# Leveraging Interactions in Microfluidic Droplets for Enhanced Biotechnology
Screens

Carolus Vitalis  Tobias Wenzel

###### Abstract

Microfluidic droplet screens serve as an innovative platform for high-
throughput biotechnology, enabling significant advancements in discovery,
product optimization, and analysis. This review sheds light on the emerging
trend of interaction assays in microfluidic droplets, underscoring the unique
suitability of droplets for these applications. Encompassing a diverse range
of biological entities such as antibodies, enzymes, DNA, RNA, various
microbial and mammalian cell types, drugs, and other molecules, these assays
demonstrate their versatility and scope. Recent methodological breakthroughs
have escalated these screens to novel scales of bioanalysis and
biotechnological product design. Moreover, we highlight pioneering
advancements that extend droplet-based screens into new domains: cargo
delivery within human bodies, application of synthetic gene circuits in
natural environments, 3D-printing, and the development of droplet structures
responsive to environmental signals. The potential of this field is profound
and only set to increase.

###### keywords:

biotechnology , microfluidics , droplets , droplet screen , interactions ,
single cell , molecular interactions

###### PACS:

87.80.-y , 87.80.Rb , 87.14.-g , 47.80.-v

###### MSC:

76T30 , 78A70 , 68U10

††journal: XXX

[puc]organization=Institute for Biological and Medical Engineering, Schools of
Engineering, Medicine and Biological Sciences, Pontificia Universidad Católica
de Chile,addressline=Vicuña Mackenna 4860, city=Macul, postcode=7820244,
state=Santiago, country=Chile

##  1 The Power of Interaction Screens in Droplets

In the rapidly advancing field of biotechnology, interaction screens in
droplets are emerging as an increasingly versatile and powerful tool.
Microfluidic droplets stand out for their ability to facilitate strong and
rapid interactions [1, 2], and interaction assays are crucial for the
discovery, optimization, and analysis of various biotechnology application
fields, including metabolic engineering [3, 4, 5, 6], synthetic biology
genetic circuits [7, 8, 9, 10], the discovery of functional antibodies [11,
12, 13], enzymes [14, 15], and drugs such as antibiotics [16, 17, 18].

Droplet screens are an ultra-high throughput technology that enables
researchers to encapsulate interacting cells and reagents into thousands to
tens of millions of microfluidic droplets with a pL to low-nL volume and
efficiently analyze experimental results from each droplet, for example by
fluorescence-based droplet sorting [19, 20], DNA sequencing [11], cultivation
[18], or imaging [21]. Droplet-based screens, in general, offer several
advantages over more traditional methods, such as their ultra-high throughput,
reduced reagent and plastic consumable use, cost-effectiveness, and minimized
contamination risk [22], while improving the cultivability of microorganisms
[23, 22, 24], spheroids and organoids [25, 1].

In the context of automated screening methods, droplets combine the advantages
of the high-throughput and single-cell resolution of flow cytometry with the
versatility of liquid-handling robots to use non-cell-bound reagents and dyes
in assays [26]. Despite these advantages, the implementation of droplet
screens also requires development time investment, and should ideally be
chosen when more traditional screens are not well suited for the
biotechnologically desired process. There are, however, many notable droplet-
based biotechnology screens that harness the above advantages but are not
necessarily based on interactions, such as different types of single-cell
analysis [27, 28] including quantification [29], transcriptomics [30] and
genomics [31, 32, 33] e.g., to identify and characterize production strains,
and directed evolution in droplets [14], which is often applied to enzyme and
pathway optimization.

Interaction assays are a unique strength of droplets, because they can usually
not be performed with any other high-throughput method. Interactions in
biotechnology are often based on combinatorial testing of a limited number of
molecules and cells, which can achieve interactions in the confined space of
microfluidic droplets [2]. Many such interactions would not take effect
quickly enough in well-plates that contain microliters instead of picoliter
volumes for the same number of molecules or cells to interact, and wells are
also much more likely to contain contaminants in their larger volume. This
volume-based advantage for interactions in droplets is illustrated in Figure
1, based on a concrete example of metabolic interactions between auxotrophic
bacterial strains by Tan et al. [2].

Figure 1: Schematic representation of the advantage of droplets to facilitate
interactions between cells. (A) The compact volume of microfluidic droplets
enables rapid elevation of concentrations of molecules secreted by cells.
Simultaneously, it promotes the swift encounter of a few encapsulated
molecules with their target. This fosters cellular and molecular interactions
to take place rapidly and intensively, such as the survival of a dependent
auxotrophic cell, symbolized here by a yellow cell. (B) In contrast, larger
assay volumes in combinatorial screens dilute the limited cell numbers
available, leading to low concentration levels. This could hinder cells from
effectively communicating or exchanging metabolites within a viable time
frame. For instance, auxotrophic strains might not survive (depicted by gray
cells) in the population due to this delayed interaction.

This review paper is a succinct exploration of the recent literature,
primarily from the last two years, that highlights diverse applications and
the significant advancements in droplet interaction screens. The aim is not to
provide an exhaustive account, but this article type rather highlights recent
examples and trends that showcase the technique’s versatility and potential in
both microbial and mammalian cell screens in biotechnology. We will also
highlight recent methodology advances that expand droplet-based interaction
screens to new environments, including cargo delivery inside the body,
application of synthetic gene circuits in the natural environment,
3D-printing, and more. It is the authors’ belief that the interaction-focused
perspective of droplet screens will lead to the discovery of previously
unavailable screens that hold significant promise for scientific and
commercial innovation.

##  2 Recent Highlights in Droplet-Based Interaction Assays

Droplet-based interaction assays have been successfully applied to study
interactions between different cell types, proteins, DNA & RNA, drugs, and
other molecules, as illustrated in Figure 2. Literature examples can be found
for all 15 interaction categories shown in the figure’s central graph, and
some methods combine more than two types of interacting entities. While a wide
spectrum of examples exists, most interaction screen types are still in the
early stages of development, and we expect more research studies to be
published in the coming years on this topic to mature biotechnological
screening applications. Nevertheless, recent breakthroughs in droplet
microfluidic screens for interaction assays have enabled new scales of
bioanalysis and biotechnological product design. In the following paragraphs,
we highlight recent examples of different types of droplet-based interaction
screens and discuss their potential uses to demonstrate the power and
versatility of this emerging method trend and stimulate its application in
research and industry.

Figure 2: Schematic representation of different kinds of droplet-based
interaction screens. The Interactions of many biological entities can be
screened in droplets, including proteins such as antibodies and enzymes,
eukaryotic cells, microorganisms, drugs, and other molecules, and DNA/RNA. At
the center, binary combinatorial interaction possibilities are shown by
connecting edges in a graph, and screens can incorporate more than two
categories. Droplet schematics A-H highlight some examples of noteworthy
interaction screens: (A) Quantitative network analysis of interactions between
microorganisms and antibiotic drugs [34]; (B) Protein-protein interaction (co-
occurrence) screens on the surface of different cell types [35]; (C) Active
merging of droplets for interaction screens where all droplets have the same
content type combination of cells and beads [36]; (D) Co-localization screen
for functional antibodies that bind antigens to magnetic beads in droplets.
The functional screen is followed by transcriptomics analysis in barcoded
droplets to sequence the functional antibody coding genes [11]; (E)
Microbiota-algae co-cultivation screen in gel-microdroplets to select and
sequence synergistic helper strains that aid the growth of algae culture [37];
(F) Yeast production cell-biosensor interaction in droplets to quantitatively
indicate excreted product concentrations [38]; (G) DNA-protein interaction
screen to detect all human viruses in multiple samples at once in a highly
multiplexed diagnostic assay in a double-droplet trap array [39]; (H) Cell-
cell interactions to form spheroids and organoids in gel-microdroplets [25].

Microfluidic droplets serve as a powerful tool for the quantitative
exploration of complex interaction networks among cells and supplementary
substances like drugs across various concentrations. The microbial network
analysis shown in Figure 2 A, for example, demonstrates an analysis involving
three distinct, labeled auxotrophic bacterial cultures [34]. The potential for
applications of such screens in modular genetic engineering approaches [40] is
notable, as it offers the possibility to widen and enrich the biochemical
synthesis space through identifying productive co-cultures [41]. Scalable
analysis of microbial metabolic interactions will further benefit the field of
microbiome analysis and its biotechnology applications [42]. It is also worth
mentioning promising methods for microfluidic co-culture, which are not
performed in droplets [43].

