-90\%$, respectively.
Recently, ALICE reported similar results for PbPb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. Measurements are presented at mid and
forward rapidity and for the same classes of centrality. Moreover, the
photoproduction at low transverse momentum has been studied in peripheral PbPb
collisions by LHCb collaboration at 5 TeV [3]. The yield of coherently
photoproduced meson has been extracted as a function of rapidity and
transverse momentum in intervals of the number of participant nucleons in
rapidity range $2.0<y<4.5$.

From the theoretical point of view the coherent photoproduction of quarkonium
in peripheral heavy ion collisions has been addressed in a number of works [5,
7, 8, 9, 10, 11, 12, 13, 14]. In Ref. [5] the coherent photonuclear cross
section has been extracted from coherent $J/\psi$ production using the ALICE
measurements from peripheral and ultra-peripheral PbPb collisions at 2.76 TeV.
The equivalent photon approximation (EPA) has been assumed and the effective
photon flux for a given centrality class was taken from Ref. [6]. A different
approach was considered in Ref. [7], where absorption effects were introduced
by modifying the photon flux in the impact parameter space. Still using EPA
approximation, this is achieved by imposing many geometrical conditions on
impact parameters between $\gamma$ and Pb and between colliding lead nuclei.
The photonuclear cross section, $\sigma(\gamma+A\rightarrow V+A)$, is
described through vector dominance model and nuclear effects given by Glauber
approach. Similar study was considered in Refs. [8, 9], where on the other
hand the photonuclear cross section is computed in the QCD color dipole
framework. The investigation done in Refs. [10, 11, 12] addresses the role
played by the participation of spectator and non-spectator nucleons in the
coherent scattering. It was found that the strong interactions in the
overlapping region of incoming nuclei modify the $J/\psi$ coherent production.
Different coupling scenarios for photons and pomerons with the nucleus in
presence of hadronic interactions were considered. It was pointed out that the
destructive interference between photoproduction on ions moving in opposite
directions plays an important role. Interestingly, modifications of an
anisotropic Quark Gluon Plasma (QGP) on photoproduced mesons have been studied
in Ref. [13]. The spatial configurations of produced quarkonia at transverse
coordinate $\vec{r}_{\perp}$ is given by their distribution over the nucleus
surface with a normalized distribution, $f^{\mathrm{norm}}(\vec{r}_{\perp})$,
which is proportional to the square of target thickness function. This is
compared with the quarkonia uniformly distributed over the target nuclear
surface in order to single out the theoretical uncertainties. Study presented
in Ref. [14] considers standard EPA approximation and color dipole framework.
Parton saturation models for the dipole-proton cross section were utilized and
the numerical calculation compared to the RHIC data.

In this study coherent $J/\psi$ photoproduction in peripheral collisions at
the LHC will be scrutinized. In our calculations we consider the Vector
Dominance Model (VDM) and Glauber multiple scattering formalism which is quite
appealing due to its simplicity and direct connection with Regge phenomenology
and Reggeon Field Theory (RFT) [15]. The theoretical input is given by the
single-component Reggeometric Pomeron model [16, 17] and the Soft Dipole
Pomeron model [18, 19]. The present study extends our earlier work [20, 21,
22, 23] where we have addressed the rapidity distribution for $J/\psi$ and
$\psi(2S)$ mesons in proton-proton collisions at the LHC for the center-of-
mass energies of 7, 8 and 13 TeV. In Ref. [23] the coherent $J/\psi$ and
$\psi(2S)$ photoproduction in ultraperipheral PbPb (at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV and $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV) and
XeXe collisions (at $\sqrt{s_{\mathrm{NN}}}=5.44$ TeV and
$\sqrt{s_{\mathrm{NN}}}=5.86$ TeV) have been analyzed. Rapidity and transverse
momentum distributions were predicted and numerical results were generally
consistent with experimental data. This paper is organized as follows. In Sec.
2 we shortly review the exclusive vector meson production in the context of
the Reggeometric and Soft Dipole Pomeron models in the process
$\gamma+p\rightarrow V+p$. In Subsection 2.1 the formalism to obtain the
photonuclear cross section for the process $\gamma+A\rightarrow V+A$ is
presented. The numerical results for peripheral PbPb reactions at different
centrality classes are compared to experimental measurements in Sec. 3, where
a discussion on the theoretical uncertainties is done. In the last section we
summarize the main results.

