11institutetext: Graz University of Technology, Graz, Austria 22institutetext:
Ludwig Maximilian University, Munich, Germany

# The Application of Procedurally Generated Libraries in Immersive Virtual
Reality

Saeed Safikhani 11 Benedikt Gross 11 Johanna Pirker 1122

# The Application of Procedurally Generated Libraries in Immersive Virtual
Reality

Saeed Safikhani 11 Benedikt Gross 11 Johanna Pirker 1122

###### Abstract

While digital libraries offer essential benefits for the digital age such as
constant availability and accessibility, they often fail in providing a
similarly enjoyable browsing experience that benefits engagement and
serendipitous exploration inherent to traditional library experiences. In this
paper, we propose a virtual reality (VR) library environment that emulates a
traditional library promoting an engaging browsing experience that benefits
browsing enjoyment and serendipity. The implemented system uses procedural
content generation to dynamically generate a library-like environment, wherein
each room corresponds to a specific category of the digital library. To assess
the usability of the implemented system, we conducted an A/B study comparing
the VR environment with the web interface of Project Gutenberg. The results of
the study suggest that while the physical demand for using the VR environment
is higher, the system can significantly benefit the browsing experience and
entice users’ curiosity. Furthermore, the study suggests that the system can
increase user engagement and users generally felt that using the system was
rewarding.

###### Keywords:

Virtual Reality Digital Library Procedural Generation

##  1 Introduction

Since the inception of digital libraries, significant contributions from
diverse fields, including library and computer science, have advanced this
research area [10]. Notably, extensive digitalization initiatives like the
Google Book Project have substantially increased the availability of digital
resources [50]. In contemporary times, digital libraries have gained paramount
importance, playing a crucial role in disseminating knowledge and information,
as underscored by the COVID-19 pandemic [36]. Despite these advancements,
conventional interfaces of digital libraries often fall short in facilitating
effective browsing and serendipitous discoveries compared to their physical
counterparts [28]. Traditional libraries leverage visual recognition and
spatial reasoning, allowing users to discover thematically related items
within the same room or on adjacent bookshelves [24]. With the increasing
prevalence and affordability of virtual reality (VR) headsets, VR technology
offers an immersive and interactive solution for enhancing user experiences
[49]. Leveraging VR for digital libraries presents a unique opportunity to
emulate the traditional library environment, utilizing the entire body for
book discovery in a manner that aligns with familiar physical library
experiences [13]. This immersive approach can potentially address the
limitations of current digital library interfaces and enhance the browsing
experience and serendipity.

The main contribution of this paper is to create an immersive VR library
system that mirrors the experience of navigating a physical library, providing
users access to a diverse range of books. The system’s environment will be
procedurally generated for easy adaptation to different book catalogs,
ensuring flexibility and scalability. Users will have the opportunity to read
through the entire library collection within an immersive setting,
supplemented by additional information and summaries generated using a
language completion model. The system will also incorporate exhibition objects
related to different library categories to enhance exploration and stimulate
curiosity. To evaluate the benefits of the system, a study was conducted to
compare the VR library environment with traditional digital library
interfaces, focusing on the user’s browsing and reading experience.

##  2 Background and Related Works

A digital library is an organization that provides access to digital
resources, including databases of text, numbers, graphics, sound, video, and
other types of media. It is a service, an architecture, a set of information
resources, and a set of tools and capabilities to locate, retrieve, and
utilize the information resources available [47]. Due to the importance of
digital libraries and repositories as infrastructure, There has been
considerable research and development for them in recent years [1]. Compared
to their physical counterparts, they can provide easier access for a wide
range of users. This can be especially useful when users need to access them
remotely, either due to the distance to the physical resources or restriction
of access to them, such as the condition during the COVID-19 pandemic [36].
With the substantial increase of objects in digital libraries and the enormous
expansion of data online, information retrieval (IR) has seen high interest in
commercial applications and research communities. While not exclusive to
libraries these methods offer ways of providing users with the right
information and results even from messy search queries [22]. The process of IR
for text-based data usually consists of some pre-processing on the documents.
This includes methods like tokenizing, stemming, stop word removal, and some
form of feature extraction. Based on these features some retrieval models like
the Boolean Model or the Vector Space Model are applied to find appropriate
search results [29]. IR not only but especially in libraries also heavily
relies on cataloging and metadata to provide additional information when
searching for books. This particularly applies to Digital Libraries with large
collections of documents [34]. Although providing a list of search results is
an adequate solution for seeking certain information, the simple listing of
relevant documents or books as done by many digital libraries can stand in the
way of serendipitous discoveries often gained by traditional library shelf
browsing [16].

The barrier towards serendipitous discoveries can be due to the lack of a
visual representation of available books in digital libraries, which is a
benefit for physical libraries [28]. Browsing habits can best be examined in
traditional libraries as there has only been little effort put into
replicating physical shelf browsing in digital libraries [46]. Here, studies
show that the lending of neighboring books increases the chances of a
subsequent loan by at least 9-fold [32] and users seem to be generally
successful when browsing the physical stacks [31]. Furthermore, library
patrons often seem to prefer the physical browsing of books over searching in
digital libraries [30]. Although browsing is no replacement for classical IR,
it can be a significant complement as it makes use of visual recognition and
spatial reasoning over linguistic specification and logical reasoning [37].
[21] and [31] both examined traditional library patron’s book selection
behaviors and present features that can be implemented in Digital Libraries to
better facilitate these habits. Some of these features are seamless
transitions, support of zooming while browsing, comparison of books, and
Displaying a large collection at once. Many of these suggestions for digital
library visualizations have already been applied in implementations by either
directly emulating the physical library experience digitally, e.g. [27], or
single bookshelf [46]. The first implementation is mainly called Digital
Library Replicas, where Blended Shelf is used for the latter. The Blended
Shelf is a digital interface developed in response to the archiving of books
from the University of Konstanz’s library due to an asbestos discovery. It
mimics physical bookshelves, allowing users to select disciplines and browse
corresponding books with accurate representations. Presented on an interactive
whiteboard, users can view book covers without direct interaction. In
contrast, the Library of Apollo is a web-based e-reader offering access to
over 9.4 million digitized books from 200,000 categories, enabling browsing
and search functionality [24, 46]. Emulating the physical library may lead to
preserving browsing and serendipitous discovery in digital format as it can
provide an already familiar browsing experience.

The overgrowing application of immersive technologies shows new opportunities
for visualization and interaction in virtual environments [45]. These
immersive experiences can be a solution for visualization in different
scenarios such as digital libraries. Cook et al. [13] suggests that this
technology especially lends itself to preserving browsing and serendipitous
discovery in digital format as it can integrate the whole body, therefore,
providing an already familiar browsing experience. In this case, VR can be a
solution to improve the representation of digital libraries to provide
serendipitous browsing and reading of books. According to previous studies, VR
has been shown to provide customizable multimedia content, higher immersion
and presence, and effective information transfer in the context of cultural
heritage [9, 6]. VR technology can be particularly useful for
reconstruction/digitalization and improvement in the presentation and
preservation of cultural heritage sites and artifacts. This approach could
also be applied to VR digital libraries for visualization in similar
environments.

Although VR shows potential as a medium for digital libraries, the quality of
the reading experience can play a crucial role in the user acceptance of the
technology. When comparing reading from paper and screens the most commonly
used metrics of measuring performance are investigating text comprehension,
reading speed, and meta comprehension which describes how well the reader is
able to make judgments about their text comprehension [19]. Even though the
body of studies investigating the differences in reading is large, research is
often contradictory [41] and while [5] for example argue that studies have not
yet come to a conclusive result, many meta-studies suggest a general
inferiority of reading from screens [14, 25, 17, 11]. Most studies compared
reading plain texts digitally and on paper not making use of multimedia and
other functions of digital devices that as [42] suggest can positively affect
reading comprehension. According to [23], virtual reality (VR) can provide an
immersive experience to its users.

Although little research has currently been done on comparing VR and paper
reading, a study by [3] found that reading in VR can improve knowledge
transfer and participants considered reading less demanding. The study also
found that users often had difficulties reading small text in VR, however
other studies already suggested guidelines for dealing with this issue. An
overview of the literature regarding a guideline for better reading experience
in VR can be summarized as the following:

  * •

Reading Surface: reading on a concave surface that is only warped around one
axis is preferable. This curvature can vary between 50°to 70°for surface warp
[48].

