# An Analysis of Emotion Communication Channels in Fan Fiction: Towards
Emotional Storytelling

Evgeny Kim  Roman Klinger
Institut für Maschinelle Sprachverarbeitung
University of Stuttgart
Pfaffenwaldring 5b, 70569 Stuttgart, Germany
<EMAIL_ADDRESS>
###### Abstract

Centrality of emotion for the stories told by humans is underpinned by
numerous studies in literature and psychology. The research in automatic
storytelling has recently turned towards emotional storytelling, in which
characters’ emotions play an important role in the plot development Theune et
al. (2004); y Pérez (2007); Méndez et al. (2016). However, these studies
mainly use emotion to generate propositional statements in the form “A feels
affection towards B” or “A confronts B”. At the same time, emotional behavior
does not boil down to such propositional descriptions, as humans display
complex and highly variable patterns in communicating their emotions, both
verbally and non-verbally. In this paper, we analyze how emotions are
expressed non-verbally in a corpus of fan fiction short stories. Our analysis
shows that stories written by humans convey character emotions along various
non-verbal channels. We find that some non-verbal channels, such as facial
expressions and voice characteristics of the characters, are more strongly
associated with joy, while gestures and body postures are more likely to occur
with trust. Based on our analysis, we argue that automatic storytelling
systems should take variability of emotion into account when generating
descriptions of characters’ emotions.

##  1 Introduction and Motivation

As humans, we make sense of our experiences through stories McKee (2003). A
key component of any captivating story is a character Kress (2005) and a key
component of every character is emotion, as “without emotion a character’s
personal journey is pointless” (Ackerman and Puglisi, 2012, p. 1). Numerous
works pinpoint the central role of emotions in storytelling Hogan (2015);
Johnson-Laird and Oatley (2016); Ingermanson and Economy (2009), as well as
story comprehension and evaluation Komeda et al. (2005); Van Horn (1997); Mori
et al. (2019).

Emotion analysis and automatic recognition in text is mostly channel-agnostic,
i.e., does not consider along which non-verbal communication channels (face,
voice, etc.) emotions are expressed. However, we know that the same emotions
can be expressed non-verbally in a variety of ways (Barrett, 2017, p. 11), for
example, through internal feelings of the character, as shown in Figure 1. We
argue that automatic storytelling systems should take this information into
account, as versatility of the emotion description is a prerequisite for
engaging and believable storytelling (Ackerman and Puglisi, 2012, p. 3).

Figure 1: Example of the emotion expressed using non-verbal communication
channel. The annotation of character and emotion are available in the dataset
by Kim and Klinger (2019). Channel annotation (in blue) is an extension to the
original dataset.

There is a growing body of literature in the field of natural language
generation that uses emotions as a key component for automatic plot
construction Theune et al. (2004); y Pérez (2007); Méndez et al. (2016) and
characterization of virtual agents Imbert and de Antonio (2005); Dias and
Paiva (2011). However, these and other studies put an emphasis on emotion per
se (“A feels affection towards B”), or on the social behavior of characters “A
confronts B”) making little or no reference to how characters express
emotions, both verbally and non-verbally.

In this paper, we aim at closing this gap by analyzing how characters express
their emotions using non-verbal communication signals. Specifically, we
analyze how eight emotions (joy, sadness, anger, fear, trust, disgust,
surprise, and anticipation) defined by Plutchik (2001) are expressed along the
following channels introduced by van Meel (1995): 1) physical appearance, 2)
facial expressions, 3) gaze, looking behavior, 4) arm and hand gesture, 5)
movements of body as a whole, 6) characteristics of voice, 7) spatial
relations, and 8) physical make-up.

This paper is an extension to our previous study Kim and Klinger (2019), in
which we presented a corpus of emotion relations between characters in fan
fiction short stories. We post-annotate the corpus with non-verbal expressions
of emotions and analyze two scenarios of non-verbal emotion expression: when
the feeler of an emotion is alone, and when a communication partner, who also
plays a role in the development of emotion, is present. In our analysis, we
look into the emotions associated with each non-verbal behavior, mapping
emotions to non-verbal expressions they frequently occur with.

Our contributions are therefore the following: 1) we propose that natural
language generation systems describing emotions should take into account how
emotions are expressed non-verbally, 2) we extend the annotations presented in
Kim and Klinger (2019) and quantitatively analyze the data, 3) we show that
facial expressions, voice, eyes and body movements are the top three channels
among which the emotion is expressed, 4) based on the data, we show that some
emotions are more likely to be expressed via a certain channel, and this
channel is also influenced by the presence or non-presence of a communication
partner.

Our corpus is available at https://www.ims.uni-stuttgart.de/data/emotion.

##  2 Related Work

Emotion analysis has received great attention in natural language processing
(Mohammad and Bravo-Marquez, 2017; Mohammad et al., 2018; Klinger et al.,
2018; Felbo et al., 2017; Abdul-Mageed and Ungar, 2017; Zhou and Wang, 2018;
Gui et al., 2017, i.a.). Most existing studies on the topic cast the problem
of emotion analysis as a classification task, by classifying documents (e.g.,
social media posts) into a set of predefined emotion classes. Emotion classes
used for the classification are usually based on discrete categories of Ekman
(1970) or Plutchik (2001) (cf. Alm et al. (2005), Suttles and Ide (2013),
Mohammad (2012)). Fewer studies address emotion recognition using a
dimensional emotion representation (cf. Buechel and Hahn (2017); Preoţiuc-
Pietro et al. (2016)). Such representation is based on the valence-arousal
emotion model Russell (1980), which can be helpful to account for subjective
emotional states that do not fit into discrete categories.

Early attempts to computationally model emotions in literary texts date back
to the 1980s and are presented in the works by Anderson and McMaster (1982,
1986), who build a computational model of affect in text tracking how emotions
develop in a literary narrative.

More recent studies in the field of digital humanities approach emotion
analysis from various angles and for a wide range of goals. Some studies use
emotions as feature input for genre classification Samothrakis and Fasli
(2015); Henny-Krahmer (2018); Yu (2008); Kim et al. (2017), story clustering
Reagan et al. (2016), mapping emotions to geographical locations in literature
Heuser et al. (2016), and construction of social networks of characters
Nalisnick and Baird (2013); Jhavar and Mirza (2018). Other studies use emotion
analysis as a starting point for stylometry Koolen (2018), inferring
psychological characters’ traits Egloff et al. (2018), and analysis of the
causes of emotions in literature Kim and Klinger (2018, 2019).

To the best of our knowledge, there is no previous research that addresses the
question of how emotions are expressed non-verbally. The only work that we are
aware of is a literary study by van Meel (1995), who proposes several non-
verbal communication channels for emotions and performs a manual analysis on a
set of several books. He finds that voice is the most frequently used
category, followed by facial expressions, arm and hand gestures and bodily
postures. Van Meel explains the dominancy of voice by the predominant role
that speech plays in novels. However, van Meel does not link the non-verbal
channels to any specific emotions. In this paper, we extend his analysis by
mapping the non-verbal channels to a set of specific emotions felt by the
characters.

Emotion |  |  Face |  |  Body |  |  Appear. |  |  Look. |  |  Voice |  |  Gesture |  |  Sptrel. |  |  Sensations |  |  No channel |  |  Total
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
anger |  | 23 |  | 20 |  | 5 |  | 38 |  | 51 |  | 7 |  | 0 |  | 4 |  | 163 |  | 311
anticipation |  | 4 |  | 14 |  | 0 |  | 17 |  | 4 |  | 2 |  | 7 |  | 6 |  | 267 |  | 321
disgust |  | 3 |  | 6 |  | 1 |  | 3 |  | 0 |  | 0 |  | 0 |  | 1 |  | 149 |  | 163
fear |  | 4 |  | 28 |  | 13 |  | 16 |  | 8 |  | 1 |  | 0 |  | 25 |  | 124 |  | 219
joy |  | 76 |  | 26 |  | 7 |  | 12 |  | 52 |  | 19 |  | 5 |  | 33 |  | 268 |  | 498
sadness |  | 3 |  | 5 |  | 4 |  | 4 |  | 2 |  | 0 |  | 3 |  | 7 |  | 81 |  | 109
surprise |  | 10 |  | 5 |  | 3 |  | 13 |  | 1 |  | 0 |  | 0 |  | 2 |  | 118 |  | 152
trust |  | 4 |  | 15 |  | 1 |  | 4 |  | 1 |  | 21 |  | 3 |  | 0 |  | 144 |  | 193
Total |  | 127 |  | 119 |  | 34 |  | 107 |  | 119 |  | 50 |  | 18 |  | 78 |  | 1314 |  | 1966
Table 1: Counts of emotion and expression-channel pairs. No channel means that
instance contains no reference to how emotion is expressed non-verbally.