##  2 Theoretical formalism

Let us start by introducing the models for the quarkonium photoproduction
describing the exclusive process, $\gamma+p\rightarrow V+p$. Two models based
on Regge phenomenology are considered. The first one is the Reggeometric
Pomeron model and the second is the Soft Dipole Pomeron model. An advantage is
that they are able to describe also the electroproduction data where the
virtuality dependence is taken into account by $\widetilde{Q}^{2}$-dependent
factors. Accordingly, the measure of the hardness,
$\widetilde{Q}^{2}=Q^{2}+M_{V}^{2}$, is the sum of the squared photon
virtuality $Q^{2}$ and the squared mass $M^{2}_{V}$ of the produced vector
meson.

The scattering amplitude associated to the single-component Reggeometric
Pomeron (RP) in a given scale $\widetilde{Q}^{2}$ is given by [16, 17]:

$\displaystyle{\cal{A}}_{\mathrm{RP}}(Q^{2},s,t)=\frac{\widetilde{A_{0}}}{\left(1+\frac{\widetilde{Q^{2}}}{{Q_{0}^{2}}}\right)^{n}}e^{-\frac{i\pi\alpha(t)}{2}}\left(\frac{s}{s_{0}}\right)^{\alpha(t)}e^{(B_{0}/2)t},\quad
B_{0}(\widetilde{Q^{2}})=4\left(\frac{a}{\widetilde{Q^{2}}}+\frac{b}{2m_{N}^{2}}\right),$
(1)

where the exponent in the exponential factor in Eq. (1) reflects its
geometrical nature and where $\alpha(t)$ is the Pomeron trajectory. The
center-of-mass energy of photon-nucleon system is denoted as
$\sqrt{s}=W_{\gamma p}$ and the quantities $a/\widetilde{Q}^{2}$ and
$b/2m_{N}^{2}$ in (1) correspond to the effective sizes of upper and lower
vertices in Fig. 1-c. The elastic differential cross section takes the form:

$\displaystyle\frac{d\sigma_{el}}{dt}=\frac{A_{0}^{2}}{\left(1+\frac{\widetilde{Q^{2}}}{{Q_{0}^{2}}}\right)^{2n}}\left(\frac{s}{s_{0}}\right)^{2(\alpha(t)-1)}\exp\left[B_{0}(\widetilde{Q^{2}})\,t\right].$
(2)

Figure 1: Diagrams of Deeply Virtual Compton scattering (DVCS) (a) and vector
meson production (VPM) (b) in $e^{\pm}p$ scattering; (c) DVCS (VMP) amplitude
in a Regge-factorized form including representation for particle vertices,
$V_{1,2}$.

The Reggeometric Pomeron model naturally applies to photoproduction
($Q^{2}=0$) limit, where $\tilde{Q}^{2}=M_{V}^{2}$. The photoproduction cross
section is given by [16, 17]:

$\displaystyle\sigma_{\gamma p\to Vp}(W_{\gamma p},\widetilde{Q}^{2})$
$\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{\left(1+\widetilde{Q}^{2}/\widetilde{Q}^{2}_{0}\right)^{2n}}\frac{\left(W_{\gamma
p}/W_{0}\right)^{4(\alpha_{0}-1)}}{B_{V}\left(W,\widetilde{Q}^{2}\right)},$
(3) $\displaystyle B_{V}\left(W_{\gamma p},\widetilde{Q}^{2}\right)$
$\displaystyle=$ $\displaystyle 4\left[\alpha^{\prime}\ln(W_{\gamma
p}/W_{0})+\left(\frac{a}{\widetilde{Q}^{2}}+\frac{b}{2m_{N}^{2}}\right)\right].$
(4)

The parameters, fitted [16] to the $J/\psi$ photoproduction data, are
presented in Table 1. The analysis done in [16] also includes the production
of $\phi$ and Deeply Virtual Compton scattering (DVCS).