  * •

Field of View: An FoV of 90°from the HMD combined with a comfortable head
movement range of 30°horizontally, 20°vertically upwards and 12°downwards
provide an area for so-called the content zone [2].

  * •

Text Height: a general text height of 32 dmm111Distance Independent
Millimetre: 1 dmm equals to a height of 1 mm when viewed at a distance of 1 m
best suited with a minimum of 23 dmm [15, 33].

  * •

Text Font: text font is mostly suggested to be kept as sans serif to provide
visual clarity [15, 8].

  * •

Text Placement: it is suggested that the reading interface is at least
partially fixed as too much movement can reduce usability [39] and face
slightly down [33].

Since the design and creation of a 3D model of a library can be time-consuming
especially since new resources are continually added to their catalog,
Procedural Content Generation (PCG) can serve as a great solution for an
approach that is able to adapt automatically to new resources [12]. PCG
describes the automated creation of media such as poetry, paintings,
architecture and content for games [4]. PCG can provide large amounts of
content and provide new and different experiences each time while often also
being able to adapt to a multitude of situations [38, 43].

In this study, we used PCG to generate the content of a digital library for a
VR environment. In this way, we benefit from both the immersive experience of
VR and the automation functionality of PCG to maintain the library contents.

##  3 Procedurally Generated VR Library

In this study, we proposed a system to generate a digital library with
customizable rooms, based on the book content of the provided library. The
conceptual architecture of this application is demonstrated in Fig. 1.

Figure 1: Conceptual Architecture of the proposed VR library application.

Accordingly, The PCG system in this study was developed for two main purposes:
(1) generation of individual rooms in the library containing the books based
on their topic (2) generation of the floor layout and concatenation of rooms.

###  3.1 Procedural Room Generation

Based on the number of books in each catalog, the PCG system for rooms should
be able to set the size of the room accordingly. In addition, the room design
should provide enough space for the player to be able to move freely inside
the room and between bookshelves. In this way, we designed rooms as a cuboid
where all rooms share the same height. We defined for PCG tool to use the edge
of the room for the bookshelves. We added an entrance and exit for each to the
different side of the room to make it easier for the user to be oriented in a
VR environment. These design decisions define the main structure of the room.

In addition, we added some decorative objects to the room, such as furniture
and category-related 3D models, to improve the visual presentation of the
environment. The 3D models are interactable in such a way that users can
rotate them and read related information about them.

###  3.2 Procedural Layout Generation

Rooms have at least one entrance and exit to ensure a continuous path from the
first room to the last. To ensure that rooms are in close proximity to each
other, rooms are laid out in a clockwise or counter-clockwise pattern.
However, since the size of each room varies, rooms might still not directly
align with others. To further compress the layout, new rooms are, if possible,
placed directly at the room south of the current placement direction while
still being connected to the outgoing room. When rooms are directly adjacent
to each other, they can be connected, opening up a new path to traverse
through the environment. The size of a room thereby determines how many
connections to other rooms are possible, since too many junctions in a small
space can lead to an overly complex layout and make navigation confusing.

###  3.3 Navigation in the Procedurally Generated Library

Navigation in a procedurally generated structure can be difficult and without
additional information, users might become disorientated and lost. We used
signboards and a map to visualize the current player location in the virtual
library layout. This map can also be used as a room selection interface, and
users can teleport between rooms by selecting them either based on their
location or category. This map supports gesture-controlled scrolling and is
scalable to change the UI size. The movement of the player is provided by a
Continuous Move Provider that allows for smooth movement through inputs of the
left controller’s d-pad and a Continuous Turn Provider that similarly allows
for rotation through the left and right inputs from the right controller’s
d-pad.

###  3.4 Additional Reading Context

To further improve the browsing experience AI can provide the user with
additional resources about the currently selected book. These resources can
include additional information and summaries. OpenAI provides various
reinforcement learning models that are trained on a large corpus of web pages
and books. Their language models provide text answers given an input prompt.
We used this language model to provide additional information related to the
selected book. The book information received from the API requests can be
displayed in a similar manner as the book content. However, since the answers
provided by the models from OpenAI contain a lot less text than the book
content, the page-turning can be replaced by scrollable text. In this way,
simple buttons can be used to switch context between book content, additional
information and book summary.

###  3.5 Implementation

Development Environment: We developed our system in Unity 2020.3.4f and we
used the XR Interaction Toolkit as an interaction system for the VR
implementation. The XR Interaction Toolkit provides some valuable components
such as a room-scale VR-camera-rig, capabilities for interaction with UI
elements, grabbable objects, and more. This interaction system supports
various controller input systems and is in this case used in combination with
OpenXR OpenXR provides a general application interface that allows the use of
a single API for a multitude of different VR devices and systems such as
SteamVR , and Oculus .

Database: We used the project Gutenberg222https://www.gutenberg.org/ as a
database for our ebooks. This digital library contains over 70’000 ebooks that
focuses on older works for which US copyright has expired. While other digital
libraries such as the Internet Archive333https://archive.org/details/texts
feature a more extensive catalog with over 37 million books, Project Gutenberg
shows better support for the acquisition of their provided resources and is
therefore preferable for showcasing the proposed system.

Library Generation: Rooms are created based on how many books are present in
each category. Most rooms are created with either the width or the height as a
fixed value that primarily considers enough space for the player to traverse
the room. The other length of the room is determined by how many bookshelves
need to be placed. A newly created room is then concatenated to the previous
room, see Fig. 2 (right). After every room has its place and all connections
are made, all room objects can be instantiated beginning with the room
structure such as floor, walls, and ceiling. Furthermore, essential other
objects such as lights, door signs, and room-dependent user interface elements
are also created in this step. An example of the generated room is
demonstrated in Fig. 2 (left).

Figure 2: (left)An example of a finished room including bookshelves, windows
and furniture, (right)The room concatenation in the procedurally generated
library

Interactions: Each book contains a Grab Interactable Component that enables
the Hover and Grab interactions. The hover interaction conveys information
about the hovered book, including title, author, publication year, and
category. Pressing the grab button on the hovered book picks up the book and
by pressing the trigger button user can read it. These interactions are
visualized in Fig. 3.

###  3.6 Optimization

The implemented procedural library deals with lots of data in the form of
text, cover images and objects that are active at the same time. Optimization
of the provided system is therefore vital to ensure adequate performance.
While too much data can be limiting in terms of memory, rendering too many
objects at once can have a large impact on performance. To counteract this
problem, not all rooms and their objects are rendered at the same time.
Furthermore, interior objects are only rendered when the user is directly
inside a room.

Figure 3: (left) reading interface with corresponding UI elements (right)
hover and grab interaction

##  4 Methodology

In order to compare our system for procedurally generated library in VR with
traditional digital library interfaces, we conducted a user study. The main
objective of this user study is to figure out whether the immersion and
interactivity of the VR library can lead to user engagement, browsing
interest, and enjoyment.

Materials and Study Design: The experiment was conducted as an A-B study.
Group A started using the VR environment and later switched to the web
interface while Group B began with the web application and then switched to
the VR system. We considered a pre-questionnaire, a demographic questionnaire,
to gather general participant information (e.g. age, gender, level of
education) as well as their background related to the study (e.g. previous
experience with visual interface, and prior experience in VR). We evaluated
the user experience after each interface using post-questionnaire. This
questionnaire was a collection of standard questionnaires and a custom
questionnaire regarding browsing and reading experience in each interface. In
this case, we considered the System Usability Scale (SUS) [7], the User
Engagement Scale (UES) [35], and the NASA Task Load Index (NASA TLX) [20] as
standard questionnaires.

Setup: We used a PC with an Intel Core i5 CPU, Nvidia RTX 3080Ti GPU, and 32
GB RAM to provide a smooth VR experience. For VR setup, we considered "Meta
Quest 2"444https://www.meta.com/at/quest/products/quest-2/. In this way, we
were able to run the study without predefined tracking sensors in the room.