Protein-protein interactions constitute another major area where a variety of
droplet methods have demonstrated their value, a topic thoroughly reviewed in
another work [44]. Figure 2 B illustrates a method where the co-occurrence of
cell-surface proteins across different cell types is profiled utilizing a pool
of labeled antibodies [35].

The encapsulation of interaction partners, like single cells or molecules,
into droplets typically involves controlling the input concentration of each
sample. This practice yields a stochastic encapsulation distribution in line
with the Poisson distribution. As a result, achieving the preferred
combination of different cells and molecules can be less efficient when
dealing with larger quantities of interacting partners. To mitigate this
challenge, several methods have been developed. One such strategy highlighted
in Figure 2 C involves actively selecting and merging encapsulated entities
into the desired combinations [36]. This approach has been employed in a
screen to detect cytokines on beads released during the interaction of two
different cell types.

Interaction screens offer the potential to encompass an array of interaction
types. Consider, for instance, the functional antibody screen depicted in
Figure 2 D. In this instance, droplets contain a cell expressing antibodies
which, in turn, interact with an antigen and labeled antibodies, thereby co-
locating fluorescence onto magnetic beads within the droplets. The screening
process is continued by recovering functional cells and re-encapsulating them
into droplets with reagents and DNA-barcoded gel-microdroplets that facilitate
the recovery of the transcribed genes of the antibody heavy and light chain
[11]. Beyond the bead-based assay, the authors also demonstrated a direct
screen on target cells, akin to an earlier antibody gene-target cell screen
conducted by a different research team [13].

Numerous types of interaction screens can be instrumental in enhancing the
production of biotechnological products, be it by metabolic engineering [4, 5]
or directed evolution [14]. A prime example is the potential for increased
yield in algal culture when cells are co-cultivated syntrophically with
certain microorganisms, as opposed to monocultures. Figure 2 E features a
screen that selects synergistic bacterial strains through the utilization of
the autofluorescence of proliferating Chlorella sp. colonies in gel-
microdroplets [37]. The study also indicated that co-cultures in droplets
could be incubated for a duration of up to 60 days. An analogous co-culture
strategy was employed to culture microorganisms in the laboratory that did not
grow in conventional cultures [45].

Where autofluorescence and fluorescent labels are not readily available,
synthetic biology biosensors can be incorporated into droplet screens to
quantify the productivity of target cells by interaction [9] for metabolic
engineering applications. The method outlined in Figure 2 F leverages droplets
for yeast metabolic engineering to quantify the product (p-coumaric acid)
excretion into the droplet with biosensor reporter cells. These biosensor
cells harbor an operon with a transcription factor that responds to the
product and encodes for a fluorescent protein signal [38]. In more recent
developments, a cellular biosensor was designed to screen for the secretion of
the industrial chemical (3-hydroxypropionic acid) in droplets [46], and
another recent biosensor platform evolved the yield (of erythromycin) from
genetically difficult to engineer actinomycetes production strains [47].

Droplets do not always need to be sorted in order to be characterized. As
demonstrated by the diagnostic assay illustrated in Figure 2 G, it is possible
to screen multiple clinical samples for all 169 human-associated viruses
(i.e., viruses with $>$10 published sequences) in a protein-DNA interaction
screen. Initially, all DNA-amplified patient sample-derived droplets and the
Cas13-based virus detection mix droplets were color-barcoded. Subsequently,
droplets were trapped pairwise side-by-side in a large double-droplet trap
array. Finally, the color barcode combinations were imaged before and after
the droplet merger, measuring the color barcodes and a fluorescent readout
where the target viral sequences were present [39].

Another notable application of the DNA-protein interaction screen type in
biotechnology deserves mentioning. This involves the interplay between genetic
fragments and the enzymes and metabolites of cell-free extracts, which serves
to express and evaluate synthetic biology constructs within droplets. This
method can significantly enhance the efficiency of engineering test cycles [8,
10].

Mammalian cell-cell interactions are crucial for the formation of
physiologically relevant tissue, a critical area for the enhancement of early-
stage clinical trials. Droplets are suitable for the generation of both
spheroids and organoids, frequently in gel-microdroplets, such as Matrigel,
agarose, alginate, gelatin, among others (refer to Figure 2 H), or within gel-
shell capsules [25].

##  3 Expanding Applications of Droplet-Based Interactions Screens

As we have seen, high-throughput interaction screens based on microfluidic
droplets can be performed with a versatile range of droplet components, such
as cells and molecules, but also magnetic beads, color or DNA barcodes, and
gels. Notably, advancements in gel-microdroplet materials have paved the way
for a broader application of droplets in cell cultures, multi-step procedures,
and streamlined molecular processing protocols. In this section, we highlight
several innovative material developments that promise to broaden the reach of
droplet-based interaction screens, catering to a wider audience and novel
environments, including interactions occurring between droplets and their
surrounding milieu.

For example, coated microgels (alginate hydrogels with an alginate and
polyacrylamide coating) have been used to physically contain genetically
modified microorganisms [48], allowing them to be incorporated into the
natural environment in a controlled manner, as illustrated in Figure 3 A. Even
after 72 h, no microorganism escape was detected, but nutrients and signal
molecules were successfully exchanged between the environment and adjacent
capsules. Such capsules provide the basis for syntrophic interaction screens
with the environment.

Figure 3: Illustration of emerging droplet-based interaction screens
facilitated by using new materials. (A) Biocontainment of microorganisms is
enabled by an alginate-polyacrylamide coating that allows controlled
incorporation of these organisms into the environment [48]. (B) Cell sorting
using a flow cytometer and double emulsion [49]. (C) Bioprinting of cells
encapsulated in shell-droplets that deliver more precise spatial cell-cell
interactions [50]. (D) Hair follicles delivered by microfluidic droplets of
gelatin methacryloyl and chitosan hydrogel, present a new method of targeted
delivery [51]. (E) DNA droplets contain a genetic circuit capable of sensing
the presence of microRNAs by disrupting the homogeneous distribution of DNA,
separating them into three distinct droplets [52]. (F) Wastewater treatment
using micromotors whose generation was assisted by microfluidic droplets and
which catalyze the degradation of organic waste [53].

A widely used advantage of gel-microdroplets and double emulsion droplets is
that they can be processed in an aqueous environment and therefore sorted in a
regular flow cytometer [49, 54]. Figure 3 B illustrates the use of double
emulsion for cell sorting in a commercial flow cytometer. Such machines are
available in a wider range of research laboratories and can therefore increase
access to droplet-based interaction screens. Unfortunately, the generation of
double emulsions and gel microdroplets is more challenging than that of
regular droplets.

Cells can be enclosed in shell droplets, which can be 3D-printed and cross-
linked into cell-containing structures [50], as shown in Figure 3 C. This
process opens new possibilities to place interaction partners in space for
interactions between different encapsulated organisms and their environments.
This technology may also enable the co-design of organs on a chip with a
spatially and compositionally engineered microbiome.

A key environment for the placement and release of drugs and organisms is the
human body. Indeed, droplets can be used for injection and targeted delivery
[55], such as injecting gel-microdroplets loaded with hair formation cells and
plasma with platelets into hairless tissue for regeneration [51]. The method
illustrated in Figure 3 D utilizes specialized hydrogels (gelatin methacryloyl
(GelMA) and chitosan) for targeted delivery. Delivery systems can be designed
in several layers and with multiple functions, such as featuring a self-
renewable hydration layer (based on liposomes) for delivery into joints [56].

Finally, the droplets can be designed to actively respond to or actuate their
environment. While these examples still tend to be in the proof-of-concept
stage, they allow us to think about new types of interaction screens. Figure 3
E shows an example of a responsive droplet used for a bio-based assay, a
single droplet made of a purpose-designed DNA gel with different DNA motifs
and linkers to bind them. The linkers feature a base complementarity for four
different target miRNAs. In the presence of target miRNAs, the linkers
hybridize to them, which unbinds the motif and compartmentalizes the droplet
into three single-motive droplets [52]. Another example of a responsive
microfluidic droplet is the stimulated drug-content ejection of a multiphase
Janus microparticle [57]. Droplet-based micromotors are not responsive, but
are constantly actuated [58], which can include catalytic interaction
properties used for organic waste degradation in wastewater treatment [53]
(Figure 3 F).

