11institutetext:  ADAPT Centre, Dublin City University, Dublin, Ireland
11email<EMAIL_ADDRESS>
Ghent University – imec – IDLab, Department of Electronics and Information
Systems, Ghent, Belgium
Signatu AS, Oslo, Norway
ADAPT Centre, Dublin City University, and Uniphar PLC, Dublin, Ireland
ADAPT Centre, Trinity College Dublin, Dublin, Ireland
University of Koblenz, Koblenz, Germany

# Data Privacy Vocabulary (DPV) - Version 2

Harshvardhan J. Pandit 11 Beatriz Esteves 22 Georg P. Krog 33 Paul Ryan 44
Delaram Golpayegani 55 Julian Flake 66

###### Abstract

The Data Privacy Vocabulary (DPV), developed by the W3C Data Privacy
Vocabularies and Controls Community Group (DPVCG), enables the creation of
machine-readable, interoperable, and standards-based representations for
describing the processing of personal data. The group has also published
extensions to the DPV to describe specific applications to support legislative
requirements such as the EU’s GDPR. The DPV fills a crucial niche in the state
of the art by providing a vocabulary that can be embedded and used alongside
other existing standards such as W3C ODRL, and which can be customised and
extended for adapting to specifics of use-cases or domains. This article
describes the version 2 iteration of the DPV in terms of its contents,
methodology, current adoptions and uses, and future potential. It also
describes the relevance and role of DPV in acting as a common vocabulary to
support various regulatory (e.g. EU’s DGA and AI Act) and community
initiatives (e.g. Solid) emerging across the globe.

###### Keywords:

privacy data protection data governance compliance

##  1 Introduction

The modern technological landscape consists of ubiquitous digital devices and
services which generate vast amounts of data, which includes sensitive
information that raises privacy concerns, as well as requires the protection
of data from misuse and cybersecurity threats. Regulations across the globe
have been developed or updated to meet this challenge, most notably the
European Union’s (EU) General Data Protection Regulation (GDPR) [27] in 2016,
which requires specific activities to be carried out based on defined norms
and requirements, and require documenting governance processes for compliance.

‘Regulatory Technology’ (RegTech) has also evolved to provide information
management capabilities and automation of tasks to support evolving
regulations. However, a key barrier to their effective use is their
proprietary nature, non-interoperable information, and lack of standards. As a
result, the RegTech landscape for privacy and data protection is fragmented
and siloed and lacks any meaningful ‘metadata’ and ‘semantics’ through which
information can be reused e.g. for processes not envisioned by a tool
provider.

Based on this background, the SPECIAL H2020 project established the W3C Data
Privacy Vocabularies and Controls Community
Group111https://www.w3.org/groups/cg/dpvcg/ (DPVCG) in 2018, which developed
the Data Privacy Vocabulary222https://w3id.org/dpv (DPV) [63] as a machine-
readable interoperable vocabulary for the exchange of legally relevant
‘metadata’. Since then, DPV has continued being a state of the art resource
that is iteratively updated to match the evolving landscape of regulations and
compliance requirements. It has seen high adoption in academic, industrial,
and mixed settings, and has been referenced in standards.

The initial focus of DPV (v1) was to provide a vocabulary for describing
personal data activities based on the GDPR. Since then, the world has evolved
at a rapid pace with innovations and regulations coming to the fore – in
particular those related to ‘increasing sharing of data’ such as the EU’s Data
Governance Act (DGA) [28] as well as for ‘regulating AI’ such as the EU’s AI
Act (AI Act) [26]. To reflect these developments as well as others in domains
such as cybersecurity, the DPVCG has updated the DPV and made it capable of
representing a wide variety of legally relevant information for ‘AI and
digital regulations’. In this article, we present the design and development
of DPV – version 2.

The rest of the article is structured as follows: Section 2 describes the
requirements which guided the development of DPV, Section 3 provides an
overview of DPV, Section 4 describes the methodology and design principles
used, Section 5 compares DPV with related work, Section 6 describes the known
adoptions, and Section 7 discusses ongoing developments and future work.

##  2 Requirements for a Legal Vocabulary

###  2.1 Information and Knowledge Modelling

DPV v1 was concerned with modelling the purposes, processing operations, legal
bases, personal data categories, technical and organisational measures, and
the roles of entities based on the GDPR, and to fill specific gaps: [63] (1)
There were no ontologies to describe and exchange information about activities
involving personal data; (2) There were no agreed upon vocabularies or
taxonomies for representing practical applications and uses of personal data
e.g. different purposes such as Service Provision, Legal Bases such as
Consent; (3) There were no vocabularies that align the terminology and
requirements of regulations for developing and using machine-readable
information.

Based on these, DPV v1 was developed to provide:

  1. 1.

a formal ontology that defines concepts and relationships;

  2. 2.

a (hierarchical) taxonomy that provides instances or specialisations of the
ontological concepts to reflect practical uses and applications; and

  3. 3.

a legally relevant modelling of information to support documentation needs for
compliance and exchange of information between stakeholder.

For DPV v2, the evolving landscape of law and technologies and the initial
adoption of DPV led the DPVCG to reconsider the scope of DPV. This resulted in
the following additional requirements:

  1. 1.

supporting multiple jurisdictions and regulations (e.g. EU DGA, USA California
CCPA);

  2. 2.

providing support for risk assessment and risk management based on relevant
standards (e.g. ISO 31000 series);

  3. 3.

representing use of services and technologies e.g. using a cloud service to
implement data collection;

  4. 4.

describing how AI (as a technology) is used;

  5. 5.

providing guidance on use of DPV to meet specific legal requirements; and

  6. 6.

providing documentation, examples, and guidance for increasing adoption.