Participants: During the course of the study, 20 people participated with 15
being male and 5 female. The participants were divided equally into two groups
each starting with a different system. Group A started with the assessment of
the VR environment and group B evaluated the Project Gutenberg web system
first. The ages of participants ranged from 16 to 63 (AVG=26.65, STD: 9.43).
Out of the 20 participants, 9 completed a bachelor’s degree, 2 a master’s
degree, and 9 with lower education level. Participants answered to regularly
use a computer (AVG: 4.70/5.0, STD: 0.73) and rated themselves as high for
adjusting to new interfaces (AVG: 4.05/5.0, STD: 0.89). Rating of experience
with video games was mixed (AVG: 3.40/5.0, STD:1.64) and participants rated
themselves low when it comes to previous experience with virtual reality (AVG:
1.80/5.0, STD: 0.89).

Procedure: The study for each participant began by filling out pre-
questionnaires. After this step, according to the assigned group (A or B),
they continued with the corresponding system (procedural in VR or traditional
digital). For VR groups, as it could be the first time for them to experience
a VR environment, we explained to them the general controller schema as well
as UI functionality. They were asked to fulfill the following tasks to
complete each scenario:

  1. 1.

Find a book in the category Harvard Classics that you find interesting.

  2. 2.

Find a book you would be interested in reading in a category about a country.

  3. 3.

Find a book you would be interested in reading in a category of your choice.

Since the goal of the study was to investigate user engagement, browsing
interest and serendipity, users were always given the choice to explore the
library on their own, especially when a participant found an interesting
category they wanted to investigate further. Users were also not given a time
limit for their experience and were free to explore both versions for as long
as they liked. After finishing the tasks, users were asked to fill out the
post-questionnaire and repeat the procedure for the other scenario.

##  5 Results

To evaluate and compare user experience in both scenarios we considered system
usability, task load, and user engagement standard questionnaire. We used
T-Test for normally distributed data and Wilcoxon Signed Rank test for not-
normally distributed data.

The results of the SUS questionnaire show no significant difference between VR
procedural generated library (AVG: 68.00, STD: 14.50) and traditional digital
library (AVG: 66.62, STD: 17.61) with a p-value of 0.8. However, we found
significant differences for individual items in SUS questionnaires. The item
“I found the various functions in this system were well integrated” shows
significant higher score in VR (AVG: 4.0, STD: 0.89) compared to traditional
interface (AVG: 3.0, STD: 1.04) with a p-value of 0.01. with p-value of 0.02
and 0.04 correspondingly.

The users’ response to task load during the experience show significantly
higher "physical demand" (p-value: 0.001) in the VR environment. The other
items of NASA TLX do not show significant differences. However, we can expect
to see a significant difference in the "frustration" item by increasing the
number of participants in the study as the current p-value is borderline
(p-value: 0.055).

The UES used in this evaluation features four dimensions: Aesthetic Appeal,
Focused Attention, Perceived Usability and Reward. Participants generally
rated their experience to be significantly more rewarding (p-value: 1.5
$e^{-9}$) when using the VR environment. They also rated the "Aesthetic
Appeal" of the VR system considerably higher (p-value: 1.1 $e^{-9}$). In
addition, users of the VR system felt significantly more absorbed (p-value:
2.5 $e^{-14}$) in the experience than when using the web page. However, we do
not find any significant difference in the case of "Perceived Usability"
between the two systems.

As an outcome of the costume questionnaire users found the VR environment more
enjoyable (p-value: 0.001) and better motivated users to browse through the
presented library and explore its content (p-value: 0.0005). However, we do
not find a significant difference between the two conditions in the case of
ease of reading and legibility.

##  6 Discussion

Users expressed their experience in VR scenarios by overall enjoyment and
worthwhile application. Users generally felt that they needed to learn more
before getting going with the VR system and were more in need of assistance
from a technical person. This could likely be due to the fact that users
generally had little to no prior experience with VR. Despite this, they found
both systems equally easy to use, and felt confident using them, given their
more substantial experience with traditional computer usage. Despite the
generally positive feedback, some participants of the VR library expressed
that they sometimes confused buttons and did not know which to press for a
certain action. We can expect by the increase in the spreading of consumer-
based VR devices among users, they become more familiar with general
interaction in this immersive environment. In addition, having an onboarding
level before asking users to enter the main VR application can help them to
improve their familiarity with VR experiences.

The Physical Demand of the VR system seems to be significantly higher, which
is in line with findings from other studies, e.g. [26, 40]. One of the reasons
for this excessive physical demand can be the difference between types of
interaction in VR. In this case, mid-air interaction can lead to additional
physical load. Combining different interaction types in the environment to
reduce continuous mid-air interactions may help in reducing physical demands.
Additionally, while found to be slightly over the threshold for being
significant in this study, Mental Demand is mostly likely also more demanding
in VR. This can mostly be attributed to the gesture controls and the general
physical effort that comes from standing and moving around as well as
generally more mental strain that occurs through the usage of VR. The most
prominent differences can be seen when looking into user engagement and the
results of the User Engagement Scale. Participants showed to be significantly
more present in the VR experience which is generally to be expected from the
usage of an immersive virtual environment [44, 18]. Furthermore, the VR
environment was able to entice users’ curiosity and therefore promoted users
to further investigate the displayed resources and succeeded in bringing an
overall fun experience to the participants. The significant increase in
Aesthetic Appeal in the VR version suggests that users generally thought this
system was more visually pleasing. The VR system especially shows promising
results regarding the browsing experience as participants found the system
provided a smooth and seamless experience while finding the environment easy
to navigate. Users also found the interface to be significantly more
motivating for browsing and felt a sense of exploration and discovery which
was likely additionally contributed by the exhibition objects that
participants found interesting to interact with. Texts in both systems were
rated to be similarly legible and easy to read. Book texts in the VR
environment were only available as plain text and while Project Gutenberg
provides well-structured text files other formats such as ePUB might lead to a
better reading experience. The overall reading enjoyment was significantly
higher in the VR environment.

While the study shows very promising results, future studies are needed to
confirm these findings and provide more concise insights. This is especially
true for longer and repeated usages of the implemented system as the current
study only investigated short-term results that could be influenced by the
novelty factor of the system. Additionally, the system should be evaluated
against other digital library systems as well as the visuals of Project
Gutenberg’s web-interface were generally evaluated quite low. Furthermore, the
browsing functionalities of the system could furthermore be expanded by
providing a recommender system that suggests books based on previously read
titles or categories interesting to the user. It is imperative to include a
note-taking feature in future development stages to enhance reading context,
especially in academic settings. Additionally, the completion model from
OpenAI could be replaced with a custom one that is trained on the actual
corpus of the library and is, therefore, able to provide additional
information and summaries even about newer resources.

##  7 Conclusion

Digital libraries are innovative and dynamic information repositories that
have evolved in response to the digital age. Commonly used digital library
interfaces often fall short of providing a similar experience for browsing as
physical libraries. Virtual reality (VR) can provide immersive and interactive
experiences that already show promising results for providing increased
motivation, enjoyment, and presence in other fields such as education. This
study proposes an immersive VR environment that allows for the exploration and
reading of books in a way that emulates the traditional library experience.
The environment is generated procedurally based on a catalog of books where
each room corresponds to a category of the digital library. The implemented
system allows users to read through the available books in an immersive
library setting. Users are furthermore provided with additional information
and summaries about the books taken from a language completion model. To
assess the benefits of the user experience using this environment, a study was
conducted comparing the system against the web interface of Project Gutenberg.
The results of the study are promising as participants rated their browsing
experience in VR as significantly more enjoyable and rewarding. Additionally,
the implemented system was able to convey a sense of exploration and entice
the curiosity of users to continue browsing.

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# Can the Higgs Still Account for the $g-2$ Anomaly?

Fayez Abu-Ajamieh<EMAIL_ADDRESS>Centre for High Energy Physics,
Indian Institute of Science, Bangalore 560012, India  Sudhir K. Vempati
<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute of
Science, Bangalore 560012, India

###### Abstract

We use an Effective Field Theory (EFT) approach to evaluate the viability of
the Higgs to account for the $g-2$ anomaly. Although the SM contribution of
the Higgs to the muon’s magnetic dipole moment is negligible, using a bottom-
up EFT, we show that given the current level of experimental limits on the
Higgs sector, the Higgs can still yield a viable solution to the $g-2$ anomaly
if its couplings to the rest of the SM particles are allowed to deviate from
their SM predictions. Such a solution would only require an $O(1)$ fine-
tuning. Further, applying unitarity arguments, we show that such a solution
would indicate a scale of New Physics (NP) of $\sim 5-8$ TeV, which could be
lowered to $\sim 3.4-4$ TeV if the Higgs couplings to the $W$ and $Z$ are
assumed to conform to their SM predictions. We show that such a scenario could
yield significant enhancement to the di-Higgs production in muon colliders,
thus providing further motivation for its consideration. A key takeaway of
this study is to highlight the importance of measuring the $\bar{\mu}\mu hh$
coupling in future experiments.