##  4 Conclusion

Droplet microfluidics has firmly established itself as an unrivaled high-
throughput approach for screening in the realm of biotechnological
applications. The significance of droplet-based screens is surging in parallel
with the growing arsenal of available methods and tools. A noteworthy trend
emerging in this field are interaction assays. Given their small volume,
droplets are perfectly suited for facilitating and quantifying interactions,
as demonstrated by numerous high-impact studies recently. We’ve brought
attention to various types of interaction methods, spanning protein-protein,
cell-cell, DNA-protein, and drug-microorganism interactions, amongst a host of
other studies blossoming in this rapidly expanding area.

Beyond droplet contents, the material composition, processing protocols, and
novel strategies for introducing droplets into the environment are being
explored, such as deploying them in more widely available instruments, within
the human body for organism and drug delivery, or even in natural settings.
These innovative methods unlock new avenues for interaction studies where the
droplet content, or the responsive droplet material itself, can interact with
the biological milieu. We anticipate this will catalyze the emergence of new
biotechnological applications.

Despite the many benefits of droplet microfluidic screens, several challenges
remain, including the need for experimental knowledge, specialized equipment,
and interdisciplinary communication skills [59]. Advanced droplet selection
set-ups are particularly promising but not widely available or easily
replicated [60]. Additionally, methods often need to be re-optimized and
benchmarked, which require time investment. It is therefore particularly
practical to focus method development efforts on screens that are unique for
droplets such as interaction assays, which do not simply compete with other
methods over a cost or efficiency margin.

As a final note, it is important to recognize the growing significance of
alternative measurement methodologies in single-cell analysis, such as mass
spectrometry, Raman spectroscopy, and impedance spectroscopy, among others.
Our review does not cover these methods, because these techniques have yet to
make their major impact within droplet-based interaction studies,
nevertheless, their potential to enhance this field is clear by offering
direct label free metabolomics, proteomics and electrochemical analysis of
droplet contents. We anticipate that these complementary tools will
increasingly intersect with droplet-based systems in the future, broadening
our scope of high-throughput screening capabilities.

As we move forward, the potential of droplets for interaction screens
continues to grow. This brief review hopes to stimulate further research and
industrial use of these promising techniques, and we look forward to the
discoveries and innovations that will undoubtedly continue to emerge from this
exciting field.

##  5 Acknowledgements

This work is part of a funded project granted to T.W. from ANID FONDECYT
Iniciación 11200666. The funder had no role in study design, data collection
and analysis, decision to publish, or preparation of the manuscript.

##  6 Credit Author Statement

Tobias Wenzel: Funding acquisition, Supervision, and Writing – review &
editing. Tobias Wenzel and Carolus Vitalis: Conceptualization, Data curation,
Visualization, Writing – original draft.

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Special Interest Works

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high-throughput screening platform employing microfluidics, next-generation
sequencing, and cell-free protein synthesis (CFPS). The study is a great
example of how to use cell-free extract in droplets for versatile screens,
especially in synthetic biology.

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Y. Wang, M. Liu, Y. Zhang, H. Liu, L. Han, Recent methods of droplet
microfluidics and their applications in spheroids and organoids, Lab on a chip
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Extensive and current overview of droplet microfluidic methods to form and
trap different spheroids and organoids and a primer to their subsequent
analysis. The article is helpful for researchers wanting to start in droplet-
based organoids, as well as to readers wanting to advance using recent
methods.

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A. Pryszlak, T. Wenzel, K. W. Seitz, F. Hildebrand, E. Kartal, M. R. Cosenza,
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interaction of microbial DNA with molecular probes to select and enrich target
species in order to improve DNA sequencing quality. The study is an example of
recent method innovation in precision genomics, and helps make the microbiome
more experimentally accessible to biotechnology applications.

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J. Y. Tan, T. E. Saleski, X. N. Lin, The effect of droplet size on syntrophic
dynamics in droplet-enabled microbial co-cultivation, PLoS ONE 17 (2022).
Systematic study demonstrating the effect of droplet size on metabolomic
interactions between cells co-cultured in microfluidic droplets. It clearly
and visually demonstrates the utility that especially small droplets have for
effective and fast interaction-based screens.

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J. L. Madrigal, N. G. Schoepp, L. Xu, C. S. Powell, C. L. Delley, C. A.
Siltanen, J. Danao, M. Srinivasan, R. H. Cole, A. R. Abate, Characterizing
cell interactions at scale with made-to-order droplet ensembles (modes),
Proceedings of the National Academy of Sciences of the United States of
America 119 (2022).
This work shows a method that allows a deliberate distribution of cells and
molecules inside the microfluidic droplets of a screen. For this purpose, they
combined two types of cells and beads (1:1:2) in a deterministic manner using
a new combination of active droplet sorting and merging.

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T.-C. Tang, E. Tham, X. Liu, K. Yehl, A. J. Rovner, H. Yuk, C. de la Fuente-
Nunez, F. J. Isaacs, X. Zhao, T. K. Lu, Hydrogel-based biocontainment of
bacteria for continuous sensing and computation, Nature Chemical Biology 17
(2021) 724 – 731.
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contained cells could still perform their functions and interact with the
environment but not physically leave the shell confinement. The method opens
the possibility of incorporating synthetic genetic circuits in real-world
scenarios.

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Y. Ou, S. Cao, Y. Zhang, H. Zhu, C. Guo, W. Yan, F. Xin, W. Dong, Y. Zhang, M.
Narita, Z. Yu, T. P. J. Knowles, Bioprinting microporous functional living
materials from protein-based core-shell microgels, Nature Communications 14
(2023).
Very recent article that shows how to combine cell encapsulation in
microfluidic core-shell droplets with 3D printing. The study describes a
method to cross-link the extruded droplet emulsion and demonstrates its
application.

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Y. Zhang, P. Yin, J. Huang, L. Yang, Z. Liu, D. Fu, Z. Hu, W. Huang, Y. Miao,
Scalable and high-throughput production of an injectable platelet-rich plasma
(prp)/cell-laden microcarrier/hydrogel composite system for hair follicle
tissue engineering, Journal of Nanobiotechnology 20 (2022).
Study presenting a new strategy for tissue engineering. It describes a new
method of hair follicle delivery using microfluidic droplets, allowing the
delivery of dermal papilla cells with platelet-rich plasma.


#  LFTC-22-1/69 $\Upsilon$ and $\eta_{b}$ nuclear bound states

J. J. Cobos-Martínez  Departamento de Física, Universidad de Sonora, Boulevard
Luis Encinas J. y Rosales, Colonia Centro, Hermosillo, Sonora 83000, México
G. N. Zeminiani  K. Tsushima  Laboratório de Física Teórica e Computacional,
Universidade Cidade de São Paulo (UNICID), 01506-000, São Paulo, SP, Brazil

###### Abstract

$\Upsilon$ and $\eta_{b}$ nuclear bound state energies are calculated for
various nuclei neglecting any possible effects of the widths. Essential input
for the calculations, namely the medium-modified $B$ and $B^{*}$ meson masses,
as well as the density distributions in nuclei, are calculated within the
quark-meson coupling (QMC) model. The attractive potentials for the $\Upsilon$
and $\eta_{b}$ mesons in nuclei are calculated from the mass shifts of these
mesons in nuclear matter in the local density approximation. These potentials
originate from the in-medium enhanced $B\overline{B}$ and $BB^{*}$ loops in
their respective self energy. After an extensive analysis we conclude that our
results suggest that the $\Upsilon$ and $\eta_{b}$ mesons should form bound
states with all the nuclei considered.

##  I Introduction

Quantum chromodynamics (QCD) is the accepted theory of the strong interactions
at the fundamental level. However, a quantitative understanding of the strong
force and strongly interacting matter from the underlying theory is still
limited. The study of the interactions between heavy quarkonia and atomic
nuclei is an important tool to gain an understanding of the strongly
interacting matter properties in vacuum and extreme conditions of temperature
and density based on QCD.