Table 1: Values of the parameters for the Reggeometric Pomeron model [16] fitted to data on $J/\psi$ production at HERA.  $A_{0}$ $\left[\frac{\sqrt{\text{nb}}}{\text{GeV}}\right]$ |  $\widetilde{Q^{2}_{0}}$ $\left[\text{GeV}^{2}\right]$ | $n$ | $\alpha_{0}$ |  $\alpha^{\prime}$ $\left[\text{GeV}^{-2}\right]$ | $a$ | $b$
---|---|---|---|---|---|---
29.8 $\pm$ 2.8 |  2.1 $\pm$ 0.4 |  1.37 $\pm$ 0.14 |  1.20 $\pm$ 0.02 |  0.17 $\pm$ 0.05 |  1.01 $\pm$ 0.11 |  0.44 $\pm$ 0.08

Let us now move to the Soft Dipole Pomeron model (SDP) applied to diffractive
photoproduction of quarkonium [18, 19]. The corresponding invariant scattering
amplitude is written as follows [18]:

$\displaystyle{\cal{A}}_{\mathrm{SDP}}(s,t)=i\left(\frac{-is}{s_{0}}\right)^{\alpha(t)-1}\left\\{G_{1}(t)+G_{2}(t)\left[\ln\left(\frac{s}{s_{0}}\right)-i\frac{\pi}{2}\right]\right\\},$
(5)

where the residua of the simple, $G_{1}$, and double pole, $G_{2}$, are given
by,

$\displaystyle G_{1}(t)=A_{1}e^{bt}(1+h_{1}t),\quad
G_{2}(t)=A_{2}e^{bt}(1+h_{2}t)-\gamma.$ (6)

Concerning the quantities appearing in Eq. (6) one has the standard
$pI\\!\\!Pp$ vertex $\sim e^{bt}$, with $b=2.25~{}GeV^{-2}$ determined from
the $pp$ scattering, and the $\gamma I\\!\\!PV$ vertex which is parameterized
by a simple polynomial with a free parameter $h_{1}$. For simplicity, in Ref.
[18] a linear trajectory for the Pomeron, $\alpha(t)=1+0.25t$, was considered.

The elastic differential cross section reads [18] as

$\displaystyle\frac{d\sigma_{el}}{dt}$ $\displaystyle=$
$\displaystyle\left(\frac{s}{s_{0}}\right)^{2\alpha(t)-2}\left\\{\left[G_{1}(t)+G_{2}(t)\ln{\left(\frac{s}{s_{0}}\right)}\right]^{2}+\frac{\pi^{2}}{4}G_{2}^{2}(t)\right],$
(7) $\displaystyle\left.\frac{d\sigma_{el}}{dt}\right|_{t=0}$ $\displaystyle=$
$\displaystyle\left[A_{1}+(A_{2}-\gamma)\ln{\left(\frac{s}{s_{0}}\right)}\right]^{2}+\frac{\pi^{2}}{4}(A_{2}-\gamma)^{2}.$
(8)

The parameters $A_{1},\,A_{2},\,h_{1},\,h_{2}$ and $\gamma$ were fitted to
experimental measurements at HERA [18]. The values of the parameters for
$J/\psi$ are presented in Table 2.

Table 2: Values of the parameters for the Soft Dipole Pomeron model [18] fitted to data on $J/\psi$ production at HERA.  $A_{1}$ $\left[\frac{\sqrt{\mu\text{b}}}{\text{GeV}}\right]$ |  $A_{2}$ $\left[\frac{\sqrt{\mu\text{b}}}{\text{GeV}}\right]$ |  $h_{1}$ [GeV-2] |  $h_{2}$ [GeV-2] |  $\gamma$ $\left[\text{GeV}^{-2}\right]$ |  $s_{0}$ $\left[\text{GeV}\right]$
---|---|---|---|---|---
0.27523 | 0.091278 | -0.80606 | 0 | 0 | 30

In next subsection we investigate the coherent nuclear scattering using as
input the two models discussed above. In particular, we calculate the cross
section $\sigma(\gamma+A\rightarrow V+A)$ as a function of the centre of mass
energy for the photon-nucleus system, $W_{\gamma A}$. The nuclear effects are
based on the Glauber formalism.