###  2.2 Legal and Semantic Extensibility

Due to the nature of legal frameworks, concepts in DPV must constantly be
assessed and possibly modified to support changes in laws and case laws.
Unlike conventional ontologies, the meaning and semantics of concepts in DPV
relies on legal norms and interpretations which differ across jurisdictions
and change with laws and rulings. The design of DPV therefore requires the
possibility for it to be used in a ’jurisdiction-independent’ manner while
also having a mechanism to explicitly assert a specific law, or its concept
applies. For example, ’consent’ is a generic concept, whereas ’consent
according to GDPR’ is a jurisdiction and regulation specific concept. Such
legal customisability is an important part of DPV’s interoperability as it
enables different stakeholders to express common requirements (e.g. consent)
which can be explicitly asserted or interpreted in relevant contexts (e.g.
regulation based on location).

Semantic interoperability is a cornerstone of DPV given its emphasis on
representation and exchange of information between stakeholders. Compared to
conventional ontologies, concepts in DPV must be extensible to support
practical requirements which do not align neatly with design considerations of
only providing classes and properties. For example, a concept ’Email’ that
represents personal data associated with emails cannot be simply modelled as a
class as it may be needed as an instance to state emails are collected. At the
same time, it may also need to be extended to specifically refer to aspects
such as Email Address, Email Contents, or Attachments which also can be used
as instances or be further specialised. This fits the SKOS modelling style
better and also matches with how information is used by non semantic-web folks
(e.g. to fill in forms), but does not fit well with semantic reasoning
processes which largely use OWL (with Abox and Tbox assertions). DPV thus
needs to satisfy both styles of using concepts as extensible instances and
having support for OWL reasoning.

###  2.3 Stakeholder Interoperability

It is not possible for DPV to provide all relevant concepts in its
ontology/taxonomy given the almost infinite potential concepts (e.g. for
personal data). Instead, DPV’s community opted to provide the ’most important’
and ’most commonly required’ concepts, while requiring that DPV support
stakeholders in extending DPV internally for reflecting the peculiarities of
their own use-cases. For example, DPV can provide the purpose ’Marketing’
which a company extends to describe ’Summer Sale Offers’.

Following from this, DPV’s mission to provide interoperability across
stakeholders relies on the ’common’ concept present in the DPV taxonomy as the
basis for establishing shared understanding even if each stakeholder ends up
creating their own unique or individual ontological representation. In the
above example, Summer Sale Offers may be incomprehensible to another
stakeholder, but the use of DPV enables both entities to correctly interpret
that this is a form of Marketing \- and thus be able to identify requirements
and obligations for legal compliance associated with this concept.

##  3 Data Privacy Vocabulary (DPV)

###  3.1 Overview

The Data Privacy Vocabulary (DPV) enables expressing information related to
processing of personal data, entities involved and their roles, details of
technologies utilised, relation to laws and legal justifications permitting
its use, and other relevant concepts based on privacy and data protection.
While it uses the EU’s General Data Protection Regulation (GDPR) as a guiding
source for the creation and interpretation of concepts, the ambition and scope
of DPV is to provide a broad globally useful vocabulary that can be extended
to jurisdiction or domain specific applications.

The motivation of DPV is to provide a ’data model’ or a ’taxonomy’ of concepts
that act as a vocabulary for the interoperable representation and exchange of
information about personal data and its processing. For this, the DPV
specification represents an abstract model of concepts and relationships that
can be implemented and applied using technologies appropriate to the use-
case’s requirements.

Figure 1: Overview of concepts in DPV and their extensions

The core concepts in DPV, as shown in Figure 1, are broadly as follows:

  * •

Purpose: end-goal for why personal data is processed e.g. Service Provision

  * •

Processing: representing operations over personal data e.g. collect, store

  * •

Personal Data: categories of personal data involved

  * •

Legal Basis: justification in law for performing this activity

  * •

Legal Roles: Data Controller, Data Subject, etc.

  * •

Technical and Organisational Measures: safeguarding activities

  * •

Processing Context: storage conditions, automation, scale and scope

  * •

Context: concepts other than the above e.g. necessity, duration, frequency

  * •

Rights: legally recognised rights associated with activities

  * •

Risk and Risk Mitigation: managing risks, consequences, and impacts

Each of these ’core concepts’ are expanded into taxonomies to reflect their
application in use-cases. The taxonomies also provide ’knowledge’ by asserting
categorisation based on the core concepts. For example, Personal Data is a
core concept which is specialised into Sensitive Personal Data, and the
taxonomy expanding upon personal data contains some instances asserted as
being sensitive. An adopter therefore not only gets a taxonomy of personal
data, but is also able to utilise the categorisation of it into sensitive
personal data.

Extensions are a collection of concepts provided in a separate namespace. They
are used to represent specific concepts in jurisdictions and regulations, for
example Consent is present in the ’main’ or ’core’ DPV, and is expanded upon
in the EU-GDPR extension to represent the specific requirements for consent
under GDPR. Extensions are also used to provide a large group of concepts for
a specific topic as their inclusion in the main vocabulary would not be
practical or would introduce ambiguities between concepts. For example, the
Personal Data extension provides a taxonomy of personal data categories, which
were taken out of the main vocabulary due to ambiguity and confusion in
concepts such as Location being used for both personal data and data storage
location.