Since heavy quarkonium and nucleons do not share light ($u$, $d$) quarks (the
OZI rule suppresses the interactions mediated by the exchange of mesons made
of only light quarks), heavy quarkonium interacts with nucleon primarily via
gluons, and therefore the production of heavy quarkonium in a nuclear medium
can be of great relevance to explore the role played by gluons. If such states
are indeed found experimentally to be bound to nuclei, it is therefore
important to search for other sources of attraction which could lead to the
binding of heavy quarkonium to nuclei. The binding of heavy quarkonium to
nuclei may give an evidence that the masses of these heavy mesons decrease in
a nuclear medium.

Since the early work of Brodsky Brodsky:1989jd  that charmonium states may be
bound to nuclei, a large amount of research, looking for alternatives to the
light meson exchange mechanism, has accumulated over the years to investigate
the possible existence of such exotic states Hosaka:2016ypm ; Krein:2016fqh ;
Metag:2017yuh ; Krein:2017usp ; Ko:2000jx ; Krein:2010vp ; Tsushima:2011kh ;
Tsushima:2011fg ; Krein:2013rha ; Klingl:1998sr ; Hayashigaki:1998ey ;
Kumar:2010hs ; Belyaev:2006vn ; Yokota:2013sfa ; Peskin:1979va ;
Kharzeev:1995ij ; Kaidalov:1992hd ; Luke:1992tm ; deTeramond:1997ny ;
Brodsky:1997gh ; Sibirtsev:2005ex ; Voloshin:2007dx ; TarrusCastella:2018php .
In addition to these, lattice QCD simulations for charmonium-nucleon
interaction in free space were performed in the last decade Yokokawa:2006td ;
Liu:2008rza ; Kawanai:2010ev ; Kawanai:2010ru ; Skerbis:2018lew . Furthermore,
more recently, studies for the binding of charmonia with nuclear matter and
finite nuclei, as well as light mesons and baryons, were performed in lattice
QCD simulations Beane:2014sda ; Alberti:2016dru , albeit with unphysically
heavy pion masses.

On the experimental side, the 12 GeV upgrade at the Jefferson Lab has made it
possible to produce low-momentum heavy-quarkonia in an atomic nucleus.
Recently Ali:2019lzf , a photon beam was used to produce a $J/\Psi$ meson
near-threshold, which was identified by the decay into an electron-positron
pair. Furthermore, with the construction of the FAIR facility in Germany,
heavy and heavy-light mesons will be produced copiously by the annihilation of
antiprotons on nuclei Durante:2019hzd . Experimental studies on $\eta_{c}$
production in heavy ion collisions at the LHC were performed in Refs.
Aaij:2019gsn ; Tichouk:2020dut ; Tichouk:2020zhh ; Goncalves:2018yxc ;
Klein:2018ypk . However, nearly no experiments have yet been aimed to produce
the $\eta_{c}$ at lower energies and its binding to nuclei, perhaps hinting at
the difficulty to produce and detect such states. In the case of bottomonium,
studies were made for $\Upsilon$ photoproduction at the Electron-Ion Collider
Xu:2020uaa ; Gryniuk:2020mlh , $\Upsilon$ production in $p$Pb collisions
Aaij:2018scz , and $\Upsilon(nl)$ (excited state) decay into
$B^{(*)}\bar{B}^{(*)}$ Liang:2019geg . With studies like these on heavy
quarkonium and future planned ones, we will improve our understanding of the
strong force and strongly interacting matter.

Returning to the phenonenological studies, the interactions frequently
considered between the heavy quarkonium and the nuclear medium are the so
called QCD van der Waals (multigluon exchange) interactions Kaidalov:1992hd ;
Luke:1992tm ; deTeramond:1997ny ; Brodsky:1997gh ; Ko:2000jx ;
Sibirtsev:2005ex ; Voloshin:2007dx ; TarrusCastella:2018php . One might think
that this must be the case, since heavy quarkonium has no light quarks,
whereas the nuclear medium is composed of light quarks, and thus the exchange
of mesons composed of light quarks do not occur at the lowest order.

However, another possible mechanism, which we consider in this paper, for the
heavy quarkonium to interact with the nuclear medium is through the excitation
of the intermediate state hadrons which do contain light quarks ($B$ and
$B^{*}$ in this work). There is a great amount of evidence that the internal
structure of hadrons changes in medium and this must be taken into account
when addressing, for example, charmonium in nuclei. For instance, Refs.
Krein:2010vp ; Tsushima:2011kh  have shown that the effect of the nuclear mean
fields on the intermediate $DD$ state is crucial when considering the $J/\Psi$
interactions with atomic nuclei. The modifications driven by the strong
nuclear mean fields on the $D$ mesons’ light-quark component enhanced the
self-energy such that it provides attraction to the $J/\Psi$. Furthermore,
only recently the in-medium properties of $\eta_{c}$ meson were renewed
theoretically Cobos-Martinez:2020ynh  using this mechanism.

In a recent paper Zeminiani:2020aho , we estimated the mass shifts of the
$\Upsilon$ and $\eta_{b}$ mesons by considering the excitations of
intermediate state hadrons with light quarks in their self-energy. The
estimates were made using an SU(5) effective Lagrangian density which contains
both the $\Upsilon$ and $\eta_{b}$ mesons with one universal coupling
constant, and an anomalous coupling that respects SU(5) symmetry in the
coupling constant. After expansion of the SU(5) effective Lagrangian with
minimal substitutions, we obtained the interaction Lagrangians for calculating
the the $BB$, and $B^{*}B^{*}$ meson loops contributions to the $\Upsilon$ and
$\eta_{b}$ self energies. As an example we show in Fig. 1 the $BB$ meson loop
contribution for the $\Upsilon$ self-energy.

Figure 1: BB meson loop contribution for the $\Upsilon$ self-energy.

For the study of the heavy quarkonium ($\Upsilon$ and $\eta_{b}$) interaction
with the nuclear medium through the excitation of the intermediate state
hadrons we need to have knowledge on the in-medium properties of the $B$ and
$B^{*}$ mesons, in particular their medium-modified masses. For this we used
the quark-meson coupling (QMC) model Guichon:1987jp , which has been
successfully applied for various studies in nuclear matter and nuclei
Krein:2017usp ; Krein:2010vp ; Tsushima:2002cc ; Tsushima:1997df ;
Guichon:1995ue ; Saito:1996sf ; Tsushima:2019wmq ; Saito:2005rv ;
Guichon:1989tx ; Stone:2016qmi .

In Ref. Zeminiani:2020aho  we did an study of the $BB,BB^{*}$ and $B^{*}B^{*}$
meson loop contributions to the $\Upsilon$ self-energy in nuclear matter
neglecting for the moment any possible imaginary part. After a detailed
analysis, our predictions for the $\Upsilon$ and $\eta_{b}$ mass shifts were
given by including only the lowest order $BB$ meson loop contribution for the
$\Upsilon$, and only the $BB^{*}$ meson loop contribution for the $\eta_{b}$,
where the in-medium masses of the $B$ and $B^{*}$ mesons were calculated by
the QMC model. We note that the in-medium $B^{*}$ meson mass was calculated
for the first time in Ref. Zeminiani:2020aho . In this work, we apply the
mechanism described above, by first extending our results in nuclear matter to
finite nuclei, and then considering the interactions between the $\Upsilon$
and $\eta_{b}$ mesons and a wide mass range of atomic nuclei.

This article is organized as follows. In Sec. II we summarize the
computational procedure used and discuss our results for the mass shifts of
the $\Upsilon$ and $\eta_{b}$ mesons in nuclear matter. These results indicate
that nuclear medium provides attraction to these mesons and therefore in Sec.
III we consider the nuclear bound states for the $\Upsilon$ and $\eta_{b}$
mesons when these mesons are produced nearly at rest inside a nucleus.
Finally, in Sec. IV we give a summary and conclusions.

##  II  Mass shifts in nuclear matter

In this section we summarize the results obtained for the mass shifts of the
$\Upsilon$ and $\eta_{b}$ mesons in nuclear matter. The details of this
analysis can be found in Ref. Zeminiani:2020aho .