##  3 Corpus Creation

We post-annotate our dataset of emotion relations between characters in fan
fiction Kim and Klinger (2019) with non-verbal communication channels of
emotion expressions. The dataset includes complete annotations of 19 fan
fiction short stories and of one short story by James Joyce. The emotion
relation is characterized by a triple
$(C_{\textrm{exp}},e,C_{\textrm{cause}})$, in which the character
$C_{\textrm{exp}}$ feels the emotion $e$. The character $C_{\textrm{cause}}$
(to which we refer as a “communication partner”) is part of an event which
triggers the emotion $e$. The emotion categorization presented in the dataset
follows Plutchik’s (2001) classification, namely joy, sadness, anger, fear,
trust, disgust, surprise, and anticipation.

Given an annotation of a character with an emotion, we annotate non-verbal
channels of emotion expressions following the classification proposed by van
Meel (1995), who defines the following eight categories: 1) physical
appearance, 2) facial expressions, 3) gaze, looking behavior, 4) arm and hand
gesture, 5) movements of body as a whole, 6) characteristics of voice, 7)
spatial relations (references to personal space), and 8) physical make-up. To
clarify the category of physical make-up, we redefine it under the name of
physical sensations, i.e., references to one’s internal physiological signals
perceived by the feeler of the emotion.

The task is exemplified in Figure 1. Labels for emotion (Anger) and character
(Farrington) are given. Physical sensation is an example of a channel
annotation we use to extend the original dataset.

The annotations were done by three graduate students in our computational
linguistics department within a one-month period. The annotators were asked to
read each datapoint in the dataset and decide if the emotion expressed by the
feeler (character) has an associated non-verbal channel of expression. If so,
the annotators were instructed to mark the corresponding textual span and
select a channel label from the list of non-verbal communication channels
given above.

| a1–a2 | a1–a3 | a2–a3
---|---|---|---
Span | 31 | 29 | 45
Sentence | 49 | 45 | 59
Table 2: F1 scores in % for agreement between annotators on a span level. a1,
a2, and a3 are different annotators. Span: label of channel and offsets are
considered. Sentence: only label of the channel in the sentence is considered.

The results of inter-annotator agreement (IAA) are presented in Table 2. We
measure agreement along two dimensions: 1) span, where we measure how well two
people agree on the label of a non-verbal emotion expression, as well as on
the exact textual position of this expression, and 2) sentence, where we
measure how well two people agree on the label of non-verbal emotion
expression in a given sentence (i.e., the exact positions of the channel are
not taken into account). The agreement is measured using the F1 measure, where
we assume that annotations by one person are true, and annotations by another
person are predicted. As one can see, the agreement scores for spans (i.e.,
channel label and exact textual positions) are relatively low (lowest 29%,
highest 45% F1 respectively). The IAA scores on a sentence level are higher
(lowest agreement is 45%, highest 59% F1 respectively), as we only consider
the label of the non-verbal channel in a sentence without looking into the
exact textual positions of the annotations.

Channel |  | Emotion |  | Examples
---|---|---|---|---
Facial expressions |  | anger |  | rolled his eyes
|  | fear |  | smiled nervously
Body movements |  | anger |  | stormed back out
|  | trust |  | slumped his shoulders
Physical appearance |  | fear |  | blushed crimson red
Looking behavior |  | fear |  | averted her eyes
|  | anticipation |  | pause to look back
Voice |  | joy |  | purred
|  | fear |  | voice getting smaller and smaller
Arm and hand gestures |  | trust |  | opened her arms
Spatial relations |  | joy |  | leaping into her arms
|  | trust |  | pulled him closer to his chest
Physical sensations |  | joy |  | tingling all over
|  | fear |  | hear in his throat
Table 3: Textual examples of non-verbal emotion expressions.

##  4 Analysis

Table 1 summarizes the results of the annotation of non-verbal channels of
emotion expressions, Table 3 gives examples of these expressions in the
dataset.

In total, there are 652 annotations of non-verbal emotion expressions. By
absolute counts, facial expressions (Face, 127 occurrences), body movements
(Body, 119), voice (Voice, 199), and looking behavior (Look., 107) have the
highest number of annotations. Spatial relations (Sptrel., 78) and physical
appearance (Appear., 34) have the lowest number of annotations.

###  4.1 Emotion-Channel associations

Figure 2: Distribution of non-verbal channels with all emotions. Darker color
indicates higher value. Values in the cells are percentage ratios. Each cell
is normalized by the row sum of absolute frequencies.

We start our analysis by looking into the emotion-channel associations.
Namely, we analyze which non-verbal channels are associated with which
emotions. To that end, we plot a heatmap of the emotion–non-verbal-channel
matrix. The value of each cell in the heatmap is normalized by the row sum
(i.e., total counts of channel annotations) and represents the likelihood of
emotion-channel association in the dataset, for each emotion, respectively.

As Figure 2 shows, anger is more likely to be expressed with voice, while joy
is more likely to be expressed with face. In contrast to all other emotions,
sadness is more likely to be experienced internally (sens. column in Figure 2)
by the feeler, as opposed to being communicated non-verbally. Some channels
and emotions show no association, such as gestures (gest.) and disgust or
spatial relations (sptrel.) and anger. Given the relatively small size of the
dataset, we do not argue that these associations are not possible in
principle. For example, fear and spatial relations have zero association in
our analysis, however, it is likely that somebody expresses this emotion by
moving away subconsciously (increasing personal space) from the source of
danger. At the same time, fear is most strongly associated with body movements
as a whole, which can be difficult to distinguish from spatial relations.
However, we believe that these associations still reflect the general trend:
emotions that require immediate actions and serve evolutionary survival
function, such as anger, disgust, and fear, are more likely to be expressed
with actions. For example, anger is an unpleasant emotion that often occurs as
a response to an appraisal of a blocked goal Harmon Jones and Harmon-Jones
(2016), which can be resolved by using voice characteristics (commanding or
shouting at someone who prevents you from achieving your goal).

Figure 3: Emotion-channel map. Each branch is an emotion, whose branches are
the top three non-verbal channels associated with the emotion.

Overall, we observe that face, look., voice, and body channels are
predominantly used with all emotions. We visualize the strongest associations
of emotions and non-verbal channels in Figure 3. For each emotion, the figure
shows the top three (in a descending order) strongly associated non-verbal
channels. As one can see, the branches are dominated by face, look., voice,
and body channels. The only exception is trust, for which the predominant way
to express emotions non-verbally is through gestures, and sadness, which is
predominantly felt “internally” (sensations).

###  4.2 Presence of a communication partner

Figure 4: The difference between situations, in which a character feels an
emotion and the communication partner is present, and situations in which the
communication partner is not present (normalized by row sum). Darker
color/higher values indicates that the channel is more likely to be used when
there is a communication partner. Figure 5: Distribution of non-verbal emotion
expressions in the narrative. Markers on the plot lines indicate the text
chunk. The plots are given for ten chunks. Light area around the solid line
indicates confidence interval of 95%. y-axis shows percentage.

The original dataset contains information whether an emotion of the character
is evoked by another character (communication partner). We use this
information to understand how the presence of a communication partner affects
the choice of a non-verbal channel.

To that end, we plot a heatmap (Figure 4) from the delta values between
situations, in which the communication partner is present, and situations in
which the communication partner is not present. As it can be seen from Figure
4, trust is more strongly associated with body movements when a communication
partner is present. Sadness, which is more likely to associate with inner
physical sensations in the feeler, is expressed through the physical
appearance and looking behavior when the communication partner is present.
Likewise, disgust is more likely to be expressed with facial expressions, and
anticipation is more likely to be expressed with looking behavior.