##  I Introduction

The anomalous magnetic dipole moment of the muon, known as the $g-2$ anomaly,
remains one of the best hints for physics Beyond the Standard Model (BSM). The
E821 experiment at BNL Muong-2:2006rrc  first showed a discrepancy compared
with the theoretical predictions, at a significance of $\sim 3.7\sigma$. This
was confirmed in 2021 by the E989 experiment at Fermilab Muong-2:2021ojo ;
Muong-2:2021ovs ; Muong-2:2021vma , which yielded a measurement of
$a_{\mu}^{\text{EXP}}=116592061(41)\times 10^{-11}$. The Fermilab measurement,
in comparison with the latest theoretical SM prediction of
$a_{\mu}^{\text{SM}}=116591810(43)\times 10^{-11}$ Aoyama:2020ynm ;
Davier:2017zfy ; Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019mqg
; Davier:2019can ; Keshavarzi:2019abf ; Kurz:2014wya ; FermilabLattice:2017wgj
; Budapest-Marseille-Wuppertal:2017okr ; RBC:2018dos ; Giusti:2019xct ;
Shintani:2019wai ; FermilabLattice:2019ugu ; Gerardin:2019rua ; Aubin:2019usy
; Giusti:2019hkz ; Melnikov:2003xd ; Masjuan:2017tvw ; Colangelo:2017fiz ;
Hoferichter:2018kwz ; Gerardin:2019vio ; Bijnens:2019ghy ; Colangelo:2019uex ;
Pauk:2014rta ; Danilkin:2016hnh ; Jegerlehner:2017gek ; Knecht:2018sci ;
Eichmann:2019bqf ; Roig:2019reh ; Colangelo:2014qya ; Blum:2019ugy ;
Aoyama:2012wk ; Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ,
further increases the discrepancy to $4.2\sigma$

$\Delta a_{\mu}=a_{\mu}^{\text{EXP}}-a_{\mu}^{\text{SM}}=251(59)\times
10^{-11}.$ (1)

On the other hand, some of the latest high-precision QCD lattice simulations
Borsanyi:2020mff ; Ce:2022kxy ; Alexandrou:2022amy  appear to agree with the
experimental measurements, thus reducing the anomaly. However, these results
could be in conflict with the $e^{+}e^{-}$ data Colangelo:2022vok ;
DiLuzio:2021uty . This discrepancy arises mainly from the Hadronic Vacuum
Polaization (HVP) corrections, which are difficult to calculate (see for
instance Crivellin:2020zul ; Keshavarzi:2020bfy ; Colangelo:2020lcg ).
Therefore, it seems that further confirmation is needed. On the other hand, a
new measurement of the cross section of $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$
from the CMD-3 group CMD-3:2023alj , reveals a larger contribution to
$a_{\mu}^{\text{had,LO}}$ compared to the previous values in the literature,
which makes $a_{\mu}^{\text{SM}}$ closer to the experimental value, thereby
lowering the significance of the discrepancy in the data-driven approach as
well. Nonetheless, this result too is inconsistent with the previous results
in the literature and still needs independent corroboration.

The upcoming runs of the E989 experiment are expected to lower the uncertainty
by a factor of 4, whereas the E34 collaboration at J-PARC Saito:2012zz ;
Mibe:2011zz ; Abe:2019thb  plans on measuring $a_{\mu}$ using a completely
different method. As for the SM predictions, various theory and lattice groups
are expected to present updated results from those in Borsanyi:2020mff ;
Ce:2022kxy ; Alexandrou:2022amy . For the purposes of this study, we will
assume that the anomaly exists and is given by eq. (1).

We should also point out that another discrepancy between the predicted and
measured magnetic dipole moments of the electron was reported, however, there
is disagreement between the two experiments that reported the discrepancy on
the sign and significance of the electron $g-2$ anomaly. The first group
Aoyama:2019ryr  reported an electron anomaly $\Delta a_{e}=-88(36)\times
10^{-14}$ corresponding to a significance of $2.4\sigma$, whereas the second
group Morel:2020dww  reported $\Delta a_{e}=48(30)\times 10^{-14}$,
corresponding to a significance of only $1.6\sigma$. Given this disagreement,
it is not clear whether the electron anomaly indeed exists or not, and if it
does which sign it has. As the sign of the discrepancy is crucial in
determining the nature of the physics behind the anomaly, we will ignore it in
this work, waiting for more accurate measurements to be made, and for an
agreement to be reached.

In this paper, we seek to investigate whether or not the Higgs boson can still
accommodate a solution to the $g-2$ anomaly for the muon. In the SM, the
contribution of the Higgs to $a_{\mu}$ at one loop is $O(10^{-14})$ and is
thus negligible, however, the current level of experimental bounds on the
Higgs couplings to the rest of the SM particles still leaves ample room for
significant deviation from the SM predictions. For instance, the coupling of
the Higgs to $\bar{\mu}\mu$ is only constrained at $O(1)$, whereas higher-
dimensional Higgs operators remain essentially unconstrained. This suggests
using an EFT approach in the Higgs interaction with the muon, in order to
determine whether or not the muon anomaly can still be solved by the Higgs
boson within the experimental constraints. The strategy is simple: We write
down the most general Higgs interactions with the muon, and then calculate its
contribution to $a_{\mu}$ given the experimental constraints, then we
determine whether this contribution can account for the measured discrepancy
in eq. (1) or not. Similar work using the Standard Model EFT (SMEFT) already
exists in the literature Buttazzo:2020ibd ; Yin:2020afe ; Fajfer:2021cxa ;
Aebischer:2021uvt ; Allwicher:2021jkr ; Cheung:2021iev , however, here we
utilize a more phenomenologically-transparent, completely model-independent
approach Chang:2019vez ; Abu-Ajamieh:2020yqi ; Abu-Ajamieh:2021egq ; Abu-
Ajamieh:2022ppp ; Abu-Ajamieh:2021vnh  in order to calculate the Higgs BSM
contributions to the anomaly. We find that given the current level of
experimental constraints, the Higgs sector can still provide a solution to the
muon $g-2$ anomaly. We show that the viable part of the parameter space can be
significantly probed by future experiments, especially the proposed $10$-TeV
muon collider MuonCollider:2022cre .

This paper is organized as follows: In Sec. II, we present a review of the
model-independent EFT used in Chang:2019vez ; Abu-Ajamieh:2020yqi ; Abu-
Ajamieh:2021egq ; Abu-Ajamieh:2022ppp ; Abu-Ajamieh:2021vnh ; Abu-
Ajamieh:2023qvh . In Sec. III, we present the contributions to $\Delta
a_{\mu}$ in this framework at 1 and 2 loops, and we present the current
experimental bounds and future projections on the parameter space. In Sec. IV
we discuss the phenomenological implications of the model. In Sect. V, we
introduce a simple UV extension that can be matched to the EFT, and finally we
conclude in Sec. VI.

##  II Overview of the Model-independent EFT

In this paper, we adopt the EFT framework employed in Chang:2019vez ; Abu-
Ajamieh:2020yqi ; Abu-Ajamieh:2021egq ; Abu-Ajamieh:2022ppp ; Abu-
Ajamieh:2021vnh ; Abu-Ajamieh:2023qvh . In this framework, we avoid any power
expansion in writing down higher-dimensional operators, as is the case in the
SMEFT. Instead, we parameterize NP as deviations from the SM predictions.
Therefore, we write the effective Lagrangian of the Yukawa interaction of the
lepton sector as follows111This methodology of expanding the Higgs couplings,
is reminiscent of the chiral expansion of the Higgs Lagrangian (see for
example,Buchalla:2020kdh  and the references therein).