###  II.1 $\Upsilon$ mass shift

The $\Upsilon$ mass shift in nuclear matter originates from the modifications
of the $BB$, $BB^{*}$, and $B^{*}B^{*}$ meson loops contributions to the
$\Upsilon$ self-energy, relative to those in free space; see, for example,
Fig. 1. The self-energy is calculated using effective $SU(5)$-flavor symmetric
Lagrangians at the hadronic level Zeminiani:2020aho ; Lin:2000ke  for the
interaction vertices $\Upsilon BB$, $\Upsilon B^{*}B^{*}$, and $\Upsilon
BB^{*}$ neglecting any possible imaginary part. In Ref. Zeminiani:2020aho  we
have made an extensive analysis of these contributions to the $\Upsilon$ self-
energy and have found that, for example, the $B^{*}B^{*}$ loop gives an
unexpectedly large contribution. For this reason, and to be consistent with
the $\eta_{b}$ case studied below, we have decided to be conservative and
consider only the $BB$ loop contribution to the $\Upsilon$ self-energy,
leaving for the future a full study. The interaction Lagrangian for the
$\Upsilon BB$ vertex is given by

${\cal L}_{\Upsilon BB}=ig_{\Upsilon
BB}\Upsilon^{\mu}\left[\overline{B}\partial_{\mu}B-\left(\partial_{\mu}\overline{B}\right)B\right],$
(1)

where the following convention is adopted for the isospin doublets of the $B$
mesons

$\displaystyle B$ $\displaystyle=\begin{pmatrix}B^{+}\\\ B^{0}\end{pmatrix},$
$\displaystyle\overline{B}=\begin{pmatrix}B^{-}&\overline{B^{0}}\end{pmatrix}.$

The coupling constant $g_{\Upsilon BB}$ for the vertex $\Upsilon BB$ is
calculated from the experimental data for $\Gamma(\Upsilon\to e^{+}e^{-})$
using the vector meson dominance model. This gives $g_{\Upsilon BB}=13.2$; see
Refs. Zeminiani:2020aho ; Lin:2000ke  and references therein for details. A
similar approach was taken in Refs. Lin:1999ad ; Krein:2010vp  to determine
the coupling constant $g_{J/\Psi DD}=7.64$ for the vertex $J/\Psi BB$.

Including only the $BB$ loop, Eq. (1), the $\Upsilon$ self-energy
$\Sigma_{\Upsilon}$ is given by

$\Sigma_{\Upsilon}(k^{2})=-\frac{g_{\Upsilon
BB}^{2}}{3\pi^{2}}\int_{0}^{\infty}\mathrm{d}q\,\textbf{q}^{2}\,I(\textbf{q}^{2})$
(2)

for an $\Upsilon$ at rest, where

$I(\textbf{q}^{2})=\frac{1}{\omega_{B}}\left(\frac{\textbf{q}^{2}}{\omega_{B}-m_{\Upsilon}^{2}/4}\right)$
(3)

and $\omega_{B}=\left(\textbf{q}^{2}+m^{2}_{B}\right)^{1/2}$. The integral in
Eq. (2) is divergent and therefore needs to be regularized. To do this, we
introduce into the integrand of Eq. (2) a phenomenological vertex form factor

$u_{B}(\textbf{q}^{2})=\left(\frac{\Lambda^{2}_{B}+m^{2}_{\Upsilon}}{\Lambda^{2}_{B}+4\omega^{2}_{B}\left(\textbf{q}^{2}\right)}\right)^{2},$
(4)

with cutoff parameter $\Lambda_{B}$ Krein:2010vp ; Tsushima:2011kh ;
Tsushima:2011fg ; Krein:2013rha ; Cobos-Martinez:2017vtr ; Cobos-
Martinez:2017woo ; Cobos-Martinez:2017onm ; Cobos-Martinez:2017fch ; Cobos-
Martinez:2019kln , for each $\Upsilon BB$ vertex. In a later section we will
discuss the non-neglegible role played by this form factor and the cutoff
parameter $\Lambda_{B}$. For the moment, we point out that form factors are
necessary to take into account the finite size of the mesons participating in
the vertices, while the cutoff $\Lambda_{B}$, which is an unknown input to our
calculation, may be associated with energies needed to probe the internal
structure of the mesons; see Ref. Zeminiani:2020aho  for a more extensive
discussion. Thus, in order to reasonably include these effects, and to
quantify the sensitivity of our results to its value, we vary $\Lambda_{B}$
over the interval 2000-6000 MeV.

Figure 2: $B$ and $B^{*}$ meson effective Lorentz-scalar masses in symmetric
nuclear matter versus baryon density. Figure 3: $\Upsilon$ mass shift in
nuclear matter as a function of the nuclear matter densidy $\rho_{B}$

The $\Upsilon$ mass shift in nuclear matter $\Delta m_{\Upsilon}$ is computed
from the difference between its in-medium mass, $m_{\Upsilon}^{*}$, and its
value in vacuum, $m_{\Upsilon}$, namely

$\Delta m_{\Upsilon}=m_{\Upsilon}^{*}-m_{\Upsilon},$ (5)

where these masses are computed self-consistently from

$m_{\Upsilon}^{2}=(m_{\Upsilon}^{0})^{2}+\Sigma_{\Upsilon}(k^{2}=m_{\Upsilon}^{2}),$
(6)

where $m_{\Upsilon}^{0}$ is the bare $\Upsilon$ mass and the $\Upsilon$ self-
energy $\Sigma_{\Upsilon}(k^{2})$ is given in Eq. (2). The
$\Lambda_{B}$-dependent $\Upsilon$ bare mass, $m_{\Upsilon}^{0}$, is fixed
such that we reproduce the physical $\Upsilon$ mass, namely
$m_{\Upsilon}=9640$ MeV.

The in-medium $\Upsilon$ mass is obtained by solving Eq. (6) with the self-
energy calculated with the medium-modified $B$ mass, which was calculated in
Ref. Zeminiani:2020aho , together with that for the $B^{*}$ meson, using the
quark-meson coupling model (QMC) as a function of the nuclear matter density
$\rho_{B}$. The results obtained in this way are presented in Fig. 2 and show
that the QMC model gives a similar downward mass shift for the $B$ and $B^{*}$
in symmetric nuclear matter. For example, at the saturation density
$\rho_{0}=0.15\,\text{fm}^{-3}$, the mass shifts for the $B$ and $B^{*}$
mesons are respectively, $(m^{*}_{B}-m_{B})=-61$ MeV and
$(m^{*}_{B^{*}}-m_{B^{*}})=-61$ MeV, where the difference in their mass shift
values appears in the next digit. The values for the masses in vacuum for the
$B$ and $B^{*}$ mesons used are $m_{B}=5279$ MeV and $m_{B^{*}}=5325$ MeV,
respectively.

The nuclear density dependence of the $\Upsilon$ mass is driven by the
intermediate $BB$ state interactions with the nuclear medium, where the
effective scalar and vector meson mean fields couple to the light $u$ and $d$
quarks in the bottom mesons. In Fig. 3 we show the results for the $\Upsilon$
mass shift as a function of the nuclear matter density $\rho_{B}$, for five
values of the cutoff parameter $\Lambda_{B}$. As can be seen from Figs. 2 and
3, a decreasing $B$ meson mass in-medium induces a negative mass shift for the
$\Upsilon$. This happens because a decrease of the $B$ meson mass enhances the
$BB$ meson loop contribution in nuclear matter relative to that in vacuum.
Expectedly, the mass shift of the $\Upsilon$ is dependent on the value of the
cutoff mass $\Lambda_{B}$ used, being larger for larger $\Lambda_{B}$; see
Ref. Zeminiani:2020aho  for further details. For example, for the values of
the cutoff shown in Fig. 3, the $\Upsilon$ mass shift varies from -16 to -22
MeV, at $\rho_{B}=\rho_{0}$.

###  II.2 $\eta_{b}$ mass shift

Figure 4: $\eta_{b}$ mass shift in nuclear matter as a function of the nuclear
matter densidy $\rho_{B}$.