Again, we observe that body, voice, face, and look. channels are the
predominant non-verbal communication channels for emotion expressions.

###  4.3 Timeline analysis

To understand if there are patterns in the frequency of use of non-verbal
channels in a narrative, we perform an additional analysis.

For this analysis, we split each short story in the dataset in ten equally
sized chunks and get frequencies of each non-verbal channel, which are then
plotted as time-series with confidence intervals (Figure 5). The averaged
values for each channel are plotted as a dark line with circular markers. The
lighter area around the main line represents confidence intervals of 95%,
which are calculated using bootstrap resampling.

We observe the general tendency of all non-verbal channels to vanish towards
the end of the story. The only exception is Facial expressions, which after
peaking in the middle of a story reverts to the mean. Overall, we find no
consistent pattern in the use of non-verbal channels from beginning to an end
of a story.

##  5 Discussion and Conclusion

The results of the analysis presented in Section 4 show that emotions are
expressed in a variety of ways through different non-verbal channels. However,
the preferred communication channel depends on whether a communication partner
is present or not. Some channels are used predominantly only when the feeler
communicates her emotion to another character, other channels can be used in
any situation.

Sadness stands out from other emotions in a sense that it is predominantly not
expressed using any external channels of non-verbal communication. In other
words, it is more common for the characters in the annotated dataset to go
through sadness “alone” and feel it “in the body”, rather than show it to the
outer world. However, when another character (communication partner) is the
reason of sadness experienced by the feeler, he or she will most likely use
eyes and overall behavior to show this emotion.

In this paper, we showed that in human-written stories, emotions are not only
expressed as propositions in the form of “A feels affection towards B” or “A
confronts B”. As Table 3 shows, often there is no direct mention of the
feelings A holds towards B (“rolled his eyes”, “purred”). It is, therefore,
important, that this observation finds its place in automatic storytelling
systems. Some attempts have been done in natural language generation towards
controllable story generation Peng et al. (2018); Tambwekar et al. (2018). We
propose that emotion expression should be one of the controllable parameters
in automatic storytellers. As more and more language generation systems have
started using emotion as one of the central components for plot development
and characterization of characters, there will be a need for a more versatile
and subtle description of emotions, which is realized not only through
propositional statements. In the end, no single instance of same emotion is
expressed in the same way Barrett (2017), and emotion-aware storytelling
systems should take this information into account when generating emotional
profiles of characters.

##  6 Future Work

This paper proposes one approach to non-verbal emotion description that relies
on a rigid ontology of emotion classes. However, it might be reasonable to
make use of unsupervised clustering of non-verbal descriptions to overcome the
limitations of using a relatively small number of coarse emotion categories
for the description of character emotions. Once clustered, such descriptions
could be incorporated in the generated text (e.g., a plot summary) and would
elaborate all the simplistic descriptions of character emotions.

Other research directions seems feasible too. For example, the annotations,
which we presented in this paper, can be used for building and training a
model that automatically recognizes non-verbal channels of emotion
expressions. This might, in a multi-task learning setting, improve emotion
classification. The data we provide could also be used as a starting point for
terminology construction, namely bootstrapping a lexicon of emotion
communications with different channels. Finally, our work can serve as a
foundation for the development of an automatic storytelling system that takes
advantage of such resources.

## Acknowledgements

This research has been conducted within the CRETA project
(http://www.creta.uni-stuttgart.de/) which is funded by the German Ministry
for Education and Research (BMBF) and partially funded by the German Research
Council (DFG), projects SEAT (Structured Multi-Domain Emotion Analysis from
Text, KL 2869/1-1).

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∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: College of Physics Science and Technology, Shenyang Normal
University, Shenyang 110034, China 22institutetext: College of Physics and
Space Science, China West Normal University, Nanchong 637002, China

# Holographic Van der Waals phase transition of the higher dimensional
electrically charged hairy black hole

Hui-Ling Liaddr1,e1  Zhong-Wen Fengaddr2

(Received: date / Accepted: date)

###### Abstract

With motivation by holography, employing black hole entropy, two point
connection function and entanglement entropy, we show that, for the higher
dimensional Anti-de Sitter charged hairy black hole in the fixed charged
ensemble, a Van der Waals-like phase transition can be observed. Furthermore,
based on the Maxwell’s equal area construction, we check numerically the equal
area law for a first order phase transition in order to further characterize
the Van der Waals-like phase transition.

††journal: Eur. Phys. J. C

##  1 Introduction

According to the Anti-de Sitter space/Conformal Field Theory (AdS/CFT)
correspondence r1 , an asymptotically AdS black hole plays a key role in
comprehending black hole thermodynamics of holographic dual field theories.
One of the compelling phenomena of thermodynamics is phase transitions in AdS
spacetime. The pioneer discussion on phase transition was done by Hawking and
Page r2 . In 1983, in five dimensional non-charged AdS background, Hawking and
Page proved that, there existed a first order phase transition between the AdS
and Schwarzschild-AdS black hole. Specifically speaking, the thermal AdS is
unstable, and it has to undergo a phase transition to the stable
Schwarzschild-AdS black hole at last, which is the best known Hawking-Page
phase transition.

Later, in 1990, in charged AdS background, Chamblin et al. r3 ; r4  explored
the phase structure of an Reissner-Nordström-AdS black hole. Compared to the
non-charged AdS case, the charged black hole’s phase structure becomes richer
and is associated with the chosen statistical ensemble. It was found that the
charged AdS black hole in the entropy-temperature plane presented an analogous
Van der Waals phase transition in the canonical ensemble (fixed electric
charge), i.e. the isocharge in the temperature-entropy plane has an unstable
branch and two stable ones when the charge below a critical value. And there
exists a second-order critical point at a critical charge. Recently, the
research on a Van der Waals-like phase transition has been generalized to the
extended phase space r5 ; r6 ; r7 ; r8 ; r9 ; r10 . In this framework, the
cosmological constant is taken as a thermodynamical pressure, and its
conjugate quantity is treated as the thermodynamical volume. Consequently,
this extended phase space makes the Van de Waals description more precise.

Very recently, entanglement entropy has also been used to detect phase
structures in different AdS backgrounds. Johnson r11  proposed that, like
black hole entropy, entanglement entropy also exhibited a Van der Waals-like
phase transition in the temperature-entanglement entropy plane in both the
fixed potential ensemble and charge ensemble. It was subsequently generalized
to supergravity STU black holes that involve four charges by Caceres et al.
r12 . The result showed that, Van der Waals behavior was observed in the cases
of three charge and four charge, however, for the one-charge and two-charge
cases, the STU black holes did not present this phase transition. Caceres et
al. also verified that for charge configuration that presented a Van der
Waals-like phase transition, the entanglement entropy indeed exhibited a
similar phase transition at the same critical temperature and the same
critical exponents as the ones obtained in the black hole entropy.
Furthermore, the equal area law of entanglement entropy was also checked by
Nguyen r13  for AdS-Reissner-Nordström black hole in the canonical ensemble.
In addition, Zeng et al. have investigated phase structures of holographic
entanglement entropy in massive gravity r14 , in the Born-Infeld-Anti-de
Sitter background r15  and in the quintessence Reissner-Nordström-AdS
background r16 , and all the results showed that there existed a Van de Waals-
like phase transition in these gravity backgrounds r14 ; r15 ; r16 ;
Zeng:2017zlm ; Zeng:2016aly .

In the framework of holography, it is interesting to detect the phase
structure of a higher dimensional charged AdS black hole. In the present work,
we will focus on investigating the holographic phase transition of a five
dimensional charged hairy black hole. Since there exist hair parameters, hairy
black hole solutions become far richer than in General Relativity r17 ; r18 .
Here, employing black hole entropy, two point correlated function and
entanglement entropy, we attempt to study whether the Van der Waals-like phase
transition can be observed, and discuss on the hair parameter’s effect on
phase transition.