A list of ongoing work in DPVCG is as follows:

  * •

Data Privacy Specification (DPV) - https://w3id.org/dpv

  * •

Personal Data Concepts extension (PD) - https://w3id.org/dpv/pd

  * •

Location concepts extension (LOC) - https://w3id.org/dpv/loc

  * •

Legal concepts extension (LEGAL) - https://w3id.org/dpv/legal

  * •

EU GDPR concepts (EU-GDPR) - https://w3id.org/dpv/legal/eu/gdpr

  * •

EU DGA concepts (EU-DGA) - https://w3id.org/dpv/legal/eu/dga

  * •

EU AI Act concepts (EU-AIAct) - https://w3id.org/dpv/legal/eu/aiact

  * •

Legal Concepts for Ireland, Germany, United Kingdom, USA -
https://w3id.org/dpv/legal/IE (Replace IE with ISO 3166-1 code e.g.
IE/DE/GB/US)

  * •

Risk Concepts - https://w3id.org/dpv/risk

  * •

Technology Concepts - https://w3id.org/dpv/tech

To assist newcomers in understanding the structure of DPV and how its concepts
are organised - a Primer333https://w3id.org/dpv/primer document has been
developed. In addition, the group is in the process of enhancing the
documentation of each specification through illustrative guidance and
examples444https://w3id.org/dpv/examples.

###  3.2 Major Changes in v2

New in v2, DPV has increased the scope of these concepts to not just be
limited to ’personal data’ but provided their potential use for any ’data or
technology’. This is an important and significant change implemented to
support use-cases where data and personal data are utilised in the same case
and it would benefit to have the same semantic structure support management of
both, as well as to prepare for future support towards regulations such as the
Data Governance Act (DGA) which motivates ’sharing any data’ and the AI Act
where both include similar concepts i.e. Purpose, Rights, Risk, etc. While the
scope of the DPVCG is still limited to personal data (and associated
technologies), the expansion of scope for concepts enables the DPV to be
utilised for a much broader range of use-cases and regulations. More
importantly, it provides a common mechanism for representing information about
activities in the so called ’AI and Data’ regulations, and makes alignments
with existing standards such as ODRL easier to manage.

Another major change in v2 is the structuring of DPV and its extensions. In
v1, the extensions were simply placed in folders within the repo and this was
reflected in the namespace. In v2, the structure and the namespaces are better
designed to support and indicate the legal relevance of each extension. For
example, the path /legal/eu/gdpr represents the legal extensions for EU
jurisdiction and specifically the GDPR regulation. In this, the location or
jurisdiction is represented using ISO 3166-1 codes, and the regulations are
referred to by their acronyms. Through this, the path for other countries and
their laws can be understood and expressed easily. At the moment, the DPV’s
legal extensions include laws and associated authorities for: EU - GDPR, DGA,
and AI Act as well as the Charter of Fundamental Rights; Germany; Ireland;
United Kingdom, and the USA.

In DPV v1, the locations associated with data, activities, etc. did not have
any vocabulary provided to support their uniform expression. In v2, this has
been changed through the new Location extension which provides simple concepts
representing each country (based on ISO 3166-1) with plans to expand it to
also include country regions and subdivisions (by using ISO 3166-2). By using
these concepts - memberships are represented e.g. which countries are
currently in the EU. Further, DPV has aligned these concepts with those
provided in ISO555https://www.iso.org/iso-3166-country-codes.html as well as
EU Vocabularies666https://op.europa.eu/en/web/eu-
vocabularies/concept/-/resource?uri=http://publications.europa.eu/resource/authority/country/.

The defined locations also made possible expressing relevant laws and
authorities with links to their relevant jurisdictions, e.g. to indicate that
Ireland has a specific law and an authority associated with that law. In
addition, the locations also enable inference of jurisdictions which are
important in regulations e.g. to check whether data is crossing the border of
EU, or to identify which laws are applicable by using the locations and their
memberships.

##  4 Methodology & Design Principles

###  4.1 Management by DPVCG

The DPVCG used the W3C infrastructure to manage development of DPV, which
consisted of the mailing list, task management, and namespace management
(w3.org). After the migration of W3C to GitHub, DPVCG utilised the provided
repository777https://github.com/w3c/dpv/ for version control, task management,
discussions, and contributions. In its meetings, the group utilised
spreadsheets (using Google Sheets for collaboration) to support the (lack of)
technical knowledge of members and ease of discussion, commenting, and
sharing. Formal discussions and approvals were undertaken via the mailing list
and meetings.

###  4.2 Ontology Engineering Process

The DPVCG consists of experts from multiple disciplines - computer science,
law, sociology, authorities, and others. The primary role of its members is to
discuss and reach consensus on the scope and information to be represented in
DPV. Ontology engineers are then responsible for providing the appropriate
modelling of concepts and organisation of DPV as a semantic web resource.
While the DPVCG did not formally establish an ontology engineering
methodology, practices that were adopted and evolved in the community reflect
commonly utilised engineering methodologies of NEON [77] and [65].

The development process generally contained a member of the group proposing
addition or modification of a concept, with information shared potentially via
the mailing list and/or GitHub repo888https://github.com/w3c/dpv/issues, and
discussions and decisions within the meetings. Domain experts offered their
advice on the information being modelled and what aspects of this should be
considered, which were then formalised and shared as proposals. The group then
discussed and voted to resolve the proposal, with minutes of the meeting
reflecting the discussions and resolutions.

Compared to formal ontology engineering methods, this ad-hoc approach lacked
the explicit documentation of requirements, competency questions, and use-
cases - which would have been expensive to maintain given the limited
(regular) participation of volunteers. However, given the stability of the
group and continued iteration and adoption of DPV, the group has identified
this as an important step to undertake in the future.