$\mathcal{L}_{\text{eff}}=\mathcal{L}_{\text{SM}}-\frac{Y_{l}v}{\sqrt{2}}\Big{(}\overline{L}_{l}\tilde{\hat{H}}l_{R}+h.c.\Big{)}\Big{[}\delta_{l1}\frac{X}{v}+C_{l2}\frac{X^{2}}{2!v^{2}}+\dots\Big{]}+\ldots,$
(2)

where $\delta_{l1}$ parameterize the deviations of the leptons’ Yukawa
couplings compared to the SM predictions

$\delta_{l1}\equiv\frac{Y_{l}-Y_{l}^{\text{SM}}}{Y_{l}^{\text{SM}}},$ (3)

whereas $C_{ln}$ are Wilson coefficients that do not have SM counterparts. The
field $X$ is defined in terms of the Higgs doublet $H$ as

$X\equiv\sqrt{2H^{\dagger}H}-v,$ (4)

whereas we define the projector $\tilde{\hat{H}}\equiv\epsilon\hat{H}^{*}$,
with

$\epsilon=\begin{pmatrix}0&&1\\\ -1&&0\end{pmatrix},\hskip
28.45274pt\hat{H}\equiv\frac{H}{\sqrt{H^{\dagger}H}}=\begin{pmatrix}0\\\
1\end{pmatrix}+O(\vec{G}),$ (5)

where $\vec{G}$ are the Goldstone bosons. Notice that we are dividing $X$ by
the appropriate powers of $v$ only to keep the Wilson coefficients
dimensionless, i.e., $v$ is NOT an expansion parameter. Notice that in the
unitary gauge, $X\rightarrow h$ and the Goldstone bosons are eaten. In this
paper, we simplify our calculations by working in the unitary gauge.
Therefore, in this gauge, the Lagrangian in terms of the physical masses
$m_{l}=\frac{Y_{l}v}{\sqrt{2}}$ becomes

$\mathcal{L}_{\text{eff}}=-m_{l}\overline{l}l\Big{[}(1+\delta_{l1})\frac{h}{v}+C_{l2}\frac{h^{2}}{2v^{2}}+\dots\Big{]},$
(6)

and for the case of the muon $l=\mu$. Before we proceed, a couple of remarks
are in order. First, we are assuming that $v$ is the minimum of the Higgs
potential including all higher-order corrections, i.e., $v=246$ GeV, and
second, although eq. (6) appears similar to the Higgs EFT (HEFT)
Grinstein:2007iv , we should keep in mind that secretly we are using the Higgs
doublet in our expansion, and one can easily demonstrate that the effective
Lagrangian in eqs. (2) and (6) can be mapped to any expansion in either the
SMEFT or the HEFT (see Abu-Ajamieh:2020yqi ; Abu-Ajamieh:2021egq ; Abu-
Ajamieh:2022ppp ; Abu-Ajamieh:2021vnh  for the matching to the SMEFT
operators). In other words, the deviations and Wilson coefficients in eqs. (2)
and (6) can receive corrections from any number of higher-order operators in
either the SMEFT or the HEFT frameworks. The benefit of this construction is
twofold. First: there are fewer assumptions in this framework compared to
either the SMEFT or the HEFT. Namely, we are only assuming that there are no
light degrees of freedom below the energy scale where the EFT breaks down, and
that the deviations and Wilson coefficients are compatible with experimental
measurements. The second benefit lies in the fact that parameterizing NP this
way is more phenomenologically-transparent, as these deviations and Wilson
coefficients are what is measured experimentally as opposed to any expansion
scale.

##  III The BSM contributions to $(g-2)_{\mu}$

With eq. (6) in hand, we can readily calculate the BSM contributions to
$(g-2)_{\mu}$ with $l\rightarrow\mu$, treating the effective couplings as
contact terms. In our calculation, we limit ourselves to 2 loops. The Feynman
diagrams that contribute to the anomaly at one and two loops are shown in Fig.
1. Diagram I has already been calculated in the SM case Jackiw:1972jz  (see
also Tucker-Smith:2010wdq ; Chen:2015vqy ). Thus, the BSM contribution can be
readily calculated simply by substituting $Y_{\mu}\rightarrow\delta_{\mu
1}Y_{\mu}$. Given the latest bounds on $\delta_{\mu 1}$ ATLAS:2019nkf

$|\delta_{\mu 1}|<0.53,\hskip 14.22636pt\text{@ 95\% C.L.},$ (7)

(a) I

(b) II(a)

(c) II(b)

(d) III

(e) IV

(f) V(a)

(g) V(b)

(h) VI(a)

(i) VI(b)

(j) VII

(k) VIII(a)

(l) VIII(b)

Figure 1: The diagrams contributing to the $g-2$ of the muon at one and two
loops.

it is easy to show that this contribution is at most $O(10^{-14})$ and thus is
negligible. Diagram II is loop-suppressed compared to diagram I and therefore
is also negligible. Diagrams III and IV (of the Barr-Zee type) have also been
calculated before in the SM Czarnecki:1995sz ; Haestier:2006mg ;
Gribouk:2005ee ; Gnendiger:2013pva ; Czarnecki:1995wq . Therefore, we can
immediately extend the results to the BSM case with the above rescaling

$\displaystyle\Delta a_{\mu}^{(\text{III})}=\delta_{\mu 1}\times
a_{\mu,W}^{\text{SM}},$ (8) $\displaystyle\Delta
a_{\mu}^{(\text{IV})}=\delta_{\mu 1}\times a_{\mu,f}^{\text{SM}},$ (9)

where $a^{\text{SM}}_{\mu,W}=(-19.97\pm 0.03)\times 10^{-11}$ and
$a^{\text{SM}}_{\mu,f}=(-1.5\pm 0.01)\times 10^{-11}$. Notice here that in the
fermion loop in diagram IV, only the contributions from $f=\\{t,b,c,\tau\\}$
are retained. Also notice in the same diagram that the BSM contribution could
include the deviations in the Yukawa couplings of the above fermions, namely
$\delta_{t1},\delta_{b1},\delta_{c1}$ and $\delta_{\tau 1}$, however, given
the bounds on these deviations, we checked that their contribution to $\Delta
a_{\mu}$ is insignificant, even if we saturate all experimental bounds and
arrange for all the corresponding contributions to conspiratorially add up.
Therefore, we set $\delta_{t1},\delta_{b1},\delta_{c1},\delta_{\tau
1}\rightarrow 0$.

Figure 2: The parameter space that corresponds to the $2\sigma$ solution to
$(g-2)_{\mu}$, together with the LHC constraints (solid red), the HL-LHC
projections (dotted red), and the muon collider projections (dotted blue) on
$\delta_{\mu 1}$, superimposed. $C_{\mu 2}$ is unconstrained.

Diagram V only depends $\delta_{\mu 1}$, and explicit calculation of its
contribution shows that it is of $O(10^{-18})$ when the bound in eq. 7 is
saturated. On the other hand, the contribution from diagram VI depends on both
$\delta_{\mu 1}$ and $C_{\mu 2}$ and therefore could be significant.
Nonetheless, we checked that compared to the other diagrams, the contribution
is suppressed and does not exceed $O(10^{-13})$ at best, and thus can safely
be neglected.

This leaves us with diagrams VII and VIII. Both diagrams are somewhat
complicated, however, if we use a UV-cutoff scheme such that $\Lambda\gg
M_{h}$, then one can show that they can be approximated by

$\displaystyle\Delta a_{\mu}^{(\text{VII})}$ $\displaystyle\simeq$
$\displaystyle\frac{m_{\mu}^{4}}{128\pi^{4}v^{4}}C_{\mu
2}^{2}\log^{2}{\Big{(}\frac{\Lambda^{2}}{M_{h}^{2}}}\Big{)},$ (10)
$\displaystyle\Delta a_{\mu}^{(\text{VIII})}$ $\displaystyle\simeq$
$\displaystyle-\frac{3m_{\mu}^{4}}{64\pi^{4}v^{4}}(1+\delta_{\mu 1})C_{\mu
2}\log{\Big{(}\frac{\Lambda^{2}}{M_{h}^{2}}}\Big{)}\log{\Big{(}\frac{M_{h}^{2}}{m_{\mu}^{2}}}\Big{)}.$
(11)

Here we note in passing that we are only using a UV cutoff scheme for
convenience, and using dimensional regularization yields similar results if
the renormalization scale is defined appropriately. The results are
insensitive to the choice of the renormalization scheme or the UV cutoff/
renormalization scale, as the divergence is only logarithmic. Putting all
pieces together, the total BSM contribution to $\Delta a_{\mu}$ is thus given
by