##  2 Phase transition and Maxwell’s equal area law for the thermodynamic
entropy

Let us start by reviewing the exact solution of an electrically charged hairy
black hole in five dimensions. Einstein-Maxwell-$\Lambda$ theory conformally
coupled to a scalar field in higher dimensions can exhibit analytic solutions
r19 ; r20 . Here, we are concerned with the case of five dimensions, and the
action of the theory reads

$\displaystyle I=\frac{1}{\kappa}\int{\rm
d}^{\,5}x\sqrt{-g}\left[R-2\Lambda-\frac{1}{4}F^{2}+\kappa
L_{m}\left(\phi,\nabla\phi\right)\right],$ (1)

where

$\displaystyle\kappa=16\pi G,$ (2) $\displaystyle
L_{m}\left(\phi,\nabla\phi\right)=b_{0}\phi^{15}+b_{1}\phi^{7}S_{\mu\nu}{}^{\mu\nu}+b_{2}\phi^{-1}\big{(}S_{\mu\gamma}{}^{\mu\gamma}S_{\nu\delta}{}^{\nu\delta}{}$
$\displaystyle\qquad\qquad\quad\,\,\,-4S_{\mu\gamma}{}^{\nu\gamma}S_{\nu\delta}{}^{\mu\delta}+S_{\mu\nu}{}^{\gamma\delta}S^{\nu\mu}{}_{\gamma\delta}\big{)},$
(3) $\displaystyle
S_{\mu\nu}{}^{\gamma\delta}=\phi^{2}R_{\mu\nu}{}^{\gamma\delta}-12\delta{}^{[\gamma}_{[\mu}\delta_{\,\nu]}^{\delta]}\nabla_{\rho}\phi\nabla^{\rho}\phi{}$
$\displaystyle\qquad\quad\,\,\,\,\,-48\phi\delta{}^{[\gamma}_{[\mu}\nabla_{\nu]}\nabla^{\delta]}\phi+18\delta{}^{[\gamma}_{[\mu}\nabla_{\nu]}\phi\nabla^{\delta]}\phi.$
(4)

Here, $b_{0}$, $b_{1}$ and $b_{2}$ are real coupling constants of the
conformal field theory. The static spherically symmetric black hole solution
coming from the action (1) can be written as

$\displaystyle{\rm d}s^{\,2}=-N^{2}\left(r\right)f\left(r\right){\rm
d}t^{\,2}+\frac{{\rm d}r^{\,2}}{g\left(r\right)}+r^{\,2}{\rm
d}\Omega^{2}_{3},$ (5)

where

$\displaystyle
g\left(r\right)=f\left(r\right)=1-\frac{m}{r^{\,2}}-\frac{q}{r^{\,3}}+\frac{e^{\,2}}{r^{\,4}}+\frac{r^{\,2}}{l^{\,2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}N^{2}\left(r\right)=1.$
(6)

Here, $l$ is AdS radius $l^{2}=-6/\Lambda$, and ${\rm d}\Omega^{2}_{3}$ is the
metric of the unit 3-sphere.The integration constants $m$ and $e$ are related
the mass and the electric charge of the hairy black hole, and $q$ is given
with respect to the scalar coupling constants by the relation

$\displaystyle q=\frac{64\pi G}{5}\varepsilon
b_{1}\left(-\frac{18b_{1}}{5b_{0}}\right)^{3/2},$ (7)

where $\varepsilon=-1,~{}~{}0,~{}~{}+1$. For the five dimensional black hole
solution to exist, the scalar coupling constants must obey the following
constraint

$\displaystyle 10b_{0}b_{2}=9b^{2}_{1}.$ (8)

The Maxwell potential is

$\displaystyle A_{\mu}=\sqrt{3}\frac{e}{r^{\,2}}\delta^{\,0}_{\,\mu}$ (9)

with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. On the other
hand, the scalar field configuration takes the form

$\displaystyle\phi\left(r\right)=\frac{n}{r^{1/3}},~{}~{}~{}~{}~{}~{}n=\varepsilon\left(-\frac{18}{5}\frac{b_{1}}{b_{0}}\right)^{1/6}.$
(10)

In Ref. r19 , Galante et. al have discussed on the thermodynamic properties of
higher dimensional black holes in detail. The mass and charge of the hairy
black hole are

$\displaystyle
M=\frac{3\pi}{8}m=\frac{3\pi\left(e^{2}l^{2}-ql^{2}r_{+}+l^{2}r^{4}_{+}-r^{6}_{+}\right)}{8l^{2}r^{2}_{+}}$
(11)

and

$\displaystyle Q=-\frac{\sqrt{3}\pi}{8}e,$ (12)

respectively, where $r_{+}$ is the event horizon, which is given by the
equation $f\left(r_{+}\right)=0$. The Hawking temperature and black hole
entropy are given by r7 ; r19

$\displaystyle T=\frac{1}{\pi
l^{2}r^{4}_{+}}\left(-\frac{32Q^{2}l^{2}}{3\pi^{2}r_{+}}+\frac{ql^{2}}{4}+\frac{l^{2}}{2}r^{3}_{+}+r^{5}_{+}\right),$
(13) $\displaystyle S=\frac{\pi^{2}}{2}\left(r^{3}_{+}-\frac{5}{2}q\right).$
(14)

Note that, for the hairy black hole solution to exist, $q$ can take three
different values namely $q=0,~{}~{}\pm\left|q\right|$. For $q=0$, the solution
reduces to the five dimensional Reissner-Nordström AdS black hole, and the
phase transition has been discussed on extensively r13 ; r21 . Consequently,
here we mainly focus on studying the phase structure for the case $q\neq 0$.

Now, we begin to explore the hairy black hole’s critical behavior and phase
transition in the temperature–entropy plane. From Eqs. (13) and (14), we can
get the function $T(S,\,Q,\,q)$ by eliminating $r_{+}$

$\displaystyle T(S,Q,q)=\\!\\!\\!\\!$
$\displaystyle\frac{1}{3\sqrt[3]{2}l^{2}\pi^{5/3}(5\pi^{2}q+4S)^{5/3}}\Big{[}75\pi^{4}q^{2}{}$
(15) $\displaystyle-128l^{2}\pi^{2}Q^{2}+120\pi^{2}Sq+48S^{2}{}$
$\displaystyle+9\sqrt[3]{4}l^{2}\pi^{10/3}q(5\pi^{2}q+4S)^{1/3}{}$
$\displaystyle+6\sqrt[3]{4}l^{2}\pi^{4/3}S(5\pi^{2}q+4S)^{1/3}\Big{]}.$

Note that the phase structure of a hairy black hole is not only related to
electric charge $Q$, but also the scalar hair parameter $q$. Based on the
function $T(S,\,Q,\,q)$ above, the phase structure of the charged hairy black
hole can be detected. We find out that, by choosing some proper hair
parameters $q$, the isocharges in the $T-S$ plane can exhibit a Van der Waals-
like phase transition. For convenience, we keep the AdS radius $l=1$
throughout this paper. Here, we discuss on the phase transition in the fixed
electric charged ensemble, and take $q=\pm 0.010$ and $q=\pm 0.005$ as
examples. To plot the isocharges in the $T-S$ plane, firstly we should get the
critical values of phase transition by using the following relations

$\displaystyle\left(\frac{\partial T}{\partial
S}\right)_{Q}=\left(\frac{\partial^{2}T}{\partial^{2}S}\right)_{Q}=0.$ (16)

However, it is hard to obtain analytical values directly. We have to get them
numerically. In Table1, we tabulate the critical charge $Q_{C}$, critical
entropy $S_{C}$ and critical temperature $T_{C}$ for different hair parameter
$q$. From this table, we see that the critical entropy becomes smaller as $q$
increases. According to Eq. (14), we know that, $q$ must satisfy $q\leqslant
2r\,_{+}^{3}/5$ in order for the positivity condition of entropy.