The data in spreadsheets was structured to support ontology/taxonomy creation
while still being comprehensible to the non-semantic web members. For
generating the RDF serialisations, a custom documentation generator was
developed which downloaded the spreadsheets, extracted the concepts and
serialised them into multiple RDF formats, and generated the corresponding
HTMLs. Existing tools such as WIDOCO [35] were not used due to their
limitation in control of outputs - for example DPV has taxonomies of concepts
for purposes, personal data, etc. which would be all grouped together in a
giant list in WIDOCO outputs, and which in DPV outputs are grouped separately
to support adopters see related concepts grouped together. WIDOCO also did not
support the dual RDFS+SKOS and OWL modelling outputs of DPV, or the ReSpec
template999https://respec.org/docs/ common in W3C outputs.

DPV follows best practices and guidelines established within the semantic web
community. Namely, the W3C Best Practices for Publishing Linked Data101010Best
Practices for Publishing Linked Data (2014) https://www.w3.org/TR/ld-bp/,
WIDOCO best practices [35], W3ID for permanent IRIs, GitHub for version
control and collaboration, and Zenodo for archival.

###  4.3 Implications of Using SKOS and OWL

As described in the requirements in Section 2, DPV’s application requires
supporting semantics where its concepts are used as instances (taxonomy) or as
a schema that is instantiated (ontology). Initially, DPV was only provided as
an OWL2 ontology. This was expanded upon in DPV v1 which used custom SKOS
extensions to define the ’base’ vocabulary with serialisations in RDFS+SKOS
and OWL2. In DPV v2, the custom SKOS extensions were removed in favour of
RDFS+SKOS as the default serialisation with OWL2 as an alternative
serialisation. The RDFS+SKOS serialisation defines concepts as RDFS classes
and instances of a top-concept with SKOS used to represent the hierarchy,
whereas the OWL2 serialisation uses subclasses to represent the hierarchy.

The resulting implications of this on the adoption and use of DPV are
illustrated in the example below. The RDFS+SKOS model is suitable for when DPV
is to be used directly in both Tbox and Abox or where the underlying
ontology/schema cannot be changed as new concepts are identified in an use-
case, whereas the OWL2 model is suitable for when the Tbox and Abox are
separate or where there is freedom to change the ontology/schema. Conversely,
the OWL2 serialisation is better suited for semantic reasoners than the
RDFS+SKOS serialisation based on the underlying formal semantics. By providing
both serialisations, DPV enables the adopters to choose the most suitable
serialisation that supports their use-case and/or existing implementations,
and retains semantic interoperability based on converting between SKOS and
OWL2111111Using OWL and SKOS (2008) https://www.w3.org/2006/07/SWD/SKOS/skos-
and-owl/master.html.

⬇

# Example: A Company is running ’Direct Marketing’ campaigns for

# a purpose (CampaignA), within which it runs another special

# campaign (CampaignB). This may envolve into another campaign C.

# RDFS+SKOS modelling

dpv:Marketing a rdfs:Class, dpv:Purpose .

dpv:DirectMarketing a dpv:Purpose; skos:broader dpv:Marketing .

ex:CampaignA a dpv:Purpose; skos:broader dpv:DirectMarketing .

ex:CampaignB a dpv:Purpose; skos:broader ex:CampaignA .

# valid statements as all objects are instances of dpv:Purpose

ex:Documentation a dpv:Process; dpv:hasPurpose dpv:DirectMarketing .

\par# OWL2 modelling

dpv:Marketing a owl:Class; rdfs:subClassOf dpv:Purpose .

dpv:DirectMarketing a owl:Class; rdfs:subClassOf dpv:Marketing .

ex:CampaignA a dpv:DirectMarketing .

# problem: instance of instance

ex:CampaignB a ex:CampaignA .

# solution: change CampaignA to subclass

ex:CampaignA rdfs:subClassOf dpv:DirectMarketing .

# problem: DirectMarketing is not an instance of dpv:Purpose

ex:Documentation a dpv:Process; dpv:hasPurpose dpv:DirectMarketing .

# solution: blank nodes, but not a good solution

ex:Documentation a dpv:Process; dpv:hasPurpose [ a dpv:DirectMarketing ] .

# solution: use the instances rather than the class

ex:Documentation a dpv:Process; dpv:hasPurpose ex:CampaignA .

# problem: does not reflect the terminology used by Company/Law

##  5 Comparison with Related Work

DPV fills an unique and necessary niche within the state of the art by
providing concepts to represent legally relevant information to represent
processing of personal data and use of technologies. Here we only describe
related works that have a comparable adoption, or are standards, or are
outputs of larger projects.

The Open Digital Rights Language (ODRL) [45] is a W3C standard for the
modelling of policies and agreements which can dictate the permissions,
prohibitions, and duties associated with use of data or resources. The ODRL
vocabulary also provides specific concepts such as ’consent’ to support
commonly used agreements. While there is some overlap between DPV and ODRL,
they are complimentary in their objectives and uses. ODRL focuses solely on
providing a language for representation of policies and does not focus on
specific legal requirements or jurisdictions. DPV’s concepts can thus provide
the necessary legal and jurisdictional relevant information within ODRL
policies. This has been explored and demonstrated by existing work [20]. The
DPVCG intends to establish close collaboration with the ODRL CG through its
common members by aligning concepts between DPV and ODRL, and providing
guidance for using DPV as an ODRL profile. DPV has also been mentioned by the
International Data Spaces Association (IDSA) as a vocabulary of interest to
support the implementation of GDPR and regulations in their policies based on
ODRL [2].