For the calculation of the $\eta_{b}$ mass shift in nuclear matter, we proceed
similarly to the $\Upsilon$ case and take into account only the $BB^{*}$ loop
contribution to the $\eta_{b}$ self-energy. In Ref. Zeminiani:2020aho  we have
also studied the inclusion of the $\eta_{b}B^{*}B^{*}$ interaction in the
$\eta_{b}$ self-energy and found that its contribution to the mass shift is
essentially negligible. Thus, in order to be consistent with the $\Upsilon$
case above, i.e. in both cases we consider only the minimal contribution, here
we only give results for the $BB^{*}$ loop in the $\eta_{b}$ self-energy. The
effective Lagrangian for the $\eta_{b}BB^{*}$ interaction is

$\displaystyle{\cal L}_{\eta_{b}BB^{*}}$ $\displaystyle=$ $\displaystyle
ig_{\eta_{b}BB^{*}}\left[(\partial^{\mu}\eta_{b})\left(\overline{B}^{*}_{\mu}B-\overline{B}B^{*}_{\mu}\right)\right.$
(7) $\displaystyle-$
$\displaystyle\left.\eta_{b}\left(\overline{B}^{*}_{\mu}(\partial^{\mu}B)-(\partial^{\mu}\overline{B})B^{*}_{\mu}\right)\right],$

where $g_{\eta_{b}BB^{*}}$ is the coupling constant for the $\eta_{b}BB^{*}$
vertex. We will use its value in the SU(5) scheme Zeminiani:2020aho , namely

$g_{\eta_{b}BB^{*}}=g_{\Upsilon BB}=g_{\Upsilon
B^{*}B^{*}}=\frac{5g}{4\sqrt{10}}.$ (8)

Using Eq. (7), the $\eta_{b}$ self-energy for an $\eta_{b}$ at rest is given
by Cobos-Martinez:2020ynh

$\Sigma_{\eta_{b}}=\frac{8g_{\eta_{b}BB^{*}}^{2}}{\pi^{2}}\int_{0}^{\infty}\mathrm{d}q\,\textbf{q}^{2}K(\textbf{q}^{2}),$
(9)

where

$\displaystyle K(\textbf{q}^{2})$ $\displaystyle=$
$\displaystyle\left.\frac{m_{\eta_{b}}^{2}(-1+q_{0}^{2}/m_{B^{*}}^{2})}{(q_{0}^{2}-\omega_{B^{*}}^{2})(q_{0}-m_{\eta_{b}}-\omega_{B})}\right|_{q_{0}=m_{\eta_{b}}-\omega_{B}}$
$\displaystyle+$
$\displaystyle\left.\frac{m_{\eta_{b}}^{2}(-1+q_{0}^{2}/m_{B^{*}}^{2})}{(q_{0}-\omega_{B^{*}})\left((q_{0}-m_{\eta_{b}})^{2}-\omega_{B}^{2}\right)}\right|_{q_{0}=-\omega_{B^{*}}},$

and $\omega_{B^{*}}=(\textbf{q}^{2}+m_{B^{*}}^{2})^{1/2}$.

The mass of the $\eta_{b}$ meson, in vacuum and in nuclear matter, is computed
similarly to the $\Upsilon$ case. First, we introduce form factors, as in Eq.
(4), into each $\eta_{b}BB^{*}$ vertex, with $\Lambda_{B}=\Lambda_{B^{*}}$, in
order to regularize the divergent integral in the self-energy, Eq. (9).
Second, we fix the value of the $\eta_{b}$ bare mass using the physical
(vacuum) mass of the $\eta_{b}$, namely $m_{\eta_{b}}=9399$ MeV, using Eq. (6)
appropriately written for the $\eta_{b}$ case. Then, for the calculation of
the $\eta_{b}$ mass shift in nuclear matter, the self-energy
$\Sigma_{\eta_{b}}$ is computed using the medium-modified $B$ and $B^{*}$
masses calculated in the QMC model and shown in Fig. 2. The results for the
$\eta_{b}$ mass shift in nuclear matter are shown in Fig. 4 as a function of
the nuclear matter density $\rho_{B}$. Note that we use the same range of
values for the cutoff mass $\Lambda_{B}$ as for the $\Upsilon$. As can be seen
from Fig. 4, the mass of the $\eta_{b}$ is shifted downwards in nuclear matter
for all values of the cutoff $\Lambda_{B}$, similarly to the $\Upsilon$. For
example, at the normal density of nuclear matter $\rho_{0}$, the mass shift
varies from -75 MeV to -82 MeV when the cutoff varies from $\Lambda_{B}=2000$
MeV to $\Lambda_{B}=6000$ MeV. Similarly to the $\Upsilon$ mass shift, the
dependence of the $\eta_{b}$ mass shift on the values of the cutoff is small,
for example, just -7 MeV when the cutoff is increased by a factor of 3 at
$\rho_{B}=\rho_{0}$.

###  II.3  Discussion of the $\Upsilon$ and $\eta_{b}$ mass shifts results

Surprisingly, the mass shift for the $\eta_{b}$ is larger than that of the
$\Upsilon$ for the same range of cutoff values explored; see Figs. 3 and 4. A
similar difference in mass shifts was observed for the $J/\Psi$ and $\eta_{c}$
mesons in Refs. Krein:2010vp ; Cobos-Martinez:2020ynh , using the
corresponding Lagrangians in the $SU(4)$ flavor sector. As demonstrated in
Refs. Zeminiani:2020aho ; Cobos-Martinez:2020ynh , these differences in the
mass shifts for the $\eta_{b}$ and $\Upsilon$ are probably due to the
following reasons: (a) a badly broken $SU(5)$ symmetry such that the couplings
$g_{\eta_{b}BB^{*}}$ and $g_{\Upsilon BB}$ are very different, and not equal
as assumed here, see Eq. (8). Indeed, as shown in Ref. Cobos-Martinez:2020ynh
, the mass shift for the $\eta_{c}$ gets closer to that of the $J/\Psi$ when
$SU(4)$ flavor symmetry is broken, such that
$g_{\eta_{c}DD^{*}}=(0.6/\sqrt{2})\,g_{J/\Psi DD}\simeq 0.424\,g_{J/\Psi DD}$
Cobos-Martinez:2020ynh ; Lucha:2015dda . $SU(5)$ flavor symmetry, like
$SU(4)$, is also broken in nature, as attested by the difference is masses of
the $\Upsilon$ and $\eta_{b}$ mesons. However, since we do not have an
empirical value for $g_{\eta_{b}BB^{*}}$, which we can use to compute the
$\eta_{b}$ self-energy, we therefore resort to $SU(5)$ symmetry and use the
value for $g_{\eta_{b}BB^{*}}$ given in Eq. (8); (b) the form factors are not
equal for the vertices $\Upsilon BB$ and $\eta_{b}BB^{*}$ and we have to
readjust the cutoff values, which means $\Lambda_{B}\neq\Lambda_{B^{*}}$ and
the comparisons for the mass shifts have to be made for different values of
the cutoffs. This is also a reason why we explore a range of values for
$\Lambda_{B}$; and (c) at the $g_{\eta_{b}BB^{*}}^{2}$ order, the number of
possible contractions to give the $BB^{*}$ loop in the $\eta_{b}$ self-energy
is $4\times 4=16$ for the isodoublet $B$ and $B^{*}$ fields, and this number
is larger than that of $2\times 2=4$ to give the $BB$ loop in the $\Upsilon$
self-energy at the $g_{\Upsilon BB}^{2}$ order (see Eqs. (1) and (7)). This
may give the larger contribution for the $\eta_{b}$ potential.

The results for the mass shifts in nuclear matter, shown in Figs. 3 and 4, for
the $\Upsilon$ and $\eta_{b}$, respectively, support the argument that the
nuclear medium provides attraction to these mesons and open the possibility to
study the binding of theses mesons to nuclei since the mass shifts, for both
the $\Upsilon$ and $\eta_{b}$, at around $\rho_{B}=\rho_{0}$, are significant.
We will see in the next section that this is indeed the case and allows for
the formation of nuclear bound states for both the $\Upsilon$ and $\eta_{b}$,
and furthermore we calculate the correspoding binding energies for several
nuclei.

##  III  Nuclear bound states

The results for the mass shifts of the $\Upsilon$ and $\eta_{b}$ in nuclear
matter clearly indicate that nuclear medium provides attraction to these
mesons. Therefore, we now consider the nuclear bound states of the $\Upsilon$
and $\eta_{b}$ mesons when these mesons have been produced nearly at rest
inside nucleus $A$ and study the following nuclei in a wide range of masses,
namely 4He, 12C, 16O, 40Ca, 48Ca, 90Zr, 197Au, and 208Pb.