According to the relation in (15) and these critical values, we plot the
isocharges in the $T-S$ plane for different $q$ in Fig. 1. Each curve
corresponds to a different electric charge. Remarkably, as can be seen from
these plots, a Van der Waals-like phase transition is clearly presented in the
$T-S$ plane. For different $q$, the phase structure is similar. when
$Q>Q_{C}$, temperature is monotonically larger as entropy increases, and the
system is thermodynamically stable. As $Q$ decreases and arrive to the
critical Value $Q_{C}$, an inflection point appears and the heat capacity is
divergent at this point, which corresponds to an second order phase
transition. In the $Q<Q_{C}$ case, in addition to two stable branches, the
isocharge has an unstable branch with negative heat capacity

$\displaystyle C_{Q}=T\left(\frac{\partial S}{\partial T}\right)_{Q},$ (17)

which corresponds to a first order phase transition.

(a) $q=-0.010$

(b) $q=-0.005$

(c) $q=0.005$

(d) $q=0.010$

Figure 1: Plots of the temperature versus the black hole entropy for different
q. The red dash line corresponds to the temperature of the first order phase
transition. The values of the electric charge chosen (from top to bottom) are
as follows. Panel (a): $Q=0.0286384<Q_{C}$, $Q=0.0486384=Q_{C}$,
$Q=0.0686384>Q_{C}$. Panel (b): $Q=0.0338006<Q_{C}$, $Q=0.0538006=Q_{C}$,
$Q=0.0738006>Q_{C}$. Panel (c): $Q=0.0429513<Q_{C}$, $Q=0.0629513=Q_{C}$,
$Q=0.0829513>Q_{C}$. Panel (d): $Q=0.0470977<Q_{C}$, $Q=0.0670977=Q_{C}$,
$Q=0.0870977>Q_{C}$

(a) $q=-0.010$

(b) $q=-0.005$

(c) $q=0.005$

(d) $q=0.010$

Figure 2: Plots of the temperature versus the free energy for different q in case of $Q<Q_{C}$. In each graph, the red dash line indicates a first order phase transition temperature $T^{*}$. Panel (a): $Q=0.0286384$ and $T^{*}=0.4506$. Panel (b): $Q=0.0338006$ and $T^{*}=0.4524$. Panel (c): $Q=0.0429513$ and $T^{*}=0.4552$. Panel (d): $Q=0.0470977$ and $T^{*}=0.4561$ Table 1: The critical values of the electric charge, temperature and entropy for different $q$ $q$ | $Q_{C}$ | $T_{C}$ | $S_{C}$
---|---|---|---
$-$0.010 | 0.0486384 | 0.439591 | 1.03476
$-$0.005 | 0.0538006 | 0.440337 | 0.992571
$0.005$ | 0.0629513 | 0.441770 | 0.906233
$0.010$ | 0.0670977 | 0.442460 | 0.862219

Like the Van der Waals phase transition of the liquid-gas system, this
unstable portion should be replaced by using an isotherm $T=T^{*}$ which obeys
Maxwell’s equal-area prescription. The subcritical temperature $T^{*}$ can be
obtained from the plot of the free energy $F=M-TS$ versus the temperature. The
plots in Fig. 2 show that the relations between the temperature and free
energy for different $q$, and for $Q<Q_{C}$, we always observe a classic
swallowtail structure in each plot, which is responsible for the first order
phase transition in Fig. 1. We indicate the transition temperature $T^{*}$ by
a red dashed line in Fig. 2, which corresponds to the horizontal coordinate of
the junction.

Next, in order to further characterize the Van der Waals-like phase
transition, we turn to check Maxwell equal area law for the first order phase
transition and the corresponding statement can be written as

$\displaystyle A_{1}\equiv\int_{S_{min}}^{S_{max}}T(S,\,Q,\,q)\,{\rm
d}S=T^{*}(S_{max}-S_{min})\equiv A_{2},$ (18)

where $T(S,\,Q,\,q)$ is defined in Eq. (15), $S_{min}$ and $S_{max}$ are the
smallest and largest roots of the equation $T(S,\,Q,\,q)=T^{*}$. Now, we take
$q=-0.010$ as an example to show how to verify the Maxwell’s equal area law.
Numerically, using $q=-0.010$, $Q=0.0286384$ and $l=1$, we obtain $T^{*}$ from
the plot (a) in Fig. 2. Then, substituting $T^{*}=0.4506$ into Eq. (15), we
obtain the smallest value $S_{min}=0.306264$ and largest value
$S_{max}=2.628909$ by resolving the equation $T(S,-0.01,0.0286384)=0.4506$.
Thus,using the values of $T^{*}$, $S_{min}$ and $S_{max}$, we can get
$A_{2}=1.04658$ at right side of Eq. (18), and obtain $A_{1}=1.04538$ by
integrating left side of Eq. (18). Namely, $A_{1}$ equals to $A_{2}$ in our
numeric accuracy with these values. Repeating the procedure above, we can
obtain $A_{1}$ and $A_{2}$ for other $q$, and we tabulate these values in
Table 2. From this table, it is obvious that the Maxwell’s equal area
construction holds in the $T-S$ plane.

##  3 Phase transition and Maxwell’s equal area law for two point correlation
function

In this section, we proceed to discuss on the phase structure of two point
correlation function. In recent years, the two point correlation function
appears to be a useful tool which can be used to explore some physical
phenomena such as holographic singularities r22 ; r23 , holographic
thermalization r24 ; r25 , holographic CFTs on maximally symmetric spaces r26
, holographic butterfly effect Shenker:2013pqa ; Leichenauer:2014nxa ;
Sircar:2016old ; Cai:2017ihd  and quantum phase transition r27 . Motivated by
the mentioned above, we attempt to detect whether the two point correlation
function can present a Van der Waals-like behaviour for the charged hairy
black hole.

According to the Anti-de Sitter space/Conformal Field Theory dictionary, in
the large $\triangle$ limit, the equal time two point correlation function can
be holographically expressed as r28

$\displaystyle\left\langle O(t_{0},x_{i})\,O(t_{0},x_{j})\right\rangle\approx
e^{-\triangle L},$ (19)

where $\triangle$ is the conformal dimension of scalar operator $O$, and $L$
is the length of the buck geodesic between the point $(t_{0},\,x_{i})$ and
$(t_{0},\,x_{j})$ on the Anti-de Sitter boundary. Due to the charged hairy
black hole’s symmetry in five dimensional AdS spacetime, we can simply set
$(\theta=\theta_{\,0},\,\varphi=\pi/2,\,\psi=0)$ and
$(\theta=\theta_{\,0},\,\varphi=\pi/2,\,\psi=\pi)$ as the two boundary point.

By utilizing $\theta$ to parameterize the trajectory, the proper length takes
the form

$\displaystyle L=\int_{0}^{\theta_{0}}\mathcal{L}(r(\theta),\theta)\,{\rm
d}\theta,\quad\mathcal{L}=\sqrt{\frac{r\,^{\prime\,2}(\theta)}{g(r)}+r^{\,2}(\theta)},$
(20)

here $r^{\prime}\equiv{\rm d}r/{\rm d}\theta$.