LegalRuleML121212https://docs.oasis-open.org/legalruleml/legalruleml-core-
spec/v1.0/os/legalruleml-core-spec-v1.0-os.html is an OASIS standard providing
a ’rule interchange language’ that enables modelling and reasoning tasks based
on legal arguments. Similar to ODRL, it focuses on providing a language for
expression of ’rules’ and does not provide any taxonomies or model
regulations. Similar to ODRL, DPV is also complimentary to LegalRuleML as a
vocabulary that can be used to support representation of legal information.
There is existing work that has explored the use of LegalRuleML for modelling
requirements from the text of GDPR [68], though it has not been maintained nor
been extended to support practical requirements related to implementing GDPR.

Gist131313https://www.semanticarts.com/gist/ is a “minimalist upper ontology
for the enterprise” which provides business concepts with a focus on
minimising ambiguity. The Financial Industry Business
Ontology141414https://spec.edmcouncil.org/fibo/ (FIBO) models concepts
relevant in financial business applications, such as contracts and financial
transactions. Both Gist and FIBO represent ’industrial ontologies’ and have
been developed over a significant portion of time with the involvement of
corporate stakeholders. While neither support specific legal requirements or
jurisdictions, Gist provides concepts relevant for modelling details about an
organisation and FIBO provides modelling of contracts and contract related
processes - which DPV does not contain. Work is underway in DPVCG to study the
FIBO contract concepts and identify how DPV can support contracts related to
personal data processing.

##  6 Adoption

###  6.1 Analyses of Citations

In this Section, the adoption of DPVCG’s outputs is evaluated through a
citation analysis performed over the DPV v1 publication [63]. In this context,
81 publications that cite DPV were found through the Google Scholar service.
The gathered results underwent a review and were included in this analysis if
deemed pertinent. Duplicated publications, publications without an open-access
version and in languages other than English were excluded from this analysis,
resulting in 72 publications to be reviewed. Table 1 presents the results of
the performed publication evaluation. The publications were evaluated in
relation to their use of DPV: publications that reference DPV as a state of
the art resource are signalled in the Mention column, that use DPV towards an
application or use case in the Use column, and that extend DPV in the
Extension column. Publications that contributed their extensions back to DPV
(Contributed column) are also marked in Table 1, as well as if the work is
applied to a certain domain or sector (Domain column).

Table 1: Citation analysis of academic publications that reference DPV v1 [63]. Work | Year | Mention | Use | Extension | Contributed | Domain
---|---|---|---|---|---|---
[11] | 2020 | X |  |  |  | Health
[53] | 2020 | X |  |  |  | Media
[7, 54, 81] | 2020 | X |  |  |  |
[12, 47, 51, 71] | 2020 |  | X |  |  |
[10] | 2020 |  | X |  |  | Health
[15, 60, 74] | 2020 |  |  | X | X |
[17, 55, 29] | 2021 | X |  |  |  | Health
[37, 6, 76, 62, 39] | 2021 | X |  |  |  |
[72, 20, 16, 73, 52] | 2021 |  | X |  |  |
[34] | 2021 |  | X |  |  | Smart products
[42] | 2021 |  |  | X | X |
[44, 46, 43, 75, 59, 66, 22] | 2022 | X |  |  |  |
[21] | 2022 | X |  |  | X |
[69, 70, 57, 23] | 2022 |  |  | X | X |
[5] | 2022 | X |  |  |  | Health
[25, 14, 13, 32, 50] | 2022 |  | X |  |  |
[4] | 2022 |  | X |  |  | IoT
[40] | 2022 |  | X |  |  | Health
[58, 9, 8, 48, 1] | 2023 | X |  |  |  |
[33, 36, 3, 80, 18, 19, 79] | 2023 |  | X |  |  |
[78, 56, 30, 31] | 2023 |  | X |  |  | Health
[49] | 2023 |  | X |  |  | Smart cities
[38, 82] | 2023 |  | X |  |  | Media
[24, 64] | 2023 |  |  | X | X |
[67] | 2023 |  |  | X |  | Health
[61] | 2024 |  | X |  |  | Health
[41] | 2024 |  | X |  |  |

In this context, DPV’s specifications were compared against other state of the
art vocabularies in the data protection domain regarding their ability to
represent information related to GDPR rights and duties [21], their machine-
readability, maintenance, accessibility, GDPR support and existence of
compliance tools [1], and their capacity to aid with data interoperability and
adhere to the FAIR principles [81]. In all mentioned surveys, DPV obtained a
higher score compared with other existing solutions. When it comes to
extensions performed over DPV, most were contributed back to DPV to be
integrated into DPVCG’s outputs. Concerning work on GDPR requirements, there
were proposed extensions focusing on consent [60, 15], in particular related
to the processing of electronic health record data[67], as well as on building
semantic models to represent records of processing activities [74, 69, 70,
73], data protection impact assessments [57], data breaches’ reports [64], and
international data transfer notices [42]. Moreover, extensions focusing on
GDPR’s data subject rights and exemptions to these rights [23] and on DGA
requirements [24, 18] were also contributed back to DPVCG’s outputs.

In terms of applications to specific use cases, there is a body of work
focusing on providing tools for auditing and GDPR compliance evaluation [71,
72, 79], on data minimisation [34], as well as on the documentation and
annotation of privacy policies [51, 52, 47, 38] and privacy preferences [4].
Moreover, ML models were trained with DPV’s taxonomies to identify personal
data processing activities in code repositories [80, 41] and textual datasets
[32, 33]. DPV’s outputs were also used to model access and usage control
policies [12, 10, 16, 82], and in particular applied to Solid [20, 25, 14, 13,
30, 3, 19] and health data-sharing use cases [78, 61], as well as to describe
consent records and contracts for sensor data [49, 50]. In the context of data
spaces, DPV was used to provide descriptions of health data handling
activities [40] and to create user-centric privacy interfaces [36, 56, 31].