###  2.1 Coherent charmonium production in heavy ion collisions

In what follows we investigate the coherent quarkonium production in nuclear
targets ($J/\psi$ in particular). Here the nuclear effects for the process,
$\gamma+A\rightarrow V+A$ (coherent scattering), are described by vector
dominance model (VDM) and Glauber approach for multiple scattering. Using the
classical mechanics Glauber formula for multiple scattering of vector meson in
the nuclear medium [24] the differential cross section is given by:

$\displaystyle\left.\frac{\mathrm{d}\sigma\left(\gamma+A\to
J/\psi+A\right)}{\mathrm{d}t}\right|_{t=0}=\frac{\alpha_{em}}{4f^{2}_{J/\psi}}\left\\{\int\mathrm{d}^{2}\textbf{b}\left[1-\exp\left(-\sigma_{tot}\left(J/\psi
p\right)T_{A}\left(\textbf{b}\right)\right)\right]\right\\}^{2},$ (9)

where $T_{A}(b)$ is the nuclear thickness function and $f_{J/\Psi}$ is the
vector-meson coupling. The value $f^{2}_{J/\Psi}/4\pi=10.4$ will be
considered. The approach above can been extended in the generalized vector
dominance model (GVDM) [25]. The input for the Glauber model calculation in
Eq. (9) is the cross section for the process $J/\psi+p\rightarrow J/\psi+p$,
which is given by:

$\displaystyle\sigma_{tot}\left(J/\psi
p\right)=\sqrt{\frac{4f^{2}_{J/\psi}}{\alpha_{em}}\left.\frac{\mathrm{d}\sigma\left(\gamma+p\to
J/\psi+p\right)}{\mathrm{d}t}\right|_{t=0}},$ (10)

where the predictions from the Reggeometric Pomeron and Soft Dipole Pomeron
models will be introduced in Eq. (10) . The integrated cross section can be
written as:

$\displaystyle\sigma(\gamma+A\rightarrow
J/\psi+A)=\left.\frac{d\sigma(\gamma+A\rightarrow
J/\psi+A)}{dt}\right|_{t=0}\int\limits_{t_{min}}^{\infty}\mathrm{d}|t|\,\left|F_{A}\left(t\right)\right|^{2},$
(11)

where $F_{A}$ is the nuclear form factor. In our calculation we consider an
analytic form factor given by a hard sphere of radius, $R_{A}=1.2A^{1/3}$ fm,
convoluted with a Yukawa potential with range $a$ [26], which is sufficient
for practical use [6]:

$\displaystyle
F_{A}(|k|)=\frac{4\pi\rho_{0}}{A|k^{3}|}\left(\frac{1}{1+a^{2}k^{2}}\right)\left[\sin{(|k|R_{A})}-|k|R_{A}\cos{(|k|R_{A})}\right],$
(12)

where $A$ is the mass number, $k$ is the momentum transfer,
$\rho_{0}=A/\frac{4}{3}\pi R_{A}^{3}$ and $a=0.7$ fm.

In order to assess the robustness of the Pomeron exchange models, we compare
our predictions to the coherent nuclear scattering cross sections,
$\sigma(\gamma+A\rightarrow J/\psi+A)$, extracted from the ultra-peripheral
nucleus-nucleus collisions at RHIC and LHC. Figure 2 shows the photonuclear
cross section for $J/\psi$ production as a function of photon-nucleus energy,
$W_{\gamma A}$. The extracted cross sections are from Ref. [5] (labeled
Contreras) and Ref. [27] (labeled GKSZ) with the data description being quite
reasonable. The Soft Dipole Pomeron model describes better the normalization
and energy behavior. It is expected that models based on Glauber approach
underestimate the nuclear suppression as the nuclear shadowing is not so
intense in those cases (see discussion in Refs. [27, 28]).

Figure 2: The coherent nuclear cross section, $\sigma(\gamma+A\rightarrow
J/\psi+A)$, as a function of the corresponding photon-nucleus energy.
Theoretical predictions for Reggeometric Pomeron (solid line) and Soft Dipole
Pomeron (long-dashed line) models are shown.

##  3 Peripheral collisions - results and discussions

Figure 3: Rapidity distribution of coherently photoproduced $J/\psi$ at
forward rapidities in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. ALICE data
[1] are presented in the centrality range 30-50% (left), 50-70% (middle) and
70-90% (right), compared to the theoretical calculations. Reggeometric Pomeron
predictions are represented by solid curves and those for the Soft Dipole
Pomeron by the long-dashed curves.