(a) $q=-0.010$

(b) $q=-0.005$

(c) $q=0.005$

(d) $q=0.010$

Figure 3: Plots of isocharges in $T-\delta L$ plane for $\theta_{\,0}=0.45$.
The Blue dash line corresponds to the temperature of the first order phase
transition. The values of the electric charge chosen (from top to bottom)are
the same with ones in Fig. 1

(a) $q=-0.010$

(b) $q=-0.005$

(c) $q=0.005$

(d) $q=0.010$

Figure 4: Plots of isocharges in $T-\delta L$ plane for $\theta_{\,0}=0.50$.
The Blue dash line corresponds to the temperature of the first order phase
transition for $Q<Q_{C}$. Panel (a): $Q=0.0286384<Q_{C}$. Panel (b):
$Q=0.0338006<Q_{C}$. Panel (c): $Q=0.0429513<Q_{C}$. Panel (d):
$Q=0.0470977<Q_{C}$

(a) $q=-0.010$

(b) $q=-0.005$

(c) $q=0.005$

(d) $q=0.010$

Figure 5: Plots of isocharges in $T-\triangle S$ plane for
$\varphi_{\,0}=0.45$. The red dash line corresponds to the temperature of the
first order phase transition. The values of the electric charge chosen (from
top to bottom) are the same with ones in Fig. 1

(a) $q=-0.010$

(b) $q=-0.005$

(c) $q=0.005$

(d) $q=0.010$

Figure 6: Plots of isocharges in $T-\triangle S$ plane for $\varphi_{\,0}=0.50$. The red dash line corresponds to the temperature of the first order phase transition for $Q<Q_{C}$. Panel (a): $Q=0.0286384<Q_{C}$. Panel (b): $Q=0.0338006<Q_{C}$. Panel (c): $Q=0.0429513<Q_{C}$. Panel (d): $Q=0.0470977<Q_{C}$ Table 2: Check of the Maxwell’s equal area construction in the $T-S$ plane $q$ | $T^{*}$ | $S_{min}$ | $S_{max}$ | $A_{1}$ | $A_{2}$
---|---|---|---|---|---
$-$0.010 | 0.4506 | 0.3062640 | 2.6289090 | 1.04538 | 1.04658
$-$0.005 | 0.4524 | 0.2371192 | 2.6989999 | 1.11250 | 1.11375
0.005 | 0.4552 | 0.1098390 | 2.7624530 | 1.20665 | 1.20747
0.010 | 0.4561 | 0.0488927 | 0.7663060 | 1.22943 | 1.22924

Taking $\mathcal{L}$ as the Lagrangian and imagining $\theta$ as time, with
the Euler-Lagrange equation

$\displaystyle\frac{\partial\mathcal{L}}{\partial r}=\frac{{\rm d}}{{\rm
d}\theta}\left(\frac{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial\mathcal{L}}{\partial
r\,^{\prime}(\theta)}\right),$ (21)

we arrive to the motion equation of $r(\theta)$

$\displaystyle
g\,^{\prime}(r)r\,^{\prime}(\theta)^{\,2}-2g(r)r^{\prime\prime}(\theta)+2g^{\,2}(r)r(\theta)=0$
(22)

with the boundary condition

$\displaystyle r(0)=r_{0},\qquad r\,^{\prime}(0)=0.$ (23)

By using the condition above and resolving the Eq. (22), we obtain the mumeric
result of $r(\theta)$. Notice that the geodesic length is divergent for a
fixed $\theta_{0}$, therefore, it needs to be regularized. Here, we do it by
subtracting the geodesic length of the minimal surface in pure AdS with same
boundary $\theta=\theta_{0}$ (denoted by $L^{\prime}$). In order to accomplish
this, we first obtain $L$ by intergrating the length function in Eq. (20) from
zero to $UV$ cutoff $\theta_{C}\lesssim\theta_{\,0}$. Then, by turning off the
hair parameter $q$, mass $M$ and electric charge $Q$ of the charged hairy
black hole background, we get pure AdS in global coordinates

$\displaystyle{\rm d}s^{\,2}=-\left(1+\frac{r^{\,2}}{l^{\,2}}\right){\rm
d}t^{2}+\left(1+\frac{r^{\,2}}{l^{\,2}}\right)^{-1}{\rm d}r^{\,2}+r^{\,2}{\rm
d}\Omega_{3}^{2},$ (24)

with this metric, repeating the same procedure as calculation $L$,
numerically, we obtain $L^{\prime}$. Thus, subtracting this quantity from the
hairy black hole, we get the renormalized geodesic length $\delta
L=L-L^{\prime}$. Here, we take $\theta_{\,0}=0.45$ and $\theta_{\,0}=0.50$ as
example to discuss the charged hairy black hole’s phase structure, and the
corresponding cutoffs are chosen to be $\theta_{C}=0.449$ and $0.499$ during
the numerical computations. For the $\theta_{\,0}=0.45$ case, in Fig. 3. we
present the plots of the isocharges generated for the two point correlation
function for different hair parameter $q$. In each panel, the isocharges in
the $T-\delta L$ plane from top to bottom correspond to $Q<Q_{C}$, $Q=Q_{C}$
and $Q>Q_{C}$, respectively. From this plots, we see that the $T$ versus
$\delta L$ plots are qualitatively similar to ones in Fig. 1. We confirm that
there exists indeed a Van der Waals-like phase transition in the $T-\delta L$
plane, and the critical temperature and critical charge are also the same as
the ones obtained from Fig. 1.

Table 3: Check of the Maxwell’s equal area construction in the $T-\delta L$ plane |  | $\theta_{\,0}=0.45$ |  | $\theta_{\,0}=0.50$ |
---|---|---|---|---|---
$q=-0.010$ | $T^{*}=0.4506$ | $\delta L_{min}=0.0001181430$ | $A_{1}=0.00027069$ | $\delta L_{min}=0.0001788345$ | $A_{1}=0.00040808$
|  | $\delta L_{max}=0.0007194292$ | $A_{2}=0.00027094$ | $\delta L_{max}=0.0010853080$ | $A_{2}=0.00040846$
$q=-0.005$ | $T^{*}=0.4524$ | $\delta L_{min}=0.0001099755$ | $A_{1}=0.00028904$ | $\delta L_{min}=0.0001664403$ | $A_{1}=0.00043580$
|  | $\delta L_{max}=0.0007494337$ | $A_{2}=0.00028929$ | $\delta L_{max}=0.0011305550$ | $A_{2}=0.00043617$
$q=0.005$ | $T^{*}=0.4552$ | $\delta L_{min}=0.00009738140$ | $A_{1}=0.00031608$ | $\delta L_{min}=0.0001473907$ | $A_{1}=0.00047660$
|  | $\delta L_{max}=0.00079196240$ | $A_{2}=0.00031617$ | $\delta L_{max}=0.0011947060$ | $A_{2}=0.00047674$
$q=0.010$ | $T^{*}=0.4561$ | $\delta L_{min}=0.00009229728$ | $A_{1}=0.00032276$ | $\delta L_{min}=0.0001397911$ | $A_{1}=0.00048686$
|  | $\delta L_{max}=0.00079951970$ | $A_{2}=0.00032256$ | $\delta L_{max}=0.0012065893$ | $A_{2}=0.00048657$
Table 4: Check of the Maxwell’s equal area construction in the $T-\triangle S$ plane |  | $\varphi_{\,0}=0.45$ |  | $\varphi_{\,0}=0.50$ |
---|---|---|---|---|---
$q=-0.010$ | $T^{*}=0.4506$ | $\triangle S_{min}=0.002343243$ | $A_{1}=0.0052170$ | $\triangle S_{min}=0.00321284$ | $A_{1}=0.0081764$
|  | $\triangle S_{max}=0.013932195$ | $A_{2}=0.0052220$ | $\triangle S_{max}=0.02136945$ | $A_{2}=0.0081814$
$q=-0.005$ | $T^{*}=0.4524$ | $\triangle S_{min}=0.00227335$ | $A_{1}=0.0055859$ | $\triangle S_{min}=0.00302369$ | $A_{1}=0.0087256$
|  | $\triangle S_{max}=0.01463187$ | $A_{2}=0.0055910$ | $\triangle S_{max}=0.02232081$ | $A_{2}=0.0087300$
$q=0.005$ | $T^{*}=0.4552$ | $\triangle S_{min}=0.00201383$ | $A_{1}=0.0061112$ | $\triangle S_{min}=0.00273341$ | $A_{1}=0.0095445$
|  | $\triangle S_{max}=0.01544459$ | $A_{2}=0.0061137$ | $\triangle S_{max}=0.02369990$ | $A_{2}=0.0095440$
$q=0.010$ | $T^{*}=0.4561$ | $\triangle S_{min}=0.00193166$ | $A_{1}=0.0062515$ | $\triangle S_{min}=0.00261655$ | $A_{1}=0.0097623$
|  | $\triangle S_{max}=0.01563169$ | $A_{2}=0.0062486$ | $\triangle S_{max}=0.02400015$ | $A_{2}=0.0097531$

In order to further characterize a Van der Waals-like phase transition for the
two point correlation function, we choose two different $\theta_{0}$ to verify
the equal area law in $T-\delta L$ plane, which is similarly defined as

$\displaystyle A_{1}\equiv\int_{\delta L_{min}}^{\delta L_{max}}T(\delta
L,\,q,\,Q)\,{\rm d}\delta L=T^{*}(\delta L_{max}-\delta L_{min})\equiv A_{2},$

where $\delta L_{min}$ and $\delta L_{max}$ are the smallest and largest roots
of the equation $T(\delta L,\,q,\,Q)=T^{*}$. $T(\delta L,\,q,\,Q)$ is an
interpolating function which can be given by our numeric result.