###  6.2 Projects and Industrial use of DPV

DPV and DPVCG are outputs of the SPECIAL H2020 project and were actively
developed and used within the project. In addition to this, DPV was also used
in TRAPEZE, MOSAICrOWN, smashHit, FAIRVASC, and PROTECT ITN projects funded
under the EU’s H2020 programme which involved both academic and commercial
partners. In addition, SPECIAL and TRAPEZE also included a Data Protection
Authority who provided legal expertise in implementation and design of
ontologies. Of these, TRAPEZE actively contributed back to the DPV and was
instrumental in identifying the design structure where both RDFS+SKOS and OWL
serialisations were developed, and to assist in developing a framework to
provide DPV concepts with multilingual documentation.

In addition to the use of DPV in industrial context in above projects,
companies that actively utilise DPV include Signatu – which develops legal
compliance solutions, and Inrupt – which develops the Solid specifications and
implementations. The DPVCG has received contributions from both, with Signatu
being an active and regular contributor and providing legal expertise and
practical requirements for industrial applications.

###  6.3 Use of DPV in Standards

ISO/IEC 27560:2023 is a Technical Specification (TS) describing a ’consent
record information structure’, which defines the specific information to be
maintained in consent records. DPV’s consent modelling played a significant
part in the development of this standard based on sharing the
knowledge151515Harshvardhan J. Pandit was an active contributor to the
development of ISO/IEC 27560:2023 through the National Standards Authority of
Ireland (NSAI) of legal and semantic requirements regarding consent records
and receipts. The annex of 27560:2023 provides examples of consent records and
receipts using JSON-LD - where DPV is explicitly referenced in the document
and its concepts are used as to define the schema (e.g. hasPurpose and
instances (e.g. Service Provision).

The IEEE P7012 Working Group is developing a specification to define how
“personal privacy terms are proffered and how they can be read and agreed to
by machines”. It explicitly references DPV as the vocabulary to describe
activities regarding processing of personal data in an interoperability
manner.

##  7 Planned Future Developments

The DPV’s status as a state of the art resource depends on its ability to be
up to date with regulatory developments as well as the evolving landscape of
technologies - especially those that relate to AI. In addition, its adoption
relies on the ability for stakeholders to understand its value and have
convenient information on how to integrate it within different kinds of
projects. To support this, the DPVCG has identified the necessity to provide a
rich set of examples that illustrate not only the concepts but also the
semantic and legal implications of using them, and developing guides and
tutorials for specific applications. To assist in refining the ontology
engineering process, the group is implementing processes to document
requirements and use-cases in a more formal manner as they are discussed and
resolved.

###  7.1 Extending DPV

The DPVCG plans to refine its extension providing taxonomies of technology
related concepts such as how it was provisioned, its network connectivity
requirements, and to model practical necessities such as whether it is a
’cloud service’ or ’runs only on device’. In addition, recent discussions have
also identified the necessity to specifically model AI related concepts as a
further extension of the existing Technology extension to provide AI specific
taxonomies, for example - AI techniques, capabilities, lifecycle stages, risks
and risk sources, and to enable stakeholders to express complex concepts such
as generative AI in a manner that supports regulatory requirements and is
based on recent developments of ISO standards. The DPVCG is also working on
providing a risk assessment and risk management vocabulary based on the ISO
31000 series to support the various risk related processes.

The DPVCG is also continuing its efforts to develop vocabularies to represent
key ’data and AI regulations’ notably in EU the Digital Services Act (DSA),
Data Markets Act (DMA), Data Act, Network Information Security Directive
(NIS2), and the recently finalised AI Act. In addition, the DPVCG is keen to
demonstrate the pan-jurisdictional capabilities of DPV by modelling
regulations in specific countries such as Ireland, USA, and UK.

To support the application of DPV in regulatory environments, the DPVCG is
developing guides based on existing work that utilises DPV to support GDPR
implementations. These include Records of Processing Activities (ROPA) [70],
Data Protection Impact Assessment (DPIA) [57], Data Breaches [64], and Rights
management. To support the implementation of decentralised and user-centric
applications of DPV, such as those envisioned in the IDSA, IEEE P7012 and
Solid, the DPVCG is developing vocabularies and guides - for example
implementation of ISO Standards for (semantic) privacy notices. In addition,
the DPV is also looking to incorporate the Standard Data Protection
Model161616https://www.bfdi.bund.de/EN/Fachthemen/Inhalte/Technik/SDM.html
provided by the German data protection authority regarding implementation of
concrete technical and organisational measures.

###  7.2 Tooling Support for Validation and Reasoning

The focus of DPV so far has been on enabling information representation. To
ensure that this information is ’correct’ and ’complete’ the group is also
exploring defining specifications which guide how information should be
represented, and how it could be verified - for example through SHACL shapes.
While this is not a sufficient replacement for formal semantics or the
creation of a ’language’, it is intended to encourage development of common
tooling and infrastructure through which barriers can be reduced to utilise
DPV and semantic technologies regarding regulatory processes.

##  8 Conclusion

The Data Privacy Vocabulary (DPV), in its second iteration, thus provides a
significantly richer, extended, and state of the art resource which fills an
important niche in the current landscape regarding expression of information
associated with personal data processing and the use of technologies. This
article highlighted the motivation for its development, described its
methodology and design processes, and showcased its value evidenced through
adoption across academia, industry, and standards.

DPV version 2 represents a significant milestone in the development of a
vocabulary to support legally relevant processes, and an expansion of scope to
include multiple regulations and jurisdictions, as well as recent advances in
AI and data sharing architectures such as Solid and IDSA. The DPVCG, in
continuing to develop the DPV, welcomes more participation and contributions
to support its vision of providing an interoperable vocabulary that provides
value and supports making legal compliance processes more efficient and
aligned for all stakeholders.