In the local density approximation, the bottomonium $h$
($h=\Upsilon\,,\eta_{b}$) potential within nucleus $A$ is given by

$V_{hA}(r)=\Delta m_{h}(\rho_{B}^{A}(r)),$ (11)

where $r$ is the distance from the center of the nucleus and $\Delta m_{h}$ is
the mass shift computed in Sec II for $h=\Upsilon\,,\eta_{b}$. The nuclear
density distributions $\rho_{B}^{A}(r)$ for the nuclei listed above are
calculated using the QMC model Saito:1996sf , except for 4He, which we obtain
from Ref. Saito:1997ae .

|
---|---
|
|
|
Figure 5:  $\Upsilon$-nucleus potentials for various nuclei and values of the cutoff parameter $\Lambda_{B}$. |
---|---
|
|
|
Figure 6:  $\eta_{b}$-nucleus potentials for various nuclei and values of the
cutoff parameter $\Lambda_{B}$.

In Figs. 5 and 6 we present the bottomonium $h$-nucleus potentials for the
eigth nuclei mentioned above and the sames values of the cutoff parameter
$\Lambda_{B}$ that were used in the computation of the mass shifts in Sec II.
We can see from Figs. 5 and 6 that the $V_{hA}$ potentials, for $h=\Upsilon$
and $\eta_{b}$, respectively, are attractive for all nuclei and all values of
the cutoff mass parameter. However, for each nuclei, the depth of the
potential depends on the value of the cutoff parameter, being more attractive
the larger $\Lambda_{B}$ is. This dependence is expected and is, indeed, an
uncertainty in the results obtained in our approach.

We now compute the bottomonium $h$-nucleus bound state energies for the
potentials shown in Figs. 5 and 6 by solving the Klein-Gordon equation for
these potentials. In order to apply the Klein-Gordon equation to obtain the
$\Upsilon$-nucleus single particle energies, since the $\Upsilon$ is a spin-1
particle, we make an approximation where the transverse and longitudinal
components in the Proca equation are expected to be very similar for an
$\Upsilon$ at rest, hence it is reduced to one component, which corresponds to
the Klein-Gordon equation.

We treat the bottomonium $h$-nucleus potential as a scalar and add it to the
mass term in the Klein-Gordon equation

$\left(-\nabla^{2}+\left(m+V_{hA}(\vec{r})\right)^{2}\right)\phi_{h}(\vec{r})=\mathcal{E}^{2}\phi_{h}(\vec{r}),$
(12)

where $h=\Upsilon\,,\eta_{b}$, $m=m_{h}m_{A}/(m_{h}+m_{A})$ is the reduced
mass of the bottomonium $h$-nucleus system with $m_{h}$ $(m_{A}$) the mass of
bottomonium $h$ (nucleus $A$) in vacuum, and $V_{hA}(\vec{r})$ is the
bottomonium $h$-nucleus potential given in Eq. (11) and shown in Figs. 5 and
6.

We note that in previous works we have approximated Eq. (12) by ignoring the
$V_{hA}^{2}$ term. We now solve the full Klein-Gordon equation Eq. (12) using
momentum space methods. Here, the Klein-Gordon equation is first converted to
a momentum space representation via a Fourier transform, followed by a partial
wave decomposition. For a given value of angular momentum $l$, the eigenvalues
$\mathcal{E}_{nl}$ of the resulting equation are found by the inverse
iteration eigenvalue algorithm.

The bound state energies ($E$) of the bottomonium $h$-nucleus system, given by
$E_{nl}=\mathcal{E}_{nl}-m$, are listed in Tables 1 and 2. In Table 1 we show
the $\Upsilon$-nucleus bound state energies for all nuclei listed at the
beginning of this section and the same range of values for the cutoff mass
parameter as used in the mass shift calculation. For each nucleus we have
listed only a few bound states, since the number of bound states increases
with the mass of the nucleus and for the heaviest of these the number of bound
states quiet large. For example, for the heaviest nucleus we have $\sim$ 70
states.

Table 1:  ${}^{4}_{\Upsilon}\text{A}$ bound state energies for several nuclei
$A$. All dimensioned quantities are in MeV.

|  | Bound state energies
---|---|---
| $n\ell$ | $\Lambda_{B}=2000$ | $\Lambda_{B}=3000$ | $\Lambda_{B}=4000$ | $\Lambda_{B}=5000$ | $\Lambda_{B}=6000$
${}^{4}_{\Upsilon}\text{He}$ | 1s | -5.6 | -6.4 | -7.5 | -9.0 | -10.8
${}^{12}_{\Upsilon}\text{C}$ | 1s | -10.6 | -11.6 | -12.8 | -14.4 | -16.3
| 1p | -6.1 | -6.8 | -7.9 | -9.3 | -10.9
| 1d | -1.5 | -2.1 | -2.9 | -4.0 | -5.4
| 2s | -1.6 | -2.1 | -2.8 | -3.8 | -5.1
${}^{16}_{\Upsilon}\text{O}$ | 1s | -11.9 | -12.9 | -14.2 | -15.8 | -17.8
| 1p | -8.3 | -9.2 | -10.4 | -11.9 | -13.7
| 1d | -4.4 | -5.1 | -6.2 | -7.5 | -9.2
| 2s | -3.7 | -4.4 | -5.4 | -6.7 | -8.3
| 1f | n | -0.9 | -1.8 | -2.9 | -4.3
${}^{40}_{\Upsilon}\text{Ca}$ | 1s | -15.5 | -16.6 | -18.2 | -20.0 | -22.3
| 1p | -13.3 | -14.4 | -15.9 | -17.7 | -19.8
| 1d | -10.8 | -11.9 | -13.3 | -15.0 | -17.1
| 2s | -10.3 | -11.3 | -12.7 | -14.4 | -16.4
| 1f | -8.1 | -9.1 | -10.4 | -12.1 | -14.0
${}^{48}_{\Upsilon}\text{Ca}$ | 1s | -15.3 | -16.4 | -17.9 | -19.7 | -21.8
| 1p | -13.5 | -14.6 | -16.0 | -17.8 | -19.9
| 1d | -11.4 | -12.4 | -13.8 | -15.6 | -17.6
| 2s | -10.8 | -11.8 | -13.2 | -14.9 | -16.9
| 1f | -9.1 | -10.1 | -11.4 | -13.1 | -15.0
${}^{90}_{\Upsilon}\text{Zr}$ | 1s | -15.5 | -16.6 | -18.1 | -19.9 | -22.0
| 1p | -14.5 | -15.5 | -17.0 | -18.8 | -20.9
| 1d | -13.2 | -14.2 | -15.7 | -17.4 | -19.5
| 2s | -12.7 | -13.8 | -15.2 | -16.9 | -19.0
| 1f | -11.7 | -12.7 | -14.1 | -15.9 | -17.9
${}^{197}_{\Upsilon}\text{Au}$ | 1s | -15.3 | -16.3 | -17.7 | -19.4 | -21.5
| 1p | -14.7 | -15.8 | -17.2 | -18.9 | -20.9
| 1d | -14.0 | -15.0 | -16.4 | -18.1 | -20.1
| 2s | -13.7 | -14.7 | -16.0 | -17.8 | -19.8
| 1f | -13.2 | -14.2 | -15.6 | -17.3 | -19.3
${}^{208}_{\Upsilon}\text{Pb}$ | 1s | -15.7 | -16.8 | -18.2 | -20.0 | -22.1
| 1p | -15.2 | -16.2 | -17.7 | -19.4 | -21.5
| 1d | -14.5 | -15.5 | -16.9 | -18.7 | -20.8
| 2s | -14.1 | -15.2 | -16.6 | -18.3 | -20.4
| 1f | -13.6 | -14.7 | -16.1 | -17.8 | -19.9

In Table 2 we show the $\eta_{b}$-nucleus bound state energies for the same
nuclei and range of values of the cutoff mass parameter as in Table 1.
Furthermore, as in the case of the case of the $\Upsilon$-nucleus bound state
energies, for each nucleus we have listed only a few bound states. For the
heaviest nucleus we have $\sim$ 200 states and clearly is not practical to
show them all.