Besides $\theta_{\,0}=0.45$, in Fig. 4, for the case $\theta_{\,0}=0.50$, we
also plot the isocharge in the $T-\delta L$ plane when the charge satisfies
$Q<Q_{C}$. In table 3, for different $q$ and $\theta_{\,0}$, we tabulate the
values of $\delta L_{min}$, $\delta L_{max}$, $A_{1}$ and $A_{2}$. Obviously,
for the two point correlation function, the equal area law hods for the five
dimensional charged hairy black hole. This strengthen our conclusion that,
indeed, the isocharges in the $T-\delta L$ plane can also present the same Van
der Waals-like phase transition as the black hole entropy.

##  4 Holographic phase transition for entanglement entropy

Now, we move on to take into account entanglement entropy case. According to
Ryu-Takayanagi description, holographic entanglement entropy for a region $A$
can be expressed as r29 ; r30

$\displaystyle S=\frac{Area(\Gamma_{A})}{4},$ (26)

where $\Gamma_{A}$ is a codimension-2 minimal surface with boundary condition
$\partial\Gamma_{A}=\partial A$. Here, we choose $\varphi=\varphi_{0}$ as the
hairy black hole’s entangling surface and employ $r(\varphi)$ to parameterize
the minimal surface. With the symmetry, in this static AdS background, the
entanglement entropy is given by

$\displaystyle
S=\pi\int_{0}^{\varphi_{\,0}}r^{\,2}\sin^{2}\varphi\sqrt{\frac{r\,^{\prime\,2}(\varphi)}{f(r)}+r^{\,2}(\varphi)}\,{\rm
d}\varphi,$ (27)

in which $r\,^{\prime}(\varphi)\equiv{\rm d}r/{\rm d}\varphi$. Adopting the
similar procedure as the two point correlation function case, we first arrive
to the motion equation of $r(\varphi)$ by utilizing the Euler-Lagrange
equation, and with the boundary condition, the numerical result of
$r(\varphi)$ is obtained. Then, we integrate the entropy function $S$ in Eq.
(27) up to the $UV$ cutoff $\varphi_{\,C}$ (that is
$\varphi_{\,C}\approx\varphi_{\,0}$). Thus, substracting the pure AdS
entanglement entropy (which is denoted by $S_{0}$), we are able to get the
regularized entanglement entropy $\triangle S=S-S_{0}$ of a charged hairy
black hole.

Specifically, choosing $\varphi_{\,0}=0.45$, we present the isocharges in the
$T-\triangle S$ plane for different hair parameter $q$ in Fig. 5. Comparing
with Fig. 1 and Fig. 3 again, we find that, like the black hole entropy and
two point correlation function, the entanglement entropy also exhibits a Van
der Waals-like phase transition, moreover, the critical charge and critical
temperature are also identified with them.

We go on to verify whether Maxwell construction works for the entanglement
entropy in the $T-\triangle S$ plane. The analogous equal area law becomes

$\displaystyle A_{1}\equiv\\!\\!\\!\int_{\triangle S_{min}}^{\triangle
S_{max}}\\!\\!\\!T(\triangle S,q,Q)\,{\rm d}\triangle S\\!=\\!T^{*}(\triangle
S_{max}\\!-\\!\triangle S_{min})\equiv A_{2},$ (28)

where $\triangle S_{min}$ and $\triangle S_{max}$ are the smallest and largest
roots of $T(\triangle S,\,q,\,Q)=T^{*}$, $T(\triangle S,\,q,\,Q)$ is an
interpolating function which is given by our numeric result, and $T^{*}$ is a
transition temperature, which is equal to the first order phase transition
temperature $T^{*}$ found for black hole entropy in section 2. In order to
verify the equal area law, we take $\varphi_{\,0}=0.45$ and
$\varphi_{\,0}=0.50$ as example, we also present the isocharges in the
$T-\triangle S$ plane for $\varphi_{\,0}=0.50$ in Fig. 6. For different $q$
and $\varphi_{0}$, the numeric results of $\triangle S_{min}$, $\triangle
S_{max}$, $A_{1}$ and $A_{2}$ are listed in Table 4. According to this table,
we conclude that the Maxwell’s equal area construction in the $T-\triangle S$
plane is valid within reasonable error. Again, the result shows that like
black hole entropy, the entanglement entropy can indeed present a Van der
Waals-like phase transition for the charged hairy black hole.

##  5 Conclusion

In this paper, in the framework of holography, we discuss on the phase
structure of a charged hairy black hole in five-dimensional AdS background (in
the fixed electric charge ensemble). The result shows that a Van der Waals-
like phase transition can be observed in $T-S$ plane, $T-\delta L$ plane and
$T-\triangle S$ plane, and the critical charge and critical temperature are
equal to each other. Notice that for some $q$ value, where the gravity
background exhibits negative entropy, there no longer exists a reasonal phase
transition. In order to guarantee the entropy is positive, scalar hair
parameter $q$ must satisfy the condition $q\leqslant 2r\,_{+}^{3}/5$. Since
the corresponding expression for the critical values are too complicated in
this charged hairy AdS background, here, we proceed to compute numerically.

It is interesting to note that, for the charged hairy black hole, in Ref. r7 ,
Hennigar and Mann have first reveal a reentrant phase transition, and have
carefully studied the criticality and the Van der Waals behaviour in the $P-V$
plane. Here, in the $T-S$ plane, by choosing some proper values of $q$, we
also present the Van der Waals-like phase transition, thereby strengthening
the conclusion of Ref. [7]. It is worth emphasizing that, motivated by
holography, besides the black hole entropy, we also make use of the two point
correlation function and entanglement entropy to detect the Van der Waals-like
phase transition. In addition, we compute numerically the equal area law in
the $T-S$ plane, $T-\delta L$ plane and $T-\triangle S$ plane, and verify the
Maxwell’s equal area construction is valid in these planes, which provides
further evidence supporting our conclusion.

###### Acknowledgements.

We would like to thank Xiao-Xiong Zeng for his discussions. This work is
supported by the National Natural Science Foundation of China (Grant Nos.
11573022).

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[a]Michael Pitt

# Physics perspectives of a CMS near-beam proton spectrometer at the HL-LHC

###### Abstract

The High-Luminosity Large Hadron Collider (HL-LHC) is designed to achieve
higher instantaneous luminosities, enabling the exploration of the rarest
processes of the Standard Model (SM). The CMS collaboration has published an
Expression of Interest to pursue the study of the central exclusive production
processes, using near-beam detectors. This report details both the expected
performance and the scientific potential of the CMS near-beam proton
spectrometer at the HL-LHC.

##  1 Introduction

In hard proton–proton collisions at the LHC, final-state particles are
produced predominantly from the interactions of quarks or gluons that
originate from the colliding protons. These events typically involve the
production of a hard process accompanied by many energetic particles, which
arise from proton remnants, initial-state radiation, or multiparton
interactions. In rare occasions, particles can be produced by exchanging
color-neutral mediators, such as photons in electromagnetic interactions or
pomerons in strong interactions, allowing the protons to emerge intact but
lose a fraction of their momentum. Hard processes associated with an intact
proton cover a broad range of topologies and scales. The coherent
photoproduction of vector mesons has so far been measured in ultraperipheral
proton–lead collisions without proton tagging targeting scales mainly up to a
few GeV [3, 4, 1, 2]. The Mueller–Tang jet topology with two energetic jets
with a transverse momentum above tens of GeV with and without proton tagging
was measured by the TOTEM and CMS collaborations [5]. The central exclusive
production (CEP) of final states with masses of a few GeV, without proton
tagging, was reported by the LHCb collaboration [6, 7]. At the same time, hard
scattering events were studied by ATLAS [8, 9] and CMS [10, 11, 12, 13, 14,
15] utilizing proton tagging techniques. The CEP processes at masses of the
order of a few hundred GeV are predominantly produced by fusion of two
photons. This implies that the LHC can be treated as a photon collider,
offering a new means for probing fundamental physics and searching for new
particles.