$\Delta a_{\mu}^{\text{BSM}}\simeq\Delta a_{\mu}^{(\text{III})}+\Delta
a_{\mu}^{(\text{IV})}+\Delta a_{\mu}^{(\text{VII})}+\Delta
a_{\mu}^{(\text{VIII})}.$ (12)

We have 3 parameters, namely $\delta_{\mu 1}$, $C_{\mu 2}$ and $\Lambda$. In
the rest of this paper, we fix $\Lambda=10$ TeV. Now, we can use eq. (12) in
order to find the region in the parameter space that corresponds to solving
the anomaly in eq. 1. Figure 2 shows the viable parameter space that
corresponds to the $2\sigma$ solution to $\Delta a_{\mu}$. The plot also shows
the experimental constraints given in eq. 7 in solid red. The plot shows that
the Higgs sector can still accommodate a solution to the $(g-2)_{\mu}$
anomaly. This is mainly because the BSM Wilson coefficient $C_{\mu 2}$ is
essentially unconstrained experimentally, and we can see that a solution to
the $g-2$ anomaly is still possible even if $\delta_{\mu 1}=0$. On the other
hand, $\delta_{\mu 1}$ cannot on its own solve the anomaly if $C_{\mu
2}\rightarrow 0$. In fact, we can see that $\delta_{\mu 1}$ only mildly
affects $\Delta a_{\mu}$, which is hardly surprising given the smallness of
the muon Yukawa coupling and the bound in eq. 7.

The size of $|C_{\mu 2}|$ required to solve the anomaly is $\sim(2-4)\times
10^{3}$, which might seem large, however, we should keep in mind that this is
merely an artifact of our parameterization when we divided $C_{\mu 2}$ by $v$
to keep it dimensionless. We should note that with proper power counting (such
as in the SMEFT), $C_{\mu 2}$ is divided by a larger UV scale that is
typically $O(\text{TeV})$ which would yield lower values of $|C_{\mu 2}|$. We
shall show later on that perturbativity is maintained for this range of
$|C_{\mu 2}|$ up to safe levels. In addition, we shall discuss the fine-tuning
associated with it in the next section.

| HL-LHC | HE-LHC | ILC | CLIC | CEPC | FCC-ee | FCC | Muon (10 TeV)
---|---|---|---|---|---|---|---|---
|  |  S2 S2′ |  250 500 1000 |  380 1500 3000 |  |  240 365 | (ee/eh/hh) |  +125 GeV
$\delta_{\mu 1}[\%]$ | $9.2$ |  $5$ $3.4$ |  $30$ $18.8$ $12.4$ |  $640$ $26$ $11.6$ | $17.8$ |  $20$ $17.8$ | $0.82$ |  $3.6$ $0.19$
Table 1: Projected $2\sigma$ bounds on $\delta_{\mu 1}$ in the various future
experiments taken from the Higgs@FutureColliders study deBlas:2019rxi  which
is summarized in Abu-Ajamieh:2022dtm  (see also Dawson:2022zbb ). The most
stringent constraints come from the combined analyses in the muon collider
with COM energies of $10$ TeV and $125$ GeV. This experiment is essentially
capable of exploring the entire $\delta_{\mu 1}$ dependence of the parameter
space.

The current LHC bound in eq. 7 is not very stringent, however, there are many
proposed future experiments that are expected to probe the viable parameter
space almost fully. These are summarized in Table 1. We show the most relevant
ones in Figure 2, which are the HL-LHC projections and the projections from
the proposed muon collider with COM runs of $10\hskip
2.84526pt\text{TeV}+125\hskip 2.84526pt\text{GeV}$. We can see from the plot
that the proposed muon collider can essentially probe the entire range of
$\delta_{\mu 1}$. However, even in this scenario, the Higgs sector remains
viable as a potential solution to the $g-2$ anomaly, because $C_{\mu 2}$
remains unconstrained. To the best of our knowledge, there aren’t any proposed
measurements of $C_{\mu 2}$ either currently in the LHC or in the future. We
think that current and future experiments should probe this effective coupling
both because it can potentially provide a solution to the $g-2$ anomaly, even
when $\delta_{\mu 1}\rightarrow 0$; and because of its potential enhancement
of the di-Higgs production as we will show below.

##  IV Phenomenological implications

In this section, we will discuss the phenomenological implications of the
viable parameter space, including the implications of nonzero $\delta_{\mu 1}$
and $C_{\mu 2}$ on unitarity; corrections to the muon mass and the
corresponding fine-tuning, the impact on other couplings; and the enhancement
to the di-Higgs production.

###  IV.1 Unitarity-Violation and the Scale of New Physics

As explained in detail in Chang:2019vez ; Abu-Ajamieh:2020yqi ; Abu-
Ajamieh:2021egq ; Abu-Ajamieh:2022ppp ; Abu-Ajamieh:2021vnh , any non-
vanishing deviations from the SM predictions will eventually lead to a
breakdown in unitarity at some high energy scale. This is because the SM is
the unique UV-complete theory with the observed particle content, and any
deviation from the SM predictions will lead to UV-incompleteness that
manifests itself as processes with energy-growing amplitudes that eventually
lead to a breakdown in unitarity at some high energy scale, which signals the
onset of NP.

This implies that each point in the parameter space that corresponds to
$\Delta a_{\mu}$ will point to an energy scale where unitarity is violated,
and thus to the scale at which NP should come into play. It is fairly easy to
adapt the results of the top sector in Abu-Ajamieh:2022ppp  to the muon
sector. There is only one model-independent process, namely

$\mu\bar{\mu}\rightarrow hh:E_{\text{max}}=\frac{8\sqrt{2}\pi
v^{2}}{m_{\mu}C_{\mu 2}}.$ (13)

This process is model-independent because it only depends on one parameter,
i.e. $C_{\mu 2}$, and thus the scale of NP is independent of any other
assumptions regarding the other potential deviations in the SM. We show the
scale of NP that corresponds to the parameter space obtained from eq. 13 on
the left side of Figure 3. We can see from the plot that the scale of NP is
$\sim 4.8-8.4$ TeV, which could be probed in the LHC, although if the scale of
NP lies at the higher end, this might be difficult. However, this is well
within the range of the future $100$-TeV collider, and possibly the proposed
muon collider.

Better bounds, however, can be obtained if the deviations in the couplings of
the Higgs to the massive gauge bosons are assumed to vanish. As shown in Abu-
Ajamieh:2022ppp , there are other process that violate unitarity, however, the
amplitudes of these processes also receive contamination from the deviation in
$hhV$ and $hhVV$ couplings, with $V$ being the $W$ or the $Z$. Therefore, the
scale of NP will also depend on the sizes of these deviations. However, if
these deviations are assumed to vanish, then these processes will become
functions of $\delta_{\mu 1}$ and $C_{\mu 2}$ only, which enables us to obtain
a lower scale of NP. Up to 6-body scattering, these processes include333We
refer the interested reader to Abu-Ajamieh:2022ppp  for a complete discussion
of how these bounds are obtained from unitarity.