Now we employ the equivalent photon approximation (EPA) where the transverse
electromagnetic fields are treated as a bunch of quasi-real photons moving in
the longitudinal direction. By integrating the photonuclear cross section over
the photon spectrum the total cross section is found to be:

$\displaystyle\sigma(A+A\rightarrow A+V+A)=\int
d\omega\frac{dN_{\gamma}}{d\omega}\,\sigma(\gamma+A\rightarrow V+A),$ (13)

where $\omega$ is the photon energy. The total photon flux modulated by the
non-interaction probability is given by [5, 6]:

$\displaystyle\frac{dN_{\gamma}}{d\omega}=\int_{b_{min}}^{b_{max}}d^{2}\vec{b}\,\left(1-P_{NH}(b)\right)\int_{0}^{R_{A}}\frac{rdr}{\pi
R_{A}^{2}}\int_{0}^{2\pi}d\phi\frac{d^{3}N_{\gamma}(k,b+r\cos(\phi))}{d\omega
d^{2}r},$ (14)

where the $b$ integral runs over impact parameter range $(b_{min},b_{max})$
according to the centrality determination (see below for more details).

The interaction probability at a given impact parameter is related to the
overlap function $T_{AB}(b)=\int
d^{2}\vec{r}\,T_{A}(\vec{r})T_{B}(\vec{r}-\vec{b})$. In our calculation the
nuclear thickness function $T_{A}(\vec{r})$ is obtained using nuclear density
given by the Woods-Saxon distribution. The experimental condition for
peripheral collisions is that the nuclei undergo a hadronic collision.
Accordingly, the probability of having no hadronic interactions is given by,

$\displaystyle P_{NH}(b)=\exp\left[-T_{AA}(b)\sigma_{NN}\right],$ (15)

where the values $\sigma_{NN}=61.8$ mb at 2.76 TeV and $\sigma_{NN}=67.6$ mb
at 5.02 TeV have been considered [29].

Before proceeding, some words on the technical details are in order. In the
comparison with ALICE data, the range on $b$ for a given centrality,
$(b_{min},b_{max})$, is taken from Table 1 of Ref. [30] for
$\sqrt{s_{NN}}=2.76$ TeV and Refs. [31, 32] and Table 1 of Ref. [33] for
$\sqrt{s_{NN}}=5.02$ TeV. In the comparison with LHCb data, we follow the
centrality determination presented in Ref. [34] and further details given at
the LHCb public pages [35]. For practical purposes, by using a Woods-Saxon
distribution the impact parameter range can be directly extracted from Ref.
[29] referred before.

The photon flux from a relativistic heavy nucleus is given by the Weizsäcker-
Williams approach. The photon flux at a distance $r$ from the nucleus and for
a target frame $\gamma$-factor is [6],

$\displaystyle\frac{d^{3}N_{\gamma}(\omega,r)}{d\omega
d^{2}r}=\frac{Z^{2}\alpha_{em}u^{2}}{\pi^{2}\omega
r^{2}}\left[K_{1}^{2}(u)+\frac{1}{\gamma^{2}}K_{0}^{2}(u)\right]$ (16)

where $Z$ is the nuclear charge and $K_{0,1}$ are the modified Bessel function
of second kind with $u=\omega r/\gamma$.

Figure 4: Rapidity distribution of coherently photoproduced $J/\psi$ at
forward rapidities in Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV.
Preliminary ALICE data [2] are presented in the centrality range 30-50%
(left), 50-70% (middle) and 70-90% (right), compared to theoretical
calculations. Same notation as Fig. 3.

In this section the numerical results are presented and compared to recent
experimental measurements. Let us start with the rapidity distribution of
$J/\psi$ production in lead-lead (PbPb) collision at the LHC energies. The
coherent production of $J/\psi$ in $AA$ collisions is straightforwardly
computed in the EPA approximation. The rapidity distribution is given by:

$\displaystyle\frac{d\sigma(A+A\rightarrow
A+V+A)}{dy}=\omega_{+}\frac{dN_{\gamma}(\omega_{+})}{d\omega}\sigma_{\gamma
A\rightarrow
VA}(\omega_{+})+\omega_{-}\frac{dN_{\gamma}(\omega_{-})}{d\omega}\sigma_{\gamma
A\rightarrow VA}(\omega_{-}),$ (17)

where the rapidity $y$ of the vector meson $V$ is related to the center-of-
mass energy of the photon-nucleus system, $W_{\gamma
A}^{2}=\sqrt{s_{\mathrm{NN}}}M_{V}e^{y}$. The quantity
$\sqrt{s_{\mathrm{NN}}}$ is the center-of-mass energy per nucleon pair in the
AA system and $\omega_{\pm}=\frac{M_{V}}{2}e^{\pm y}$. The photon flux for a
given centrality interval will be described by expression in Eq. (14) and the
photonuclear cross section is computed using Eq. (11).