Tagging events with two intact protons can discriminate between inclusive and
exclusive processes produced in proton–proton collisions. These protons,
having lost a fraction of their initial momentum (commonly denoted by
$\xi=\Delta p/p$), will be laterally deflected from the proton beam by the LHC
magnets and can be measured using dedicated near-beam detectors. In typical
LHC runs, several tens of proton–proton interactions occur per proton bunch
crossing. In a small percentage of these interactions, at least one proton
undergoes diffractive scattering with a large momentum transfer and remains
intact. Consequently, an inclusive process accompanied by a pair of protons
resulting from such additional interactions (pileup) can resemble the
exclusive final state. Hence, CEP studies face the significant challenge of a
large combinatorial background (i.e., an inclusively produced central system
with pileup protons detected on both sides).

The Time-of-Flight (ToF) method can be used to measure the arrival times of
protons, allowing the reconstruction of the spatial or temporal coordinates of
the vertex associated with the scattered protons [16]. By aligning the
kinematic properties of the protons with those of the central system and
central production vertex, it becomes feasible to identify CEP events at the
LHC.

##  2 The CMS Precision Proton Spectrometer

CEP events with intact protons can be tagged using specialized near-beam
proton spectrometers. These detectors are housed in a movable detector vessel,
enabling them to approach the beam center within just a few millimeters and
measure the scattered protons. Introduced in 2016 as a joint effort between
CMS and TOTEM collaborations [17], the CMS Precision Proton Spectrometer (PPS)
has since been successfully operational as a standard CMS subdetector [18].
The PPS detector, located about 200 meters from the CMS interaction point in
both arms, has tracking and timing stations that allow for precise measurement
of forward protons.

Prior to the transition to the High-Luminocity LHC (HL-LHC) phase, the
accelerator layout will be rearranged, and the CMS collaboration proposes a
new detector design for the PPS [19]. The new locations of the forward
detectors are selected at $z=\pm 196$, $\pm 220$, and $\pm 234$ meters from
the CMS interaction point, making them sensitive to protons with a momentum
loss from 1.5% to 20%, which corresponds to a mass range of centrally produced
final-state particles between 133 GeV and 2.7 TeV. Additionally, with a
further station positioned at $z=\pm 420$ meters, the $\xi$ acceptance range
will be as low as 0.325%, including central masses as light as 43 GeV.

##  3 Physics with the PPS sub-detector at the HL-LHC

###  3.1 Standard Model processes

The HL-LHC upgrade aims to deliver about 3000 fb-1 of total integrated
luminosity over a span of 10-12 years, enabling the study of the rarest SM
processes, including CEP. Table 1 shows the fiducial cross-section for
different SM photon-induced CEP processes in $pp$ collisions at $\sqrt{s}=14$
TeV, where one of the two protons will be within the PPS acceptance for two
scenarios: without the 420m station ($\xi\in\left(0.0142,0.1967\right)$), and
with all four stations ($\xi\in\left(0.00325,0.1967\right)$).

Table 1: Fiducial cross-sections of exclusive photon-initiated production processes in pp collisions at $\sqrt{s}=14$ TeV with one or both protons within the PPS acceptance for two scenarios, with and without the station at $\pm 420$ m. Process | fiducial cross section [fb]
---|---
2 tag | 1 tag
w/o 420 | all stations | w/o 420 | all stations
$\rm jj$ | 2 | 60 | 219 | 526
$\rm b\bar{b}$ | 0.04 | 1.7 | 6.3 | 15
$W^{+}W^{-}$ | 15 | 37 | 152 | 178
$\mu\mu$ | 1.3 | 46 | 172 | 417
$\rm t\bar{t}$ | 0.1 | 0.15 | 0.65 | 0.74
H | 0 | 0.07 | 0.23 | 0.30
$HW^{+}W^{-}$ | 0.01 | 0.01 | 0.06 | 0.07
$ZZ$ | 0.03 | 0.06 | 0.23 | 0.26
$Z\gamma$ | 0.02 | 0.04 | 0.15 | 0.17
$\gamma\gamma$ | 0.003 | 0.02 | 0.19 | 0.33

The cross-section was calculated with the MadGraph5_aMC@NLO event generator
[20], using the Equivalent Photon Approximation (EPA) for elastic photon
fluxes [21], assuming a gap survival probability of 90% and using the
MMHT2015qed_nlo_inelastic PDF set for inelastic photon fluxes [22], assuming a
gap survival probability of 70%. The production of the Single Higgs boson was
computed in the Higgs Effective Field Theory (HEFT) model [23, 24, 25], and
the loop-induced production of neutral di-boson final states ($ZZ$, $Z\gamma$,
and $\gamma\gamma$) was generated using the loop_qcd_qed_sm model [26]. A
central detector selection cut of $p_{T}>20$ GeV, and $|\eta|<2.5$ on the
generated objects was applied for processes with two particles in the final
state.

Despite the relatively small cross-sections, the amount of data delivered by
the HL-LHC will enable the exploration of various SM CEP processes. This
includes the exclusive production of QCD multijet events [28]; the di-lepton
production for detector calibration studies [27] and model tunes; $WW$
production, which is a particularly clean channel in its fully leptonic decay
mode [17]; and the exclusive production of top-quark pairs [29, 30, 31, 32].
The central exclusive Higgs boson production was extensively studied, with a
predicted cross-section of the order of a few fb, predominantly in the QCD-
initiated production mode [33, 34, 35, 36, 37, 38, 39, 40]. The photon-induced
cross-section for this production mode is an order of magnitude lower. For the
Higgs boson masses, tagging two protons is feasible only with the 420 m
station scenario. However, the associated production of a Higgs boson and a
$\rm W^{+}W^{-}$ vector–boson pair holds the potential for probing the Higgs
sector in CEP events, even without the $\pm$420 m stations, primarily in
processes dominated by photon-induced production.

###  3.2 Physics Beyond the Standard model

The PPS offers exceptional sensitivity to anomalous couplings and can probe
di-boson events with high invariant masses. It will play a crucial role in the
search for Axion-Like Particles (ALPs) through exclusive di-photon events at
high invariant masses [41], particularly in single tagged events, bolstered by
recent advancements in the theoretical framework [42]. Proton tagging is
instrumental in investigating the BSM $\gamma\gamma\gamma Z$ coupling in
exclusive $Z\gamma$ final states [43], as well as anomalous $\gamma\gamma WW$
coupling in exclusive $WW$ events.

The PPS is particularly attractive for searches in supersymmetric scenarios
with compressed spectra. An example of this is the process
$pp\to\tilde{\ell}\tilde{\ell}\to\ell\ell\tilde{\chi}_{1}^{0}\tilde{\chi}_{1}^{0}$,
where both neutralinos ($\tilde{\chi}_{1}^{0}$) are produced at rest. At the
LHC, such searches typically require high initial state radiation activity to
boost neutralinos and tag events with high missing transverse momentums. In
the exclusive production, where both protons are tagged, the di-slepton mass
($m_{\tilde{\ell}\tilde{\ell}}$) is measured by the PPS independently of the
outgoing particles reconstructed by the central detector [44, 45].

##  4 Summary

The PPS subsystem extends the scope of the CMS physics program at HL-LHC in
the field of central exclusive production processes, both in terms of the mass
range and the higher amount of statistics. With its enhanced timing
capabilities, the newly designed PPS meets the challenges posed by the high
radiation rates anticipated at the HL-LHC. The CMS collaboration has proposed
a staged installation strategy, beginning with installing three stations at
about 200 meters from the interaction point at the onset of LHC Run 4 (the HL-
LHC phase). The construction of the station at 420 meters from the interaction
point is forseen for LHC Run 5 or later, aligning with the evolving
requirements and objectives of the experiment.

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