## Acknowledgements of Funding

The DPVCG was established as part of the SPECIAL H2020 Project, which received
funding from the European Union’s Horizon 2020 research and innovation
programme under Grant#731601 from 2017 to 2019. Harshvardhan J. Pandit was
funded to work on DPV from 2020 to 2022 by the Irish Research Council’s
Government of Ireland Postdoctoral Fellowship Grant#GOIPD/2020/790. The ADAPT
SFI Centre for Digital Media Technology is funded by Science Foundation
Ireland through the SFI Research Centres Programme and is co-funded under the
European Regional Development Fund (ERDF) through Grant#13/RC/2106 (2018 to
2020) and Grant#13/RC/2106_P2 (2021 onwards). The contributions of Piero
Bonatti and Luigi Sauro to the DPVCG have been funded by the European Union’s
Horizon 2020 research and innovation programme under grant agreement N. 731601
(project SPECIAL) until 2019, and under Grant#883464 (project TRAPEZE) from
2020 until 2023. The contributions of Beatriz Esteves and Delaram Golpayegani
have received funding through the PROTECT ITN Project from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie Grant#813497.

For the purpose of Open Access the authors have applied a CC-BY public
copyright licence to any Author Accepted Manuscript version arising from this
submission.

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# Theoretical study in the $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ and
$\bar{B}^{0}\to J/\psi f_{1}(1285)$ decays

DaZhuang He  Xuan Luo  YiLing Xie  Hao Sun***Corresponding author:
<EMAIL_ADDRESS>Institute of Theoretical Physics, School of Physics, Dalian
University of Technology,
No.2 Linggong Road, Dalian, Liaoning, 116024, P.R.China

###### Abstract

We study the decay processes of $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ and
$\bar{B}^{0}\to J/\psi f_{1}(1285)$ to analyse the $f_{1}(1285)$ resonance. By
the calculation within chiral unitary approach where $f_{1}(1285)$ resonance
is dynamically generated from the $K^{*}\bar{K}-c.c.$ interaction, we find
that the $\bar{K}^{*0}K^{0}$ invariant mass distribution has a clear broad
peak. Such broad peak has been understood as the signal of the $f_{1}(1285)$.
Finally, we obtain a theoretical result $R_{t}=\Gamma_{\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}}/\Gamma_{\bar{B}^{0}\to J/\psi f_{1}(1285)}$ which is
expected to be compared with the experimental data.

##  I Introduction

One of the main goals of the hadron physics is to unravel the nature of mesons
or baryons. With the discovery of more and more new particles, the traditional
quark model is not enough for us to understand the nature of particles. The
general examples are $a_{0}(980)$ and $f_{0}(980)$, and several perspectives
were presented about the nature of these: $q\bar{q}$ states Li et al. (2000);
Godfrey and Napolitano (1999), glueballs, multiquark states or meson-meson
molecules Klempt and Zaitsev (2007).

In very low energy regions, the strong interactions of the light pseudoscalar
particles can be simplified by considering global symmetry. Thus, based on the
chiral symmetry of Quantum Chromodynamics (QCD) and the concept of effective
field theory(EFT), a powerful theoretical tool Chiral Perturbation Theory
($\chi PT$) has been proposed Pich (1995); Bernard et al. (1995). The $\chi
PT$ can be used for the study of QCD at low energies($\leq 500$ MeV) by means
of chiral Lagrangians using fields related to mesons and baryons. Further, the
application of the unitary approaches in $\chi PT$, unitary chiral
approach($U\chi PT$), provides a possibility to study strong interactions at
higher energy($\geq 500$ MeV) by $\chi PT$. The $U\chi PT$ has been
successfully applied to study meson-meson and meson-baryon interactions, and
also gives a new perspective to understand the nature of particles, since
several resonances can be understood as dynamically generated. Such as, the
$a_{0}(980)$ and $f_{1}(1285)$ have been viewed as dynamically generated from
the interaction of $\bar{K}K$, $\pi\eta$ Oller and Oset (1997); Locher et al.
(1998); Xie and Geng (2016); Molina et al. (2020); Duan et al. (2020); Wang et
al. (2020) and $K^{*}\bar{K}-c.c.$ Xie and Oset (2016); Miyahara et al. (2015)
respectively. As for the baryon case, the $\Lambda(1405)$ can be shown as
dynamically generated from the interaction of $\bar{K}N$ and $\eta\Lambda$
Oller et al. (2000); Xie et al. (2018); Roca et al. (2015); Miyahara et al.
(2015), $\cdots$. In this paper, we focus on the $f_{1}(1285)$ as dynamically
generated.