Figure 5: The differential $J/\psi$ photoproduction yields as a function of
the rapidity in peripheral PbPb collisions for $\langle
N_{part}\rangle=19.7\pm 9.2$. Data from LHCb collaboration at 5 TeV [3]. Same
notation as previous figures.

In Fig. 3 results are shown for coherent nuclear $J/\psi$ production for PbPb
peripheral collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV ($p_{T}<0.3$ GeV).
The rapidity distribution of the meson is depicted as a function of the
centrality bin. The experimental measurements from ALICE collaboration [1] are
presented in the centrality intervals of 30-50%, 50-70% and 70-90%. The
predictions for the Reggeometric Pomeron (RP) are labeled by the solid curves
whereas the ones for Soft Dipole Pomeron are labeled by the long-dashed
curves, respectively. Both models produce generally similar results, with RP
systematically overshooting the SDP. The predictions overestimate the
experimental measurements at semi-central centralities and are in agreement
with data for the more peripheral ones. The standard photon flux we used
systematically overestimates the cross section in semi-central collisions. The
reason is related to the absorption effects in the region of overlapping
colliding nuclei in b-space (small-$b$ for semi-central centralities). The
size of these absorption corrections is model-dependent and in general they
are implemented as a modification of the photon flux. We enumerate some of the
phenomenological approaches in literature in what follows. Actually, the
corrections in the flux suppress the differential cross section
$d\sigma/db(A+A\rightarrow A+V+A)$ for $b<<2R_{A}$. For instance, we quote the
Figs. 8 and 9 of Ref. [7] as an illustration of the referred effects. On the
other hand, in case of peripheral collisions the formalism presented here
produces numerically similar results as other approaches to compute the
effective photon flux as discussed below.

In Ref. [7] the photoproduction mechanism for peripheral and semi-central
collisions has been proposed. It is assumes that the whole nucleus produces
photons and they must interact with the other nucleus in order to produce a
heavy meson. The treatment of the region of overlapping colliding nuclei in
the $b$-space is based on absorption effects which are taken into account by
changing the effective photon fluxes. This is enforced by imposing geometrical
conditions on impact parameters between $\gamma$ and $A$ and/or between
scattering nuclei. The resulting modification describes different centrality
bins adequately. Other works follow the same procedure, however in Refs. [8,
9] an effective photonuclear cross section is proposed (comparison between the
different modifications in the photon flux and photonuclear cross section is
performed). Another approach has been presented in Refs. [10, 11, 12] where
two scenarios are assumed in which the coherence of the vector meson
production is influenced or not by scattering with the overlap region of the
colliding nuclei.

In Fig. 4 results are shown for the energy of $\sqrt{s_{\mathrm{NN}}}=5.02$
TeV. The preliminary measurements from ALICE collaboration [2] are presented
with the cut $p_{T}<0.3$ GeV. The notation is the same as previous figure. For
the centralities bins of 50-70% and 70-90% both models describe data within
the errors (summed into quadrature), with SDP being somewhat preferred. The
predictions follow similar trend as observed at 2.76 TeV. In general, models
are suitable to predict the magnitude of the forward rapidity measurements in
peripheral collisions.

Figure 6: Differential yields of photo-produced $J/\psi$ as a function of
number of participants nucleons, $\langle N_{part}\rangle$, at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV in the rapidity range $2.0<y<4.5$.
Predictions of the Reggeometric Pomeron model (open squares) and Soft Dipole
Pomeron model (open circles) are compared to data from LHCb collaboration [3].
Figure 7:  The double differential $J/\psi$ photoproduction yields as a
function of the transverse momentum for $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV in
the rapidity range $2.0<y<4.5$. Predictions of the Reggeometric Pomeron model
(solid curve) and Soft Dipole Pomeron model (long-dashed curve) are compared
to data from LHCb collaboration [3].