$\displaystyle\mu\bar{\mu}\rightarrow W_{L}^{+}W_{L}^{-}$ $\displaystyle:$
$\displaystyle E_{\text{max}}\leq\frac{8\pi v^{2}}{m_{\mu}\delta_{\mu 1}},$
(14) $\displaystyle\mu\bar{\nu}\rightarrow W_{L}^{-}h$ $\displaystyle:$
$\displaystyle E_{\text{max}}\leq\frac{8\pi v^{2}}{\sqrt{2}m_{\mu}\delta_{\mu
1}},$ (15) $\displaystyle\mu\bar{\nu}\rightarrow W_{L}^{-}W_{L}^{+}W_{L}^{-}$
$\displaystyle:$ $\displaystyle
E_{\text{max}}\leq\Big{[}\frac{32\pi^{2}v^{3}}{\sqrt{2}m_{\mu}\delta_{\mu
1}}\Big{]}^{\frac{1}{2}},$ (16) $\displaystyle\mu\bar{\mu}\rightarrow
W_{L}^{+}W_{L}^{-}h,\hskip 2.84526pt\mu\bar{\nu}\rightarrow W_{L}^{-}h^{2}$
$\displaystyle:$ $\displaystyle
E_{\text{max}}\leq\Big{[}\frac{64\pi^{2}v^{3}}{\sqrt{2}m_{\mu}|C_{\mu
2}-\delta_{\mu 1}|}\Big{]}^{\frac{1}{2}},$ (17)
$\displaystyle\mu\bar{\nu}W_{L}^{+}\rightarrow W_{L}^{+}W_{L}^{-}h$
$\displaystyle:$ $\displaystyle
E_{\text{max}}\leq\Big{[}\frac{128\sqrt{3}\pi^{3}v^{4}}{\sqrt{2}m_{\mu}|C_{\mu
2}-3\delta_{\mu 1}|}\Big{]}^{\frac{1}{3}}.$ (18)

where $W_{L}^{\pm}$ are the longitudinal modes of the $W$ boson. We plot the
lowest unitarity-violating scale from eqs. (14-18) on the right-hand side of
Figure 3. We can see now that the scale of NP drops to $\sim 3.4-4$ TeV, which
is indeed within the reach of the LHC. This suggests that the HL-LHC could
shed more light on the potential BSM physics in the Higgs sector and its
potential solution of the $g-2$ anomaly.

Figure 3: The model-independent unitarity-violating scale that corresponds to
the parameter space that can solve the $g-2$ anomaly when the Higgs couplings
to the $W$ and $Z$ are allowed to deviate from the SM predictions (left) and
when they conform to the SM predictions (right). In the former case, the scale
of NP is $\sim 4.8-8.4-$ TeV, whereas in the latter case, it could be as low
as $\sim 3.4-4$ TeV.

###  IV.2 Corrections to the Muon Mass and Fine-tuning

Nonzero $\delta_{\mu 1}$ and/or $C_{\mu 2}$ will yield corrections to the muon
mass. Given the bound in eq. 7, it is easy to see that $\delta_{\mu 1}$ will
at most lead to a mass correction $\delta m_{\mu}\sim 2\delta_{\mu 1}m_{\mu}$,
and thus to fine-tuning $\sim O(1)$. On the other hand, the required size of
$C_{\mu 2}$ to solve the anomaly is larger, and thus the corresponding mass
correction could be significantly larger.

The correction to the muon mass due to $C_{\mu 2}$ arises when the Higgs loop
is closed. Thus, calculating the mass correction can easily be done by
integrating out that loop. The corresponding fine-tuning can be estimated as

$\frac{\delta m_{\mu}}{m_{\mu}}\sim C_{\mu
2}\frac{M^{2}_{h}}{32\pi^{2}v^{2}},$ (19)

Given that the size of $C_{\mu 2}$ required to solve the anomaly is
$\sim(2-4)\times 10^{3}$, is is easy to see that although the fine-tuning is
larger in this case, it is nonetheless still of $O(1)$, and thus remains
acceptable.

###  IV.3 Modification to Muon’s Coupling to the $W$ and $Z$

The Lagrangian in eq. (2) might suggest that the couplings of the muon to
other SM particles might be significantly modified. In particular, when eq.
(2) is expressed in a general gauge where the Goldstone bosons are manifest,
one would expect from the equivalence theorem, that the coupling of $\mu$ to
$W$ and $Z$ will be modified compared to their SM predictions. Nonetheless,
here we show that this is not the case, as we are defining the field $X$ to be
VEV-subtracted as can be seen in eq. (4). To show this explicitly, we write
the Higgs doublet in a general gauge

$H=\frac{1}{\sqrt{2}}\begin{pmatrix}G_{1}+iG_{2}\\\ v+h+iG_{3}\end{pmatrix}$
(20)

then we insert $H$ in eqs. (4) and (5) and expand in powers of $\vec{G}$, we
find

$\displaystyle X$ $\displaystyle=$
$\displaystyle\sqrt{(v+h)^{2}+G_{1}^{2}+G_{2}^{2}+G_{3}^{2}}-v\simeq
h+O(\vec{G)},$ (21) $\displaystyle\tilde{\hat{H}}$ $\displaystyle\simeq$
$\displaystyle\begin{pmatrix}\frac{-G_{1}+iG_{2}}{v+h}\\\
-1-\frac{iG_{3}}{v+h}\end{pmatrix}+O(\vec{G}^{2}),$ (22)

and the eq. (2) becomes

$\delta\mathcal{L}_{\text{eff}}\simeq\frac{Y_{\mu}\delta_{\mu
1}}{\sqrt{2}}\Big{(}\frac{h}{v+h}\Big{)}\Big{[}\sqrt{2}\overline{\nu}_{\mu
L}\mu_{R}W_{L}^{+}+\sqrt{2}\overline{\mu}_{L}\nu_{\mu
R}W_{L}^{-}+i\overline{\mu}\mu Z_{L}\Big{]}.$ (23)

We can expand eq. (23) in powers of $h/v$, where it is easy to see that at
leading order, there is no correction to either $\mu\mu Z$ or $\mu\nu W$. On
the other hand, we could have the effective couplings $\mu\mu Zh$ and $\mu\nu
Wh$. Nonetheless, it is easy to see that even when $\delta_{\mu 1}$ saturates
the bound in es. (7), these couplings are $\sim O(10^{-6})$ at most, and thus
can easily evade any experimental constraints.

###  IV.4 Di-Higgs Production Enhancement

Nonzero $\delta_{\mu 1}$ and/or $C_{\mu 2}$ could have interesting
phenomenological consequences, such as deviation in the branching fraction of
the Higgs decay to $\mu^{+}\mu^{-}$, and deviations in the Higgs signal
strength from the SM predictions. However, perhaps the most interesting
experimental signature for a nonzero $\delta_{\mu 1}$ and/or $C_{\mu 2}$ is
their enhancement of the di-Higgs production. The di-Higgs production provides
an excellent probe for NP through both investigating the Higgs couplings
compared to the SM, and searching for BSM particles through resonant
enhancement, such as additional heavy Higgses, KK gravitons etc. Thus, we
confine our discussion to the potential enhancement of the di-Higgs
production.

Figure 4: The cross-section of the di-Higgs production in the proposed
$10$-TeV muon collider as a function of $C_{\mu 2}$. The red dashed lines
bound the region of the parameter space that corresponds to solving the $g-2$
anomaly.

The strongest enhancement of the di-Higgs production through nonzero
$\delta_{\mu 1}$ and/or $C_{\mu 2}$ arises in muon colliders. In a muon
collider, the di-Higgs production proceeds at tree-level and is therefore
expected to be sizable. Specifically, $\mu\bar{\mu}\rightarrow hh$ proceeds
through the $s,t$ and $u$ channels, in addition to the contact term. The $s,t$
and $u$ channels are proportional to $\delta_{\mu 1}$ alone and are
negligible. On the other hand the contact term depends on $C_{\mu 2}$ and can
be sizable. Its cross-section is given by

$\sigma^{\text{BSM}}(\mu\bar{\mu}\rightarrow hh)\simeq\frac{C_{\mu
2}^{2}}{64\pi v^{4}}m_{\mu}^{2}\sqrt{1-\frac{4M_{h}^{2}}{s}}.$ (24)

In Figure 4, we show the cross-section of the di-Higgs production in the
proposed $10$-TeV muon collider. The plot shows that for the region of the
parameter space that corresponds to the solution to the $g-2$ anomaly, the di-
Higgs production cross-section is $\sim 30-105$ pb, which is quite sizable.
This could be the smoking gun for the viability of the Higgs solution to the
$g-2$ anomaly, and it indeed represents another motivation for building the
muon collider.

##  V A Possible UV Model

So far, we have followed a completely bottom-up model-independent approach
without reference to any UV completions. Similar work has been discussed in
the literature using the SMEFT, see for instance Dermisek:2022aec , where it
was argued that any new interaction that results in a chirally-enhanced
contribution to $(g-2)_{\mu}$, necessarily modifies the decay rate of the
Higgs to $\overline{\mu}\mu$, or leads to an electric dipole moment for the
muon.