Table 2:  ${}^{4}_{\eta_{b}}\text{A}$ bound state energies for several nuclei
$A$. All dimensioned quantities are in MeV.

|  | Bound state energies
---|---|---
| $n\ell$ | $\Lambda_{B}=2000$ | $\Lambda_{B}=3000$ | $\Lambda_{B}=4000$ | $\Lambda_{B}=5000$ | $\Lambda_{B}=6000$
${}^{4}_{\eta_{b}}\text{He}$ | 1s | -63.1 | -64.7 | -66.7 | -69.0 | -71.5
| 1p | -40.6 | -42.0 | -43.7 | -45.8 | -48.0
| 1d | -17.2 | -18.3 | -19.7 | -21.4 | -23.2
| 2s | -15.6 | -16.6 | -17.9 | -19.4 | -21.1
${}^{12}_{\eta_{b}}\text{C}$ | 1s | -65.8 | -67.2 | -69.0 | -71.1 | -73.4
| 1p | -57.0 | -58.4 | -60.1 | -62.1 | -64.3
| 1d | -47.5 | -48.8 | -50.4 | -52.3 | -54.4
| 2s | -46.3 | -47.5 | -49.1 | -51.0 | -53.0
| 1f | -37.5 | -38.7 | -40.2 | -42.0 | -43.9
${}^{16}_{\eta_{b}}\text{O}$ | 1s | -67.8 | -69.2 | -71.0 | -73.1 | -75.4
| 1p | -61.8 | -63.2 | -64.9 | -67.0 | -69.2
| 1d | -54.9 | -56.2 | -57.9 | -59.9 | -62.0
| 2s | -53.2 | -54.6 | -56.3 | -58.2 | -60.3
| 1f | -47.3 | -48.6 | -50.2 | -52.1 | -54.2
${}^{40}_{\eta_{b}}\text{Ca}$ | 1s | -79.0 | -80.6 | -82.6 | -85.0 | -87.5
| 1p | -75.4 | -77.0 | -79.0 | -81.4 | -83.9
| 1d | -71.4 | -73.0 | -74.9 | -77.2 | -79.7
| 2s | -70.5 | -72.0 | -74.0 | -76.3 | -78.8
| 1f | -67.0 | -68.5 | -70.4 | -72.7 | -75.1
${}^{48}_{\eta_{b}}\text{Ca}$ | 1s | -76.7 | -78.2 | -80.2 | -82.5 | -85.0
| 1p | -74.0 | -75.5 | -77.4 | -79.7 | -82.1
| 1d | -70.8 | -72.3 | -74.2 | -76.4 | -78.8
| 2s | -69.9 | -71.4 | -73.3 | -75.5 | -77.9
| 1f | -67.2 | -68.6 | -70.6 | -72.8 | -75.1
${}^{90}_{\eta_{b}}\text{Zr}$ | 1s | -75.5 | -77.0 | -78.9 | -81.1 | -83.5
| 1p | -74.1 | -75.6 | -77.5 | -79.7 | -82.1
| 1d | -72.3 | -73.8 | -75.7 | -77.9 | -80.2
| 2s | -71.6 | -73.0 | -74.9 | -77.1 | -79.5
| 1f | -70.2 | -71.7 | -73.6 | -75.8 | -78.1
${}^{197}_{\eta_{b}}\text{Au}$ | 1s | -72.8 | -74.2 | -76.1 | -78.2 | -80.5
| 1p | -72.3 | -73.7 | -75.6 | -77.7 | -80.0
| 1d | -71.3 | -72.8 | -74.6 | -76.7 | -79.0
| 2s | -70.7 | -72.1 | -74.0 | -76.1 | -78.4
| 1f | -70.2 | -71.7 | -73.5 | -75.6 | -77.9
${}^{208}_{\eta_{b}}\text{Pb}$ | 1s | -74.7 | -76.2 | -78.1 | -80.3 | -82.6
| 1p | -74.2 | -75.7 | -77.5 | -79.7 | -82.1
| 1d | -73.2 | -74.7 | -76.6 | -78.8 | -81.1
| 2s | -72.7 | -74.1 | -76.0 | -78.2 | -80.5
| 1f | -72.1 | -73.6 | -75.5 | -77.6 | -80.0

We can now give some general conclusion concerning the results given in Tables
1 and 2. These results show that the $\Upsilon$ and $\eta_{b}$ mesons are
expected to form bound states with all the nuclei studied, independent of the
value of the cutoff parameter $\Lambda_{B}$. However, the particular values
for the bound state energies are dependent on the cutoff parameter, increasing
in absolute value as the cutoff parameter increases. This dependence was
expected from the behavior of the bottomonium $h$-nucleus potentials, since
these are more attractive for larger values of the cutoff parameter. Note also
that bottomonium $h$ binds more strongly to heavier nuclei and therefore a
richer spectrum is expected for these nuclei.

###  III.1 Discussion of the $\Upsilon$ and $\eta_{b}$ sigle particle energies

The discussion of the mass shifts results for the $\Upsilon$ and $\eta_{b}$
carried out in section II.3 can be translated to the $\Upsilon$ and $\eta_{b}$
sigle particle energies. From Tables 1 and 2, we see that the bound state
energies for the $\eta_{b}$ are larger than those of the $\Upsilon$ for the
same nuclei and range of cutoff values explored. As before, these differences
are probably due to two reasons: (a) the couplings $g_{\eta_{b}BB^{*}}$ and
$g_{\Upsilon BB}$ are very different. Indeed, the results obtained in Ref.
Cobos-Martinez:2020ynh  on the $\eta_{c}$ nuclear bound state energies are
closer to those the $J/\Psi$ when the $SU(4)$ flavor symmetry is broken, such
that $g_{\eta_{c}DD^{*}}=(0.6/\sqrt{2})\,g_{J/\Psi DD}\simeq 0.424\,g_{J/\Psi
DD}$ Cobos-Martinez:2020ynh ; Lucha:2015dda . Thus a reduced coupling
$g_{\eta_{b}BB^{*}}$ can bring the $\eta_{b}$ nuclear bound state energies
closer to those the $\Upsilon$, since the $\eta_{b}$ self-energy is
proportional to $g_{\eta_{b}BB^{*}}^{2}$. (b) the form factors are not equal
for the vertices $\Upsilon BB$ and $\eta_{b}BB^{*}$ and we have to readjust
the cutoff values, which means $\Lambda_{B}\neq\Lambda_{B^{*}}$, and the
comparisons for the mass shifts have to be made for different values for the
cutoff parameters.

##  IV  Summary and discussion

We have calculated the $\Upsilon$\- and $\eta_{b}$-nucleus bound state
energies for various nuclei neglecting any possible effects of the widths and
various values of the cutoff parameter $\Lambda_{B}$ that was introduced to
regularize the divergent integral in the self energies for these mesons. The
bottomonium $h$-nucleus potentials were calculated using a local density
approximation, with the inclusion of the $BB$ ($BB^{*}$) meson loop in the
$\Upsilon$ ($\eta_{c}$) self-energy. The nuclear density distributions and the
in-medium $B$ and $B^{*}$ meson masses were calculated using the quark-meson
coupling model. Using the bottomonium $h$ potentials in nuclei, we have solved
the Klein-Gordon equation and obtained bottomonium $h$-nucleus bound state
energies.

Our results show that the $\Upsilon$ and $\eta_{b}$ mesons are expected to
form bound states with all the nuclei studied, independent of the value of the
cutoff parameter $\Lambda_{B}$. However, the particular values for the bound
state energies are dependent on cutoff parameter $\Lambda_{B}$. The
sensitivity of our results to the cutoff parameter $\Lambda_{B}$ has also been
explored. However, an study needs to be done where we use a more properly
determined coupling $g_{\eta_{b}BB^{*}}$ and different functional forms for
the form factors. Furthermore, it is certainly necessary to include the
possible effects of the widths for the $\Upsilon$ and $\eta_{b}$. Such
elaborated studies are underway and will be reported elsewhere.

## Acknowledgements

We thank Prof. Tomoi Koide for useful conversations. GNZ was supported in part
by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior \- Brazil
(CAPES), and KT was supported by the Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq) Process, No. 313063/2018-4, and No.
426150/2018-0, and Fundação de Amparo à Pesquisa do Estado de São Paulo
(FAPESP) Process, No. 2019/00763-0, and this work was also part of the
projects, Instituto Nacional de Ciência e Tecnologia — Nuclear Physics and
Applications (INCT-FNA), Brazil, Process. No. 464898/2014-5.

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