Motivated by the recent LHCb measurement for photoproduction of $J/\psi$ at
low transverse momentum in peripheral PbPb collisions at 5 TeV [3], we
calculate the rapidity distribution for fixed average number of participant.
This is presented in Fig. 5 with $\langle N_{part}\rangle=19.7\pm 9.2$ in the
rapidity range $2.0<y<4.5$. In this case both models are consistent with data
within the experimental errors. The Soft Dipole Pomeron model seems to be
preferred though, similarly to Figures 3 and 4. Moreover, in Fig. 6 is shown
the differential yields of photo-produced $J/\psi$ as a function of number of
participants nucleons, $\langle N_{part}\rangle$, in the same rapidity
interval. The numerical results for the Reggeometric Pomeron model are
represented by the open square symbols whereas the ones for Soft Dipole
Pomeron model are labeled by open circle symbols. As expected, the present
approach is reasonable for more peripheral collisions (the photon flux has
been not modified to include overlap or inteference effects). The predictions
are compared to data from LHCb collaboration [3]. The number of binary
nucleon-nucleon collisions and the number of participating nucleons are
calculated for a given impact parameter using the Glauber approach in
relativistic heavy ion physics [36]. The input for the nuclear density is
given by the Woods- Saxon distribution for lead nuclei. The centrality
percentile intervals follows the supplementary material and additional
information available in the LHCb public pages [35].

As a final investigation, we present the predictions for the transverse
momentum distribution for coherent $J/\psi$ photoproduction and compare them
to the double differential photoproduction yields measured by LHCb
collaboration [3] in PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=5$ TeV. The
$p_{T}$-distribution is obtained theoretically from the momentum transfer
distribution as follows:

$\displaystyle\frac{d^{2}\sigma(A+A\rightarrow
A+V+A)}{dydp_{T}^{2}}=\omega_{+}\frac{dN_{\gamma/A}(\omega_{+})}{d\omega}\frac{d\sigma_{\gamma
A\rightarrow
VA}(\omega_{+})}{d|t|}+\omega_{-}\frac{dN_{\gamma/A}(\omega_{-})}{d\omega}\frac{d\sigma_{\gamma
A\rightarrow VA}(\omega_{-})}{d|t|},$ (18)

where the $t$-dependence is obtained by Eq. (11) after removing the
$|t|$-integration. In Fig. 7 the results are presented and compared to LHCb
data at the rapidity range $2.0<y<4.5$. The Reggeometric Pomeron model (solid
curve) and the Soft Pomeron model (long-dashed curve) overestimate the data at
very low-$p_{T}$ but provide a better agreement for intermediate transverse
momenta values. The Reggeometric Pomeron model was already considered in Ref.
[23], where it has been compared with the first measurement of ALICE for
$t$-dependence in ultra-peripheral PbPb collisions [37]. Based in that
analyzes the overestimation at very small $t$ ($p_{T}$) would be expected. The
$t$-dependence is related to the coupling Pomeron-nucleus and the low-$t$
behavior may be related to the two-pion threshold in the non-linear Pomeron
trajectory, manifest as a break in proton-proton differential cross section
(pion atmosphere around the nucleon). We quote for instance Ref. [38] for a
detailed discussion on this issue. Our results have similar shape but larger
normalization than the predictions of Refs. [10, 12] as pointed out in the
LHCb paper [3].

##  4 Conclusions

In this paper predictions for coherent $J/\psi$ meson photoproduction in semi-
central and peripheral collisions at the LHC are presented and compared with
the recent experimental data collected by different collaborations. The
theoretical input is based on Regge phenomenology for the production cross
section and the effective photon flux for a given centrality class. We
considered two models which describe the exclusive vector meson measurements
for proton targets: i) the single-component Reggeometric Pomeron model and ii)
the Soft Dipole Pomeron model. By using the Glauber multiple scattering model
and vector meson dominance in order to obtain the photonuclear cross section,
predictions for semi-central and peripheral PbPb collisions at 2.76 and 5.02
TeV are presented. Concerning the rapidity distributions, the models are in a
reasonable agreement with the peripheral collisions especially for forward
rapidities. We provide predictions for the differential yields as a function
of number of participants nucleons,transverse momentum and rapidity which are
in agreement with experimental results for peripheral classes of centrality.

## Acknowledgements

L.J. was supported by the NASU grant 1230/22-1 Fundamental Properties of
Matter. MVTM was supported by funding agencies CAPES (Finance Code 001) and
CNPq (grant number 306101/2018-1), Brazil.

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