In our paper, we have seen that a proper solution would require sizable
contributions to the $\mu\mu h^{2}$, while at the same time keep the
contribution to $\mu\mu h$ small to evade the LHC constraints in eq. (7).
Building such a model is challenging as it would require conspiratorial
cancellations to achieve such a requirement. Here we present a model than can
achieve this using vectorlike leptons. The use of vectorlike leptons to solve
the $g-2$ anomaly is not new, and has been studied extensively in the
literature (see for instance Dermisek:2021mhi ; Kannike:2011ng ;
Dermisek:2013gta ; Poh:2017tfo ; Dermisek:2021ajd ; Arkani-Hamed:2021xlp ).

We adopt a model similar to the one presented in Dermisek:2021ajd . The SM is
extended by two vectorlike $SU(2)$ doublets $L_{L,R}$ and two vectorlike
$S(2)$ singlets $E_{L,R}$. $L_{L}$ and $E_{R}$ have the same quantum numbers
as the SM leptons. In addition, we also introduce a heavy Higgs-like scalar
doublet $\Phi$ that develops a VEV $v_{\phi}$

$\Phi=\begin{pmatrix}0\\\ v_{\phi}+\phi\end{pmatrix}.$ (25)

Figure 5: The relevant UV processes. After integrating out the fields $L$ and
$E$, we obtain the effective Lagrangian in eq. (27).

The most general Lagrangian can be written as

$\begin{split}\mathcal{L}_{\text{UV}}=&-M_{L}\overline{L}_{L}L_{R}-M_{E}\overline{E}_{L}E_{R}-y_{\mu}\overline{l}_{L}\mu_{R}H-Y_{\mu}\overline{l}_{L}\mu_{R}\Phi-\lambda_{E}\overline{l}_{L}E_{R}H-\lambda_{L}\overline{l}_{L}\mu_{R}H\\\
&-\lambda\overline{L}_{L}E_{R}H-\overline{\lambda}H^{\dagger}\overline{E}_{L}L_{R}-\kappa_{E}\overline{l}_{L}E_{R}\Phi-\kappa_{L}\overline{L}_{L}\mu_{R}\Phi+\kappa\overline{L}_{L}E_{R}\Phi-\overline{\kappa}\Phi^{\dagger}\overline{E}_{L}L_{R}+\text{h.c.},\end{split}$
(26)

where we have neglected the interaction of $\Phi$ with $H$ as they are
irrelevant for our purposes. However, we assume that the parameters of the
scalar potential are such that they are consistent with experiment. In our
model, we set $\overline{\lambda}\rightarrow 0$. Assuming that $M_{L},M_{E}\gg
M_{\text{EW}}$, we can integrate out $L$ and $E$ (see Figure 5) to arrive at
the effective Lagrangian

$\mathcal{L}_{\text{EFT}}\supset-
y_{\mu}\overline{l}_{L}\mu_{R}H-Y_{\mu}\overline{l}_{L}\mu_{R}\Phi-\frac{\lambda_{L}\overline{\kappa}\lambda_{E}}{M_{L}M_{E}}\overline{l}_{L}\mu_{R}H\Phi^{\dagger}H-\overline{\kappa}\Bigg{(}\frac{\kappa_{E}\lambda_{L}+\lambda_{E}\kappa_{L}}{M_{L}M_{E}}\Bigg{)}\overline{l}_{L}\mu_{R}\Phi\Phi^{\dagger}H+h.c.,$
(27)

Defining $\overline{\kappa}/M_{L}M_{E}\equiv 1/\overline{M}$ and matching the
Lagrangian in eq. (27) with the effective Lagrangian in eq. (6), we obtain the
following matching conditions:

$\displaystyle\delta m_{\mu}$ $\displaystyle=$ $\displaystyle
v_{\phi}Y_{\mu}+\frac{vv_{\phi}}{2\overline{M}^{2}}\Big{(}v\lambda_{L}\lambda_{E}+\sqrt{2}v_{\phi}(\kappa_{E}\lambda_{L}+\lambda_{E}\kappa_{L})\Big{)},$
(28) $\displaystyle\delta_{\mu 1}$ $\displaystyle=$
$\displaystyle\frac{vv_{\phi}}{\sqrt{2}\hskip
2.84526pt\overline{M}^{2}m_{\mu}}\Big{(}\sqrt{2}v\lambda_{L}\lambda_{E}+v_{\phi}(\kappa_{E}\lambda_{L}+\lambda_{E}\kappa_{L})\Big{)},$
(29) $\displaystyle C_{\mu 2}$ $\displaystyle=$
$\displaystyle\frac{v^{2}v_{\phi}\lambda_{L}\lambda_{E}}{\overline{M}^{2}m_{\mu}}.$
(30)

In order to be consistent with experimental measurements, we require that
$\delta m_{\mu}=0$, which leads to

$C_{\mu 2}=2\Big{(}\frac{v_{\phi}Y_{\mu}}{m_{\mu}}+\delta_{\mu 1}\Big{)}.$
(31)

and we can see that we can easily generate the required hierarchy in the
scales of the $\mu\mu h$ and $\mu\mu h^{2}$ couplings with $v_{\phi}\sim$ TeV.
For instance, setting $\delta_{\mu 1}=0$, $Y_{\mu}=0.1$, we can generate
$C_{\mu 2}\sim(2-4)\times 10^{3}$ by selecting $v_{\phi}\sim 1-2$ TeV.

We can see from the proposed UV completion, that maintaining small corrections
at LO while allowing for sizable corrections at NLO, would require some
conspiratorial cancellations. Nonetheless, one can envisage that such
cancellations might arise from a hidden symmetry that protects the LO from
receiving large corrections. We will not pursue this issue any further.

##  VI Conclusions and outlook

In this paper, we analyzed the viability of a BSM Higgs solution to the $g-2$
anomaly using a bottom-up EFT approach. In the SM, the Higgs contribution to
the muon’s magnetic dipole moment is negligible. However, given that the
current level of experimental limits on the Higgs sector still leaves ample
room for NP, it is possible for this contribution to be significantly enhanced
compared to the SM.

We saw that while the deviation in the Higgs Yukawa coupling to the muon is
not enough to account for the anomaly, the effective coupling $\bar{\mu}\mu
hh$, which is unconstrained experimentally, can account for the entire
discrepancy. We surveyed the proposed projections on the parameter space and
found that the proposed muon collider could provide the best sensitivity for
exploring the parameter space. We also showed that unitarity can be used in
order to estimate the scale of NP that corresponds to non-vanishing deviations
from the SM, i.e. $\delta_{\mu 1}$ and $C_{\mu 2}$. We found that the model-
independent scale of NP that corresponds to the region of the parameter space
that can solve the anomaly to be $\sim 5-8$ TeV. This scale is lowered to
$\sim 3.4-4$ TeV if the Higgs couplings to massive gauge bosons are assumed to
be equal to their SM predictions. We also showed that the corrections to the
muon mass that correspond to the size of $\delta_{\mu 1}$ and $C_{\mu 2}$
required to solve the anomaly, correspond to a fine-tuning level of $O(1)$
only.

We also discussed the possible enhancement of the di-Higgs production cross-
section, and found that it could be sizable in the proposed $10$ TeV muon
collider as a result of the contact term interaction through $C_{\mu 2}$. A
key takeaway of this study is that current and future experiments should try
to measure and set limits on $C_{\mu 2}$ due to its potential significance in
solving the $g-2$ anomaly. Current searches focus of the $hh\bar{t}t$ coupling
$C_{t2}$, which if non-vanishing, is expected to be the largest of the
effective couplings of the type $hh\bar{f}f$. Nonetheless, we believe that
$C_{\mu 2}$ should be given similar interest.

In this paper, we confined our analysis to flavor-conserving operators and
avoided Flavor Violation (FV). FV Yukawa couplings
$Y_{ij}\neq\delta_{ij}\sqrt{2}m_{i}/v$ have been discussed extensively in the
literature, see for instance Harnik:2012pb  and the references therein,
however, to the best of our knowledge, FV effective operators such as $C_{ij}$
have never been properly investigated before, although they were briefly
discussed in Babu:1999me ; Giudice:2008uua ; Goudelis:2011un . We will study
these types of FV coupling in a future work.

## Acknowledgments

FA would like to that Stephen Martin for the valuable correspondence. The work
of FA is supported by the C.V. Raman fellowship from CHEP at IISc. S.K.V.
thanks SERB Grant CRG/2021/007170 ”Tiny Effects from Heavy New Physics” from
Department of Science and Technology, Government of India.

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