The $f_{1}(1285)$ resonance is an axial-vector state with quantum numbers
$I^{G}(J^{PC})=0^{+}(1^{++})$, mass $M_{f_{1}}=1281.9\pm 0.5$ MeV and total
decay width $\Gamma_{f_{1}}=22.7\pm 1.1$ MeV (Zyla et al., 2020). It’s hard
for $f_{1}(1285)$ to decay to two-body, since the mass of the $f_{1}(1285)$ is
$100$ MeV below the $K^{*}\bar{K}$ threshold. The main decay modes of
$f_{1}(1285)$ are the $\eta\pi\pi$ ($52.2\%$) and $4\pi$ ($32.7\%$). In
Ref.(Roca et al., 2005), the authors found a pole of the amplitude of
pseudoscalar-vector meson interaction in the complex plane at the $1288-i0$
MeV and assigned it to the $f_{1}(1285)$ resonance. At the same time, similar
results were obtained in Zhou et al. (2014) when including higher order
Lagrangians. Thus, in the chiral unitary approach, the $f_{1}(1285)$ can be
qualified as dynamically generated from the pseudoscalar-vector meson
interaction. Although, the PDG Zyla et al. (2020) has reported ”not seen” for
the $K^{*}\bar{K}$ decay mode of the $f_{1}(1285)$, inspired by the
$K\bar{K}\pi$ decay mode of the $f_{1}(1285)$, the authors in Ref.Aceti et al.
(2015a) explained the possibility of the $K^{*}\bar{K}$ decay mode of the
$f_{1}(1285)$ and gave the coupling $g_{f_{1}}=7555$ MeV of the $f_{1}(1285)$
to $K^{*}\bar{K}$ channel. Some works in Refs. Debastiani et al. (2017); Oset
et al. (2018) also show the possibility of the $K^{*}\bar{K}$ decay mode of
the $f_{1}(1285)$.

In Refs.(Xie and Oset, 2016; Molina et al., 2016), the role of the
$f_{1}(1285)$ that was dynamically generated from the interaction of
$K^{*}\bar{K}-c.c.$, has been studied in the $J/\psi\to\phi f_{1}(1285)$ and
$\bar{B}_{s}^{0}\to J/\psi f_{1}(1285)$ decay processes, respectively. Unlike
the latter which showed only one tail of the resonance peak, the former showed
a clear threshold enhancement of invariant mass distribution of
$\bar{K}K^{*}$, which is caused by the production of the $f_{1}(1285)$. In
Ref.Aceti et al. (2015b), they obtained a consistent result with the
experiment for the $f_{1}(1285)\to\pi^{0}a_{0}(980)$ decay branching fraction,
where the $f_{1}(1285)$, $a_{0}(980)$ are also treated as dynamically
generated. Also, the process $\tau\to f_{1}(1285)\pi\nu_{\tau}$ has been
researched by considering a triangle loop mechanism with $K^{*}(\bar{K}^{*})$
and $\bar{K}(K)$ as internal lines Oset and Roca (2018).

On the other hand, in the experiments, the $B^{0}_{(s)}$ provides a good
platform for the study of some resonances $f_{0},a_{0},f_{1},\cdots$ Bayar et
al. (2014); Xie and Oset (2014); Liang et al. (2015); Dai et al. (2016).
Following the original ideas in Refs.(Xie and Oset, 2016; Molina et al.,
2016), we study the decays of $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ and
$\bar{B}^{0}\to J/\psi f_{1}(1285)$ decay by considering that the
$f_{1}(1285)$ resonance is dynamically generated from $K^{*}\bar{K}$
interaction. Notice although the channel $\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}$ cannot be observed directly, it can be related to
$\bar{B}^{0}\to J/\psi K^{0}K^{-}\pi^{+}+c.c.$ by decay of the $K^{*}\to
K\pi$, which was measured by LHCb Collaboration with $Br(B^{0}\to J/\psi
K^{0}K^{-}\pi^{+}+c.c.)=2.1\times 10^{-5}$ Aaij et al. (2014a). The branching
ratio of $Br(B^{0}\to J/\psi f_{1}(1285))=(8.4\pm 2.1)\times 10^{-6}$ was also
obtained from the measurement of $[\Gamma(B^{0}\to J/\psi
f_{1}(1285))/\Gamma_{total}]\times[B(f_{1}(1285)\to 2\pi^{+}2\pi^{-})]$ by the
LHCb Collaboration Lees et al. (2012); Aaij et al. (2014b). We can therefore
define the physical quantity $R_{\Gamma}=\frac{d\Gamma_{\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}}/dM_{inv}(\bar{K}^{*0}K^{0})}{\Gamma_{\bar{B}^{0}\to
J/\psi f_{1}(1285)}}$ without any other free parameter dependence except the
energy scale, and show the distribution of $\bar{K}^{*0}K^{0}$ invariant mass.
Moreover, we can obtain a theoretical result $R_{t}=\Gamma_{B^{0}\to
J/\psi\bar{K}^{*0}K^{0}}/\Gamma_{\bar{B}^{0}\to J/\psi f_{1}(1285)}$ which is
possible to be compared with the experimental data at a certain energy scale.

This paper is organized as follows. In Sec.II, we give the theoretical
formalism for the weak decay of $b$ quark and the hadronization of quark
pairs. In Sec.III, the Lagrangian of the processes $VP\to VP$ and the detailed
calculations are presented. In Sec.IV, the numerical results and discussions
are presented. Finally, a short summary is given in Sec.V.

##  II Formalism

$b$$\bar{B}^{0}$$c$$J/\psi$$\bar{c}$$d$$\bar{d}$$\bar{d}$$W$
$b$$\bar{B}^{0}$$c$$J/\psi$$\bar{c}$$d$$\bar{d}$$\bar{d}$$W$$\bar{q}q(\bar{u}u+\bar{d}d+\bar{s}s)$

Figure 1: The $\bar{B}^{0}\to J/\psi f_{1}(1285)$ decay process at the quark
level. (a) Elementary quark arrangement for the weak decay of $b$ quark. (b)
Hadronization of the $d\bar{d}$ component.

In this section we draw the pictures of the weak decay of $b$ quark and the
hadronization. The reaction process at the quark level as shown in the FIG.1
can be divided into three steps. The first part is that the $b$ quark of the
$\bar{B}^{0}$ converts to a $c\bar{c}$ and a $d$ quark via weak interaction,
where two vertex $bcW$ and $dcW$ involved by the weak decay are Cabibbo
suppressed Chau and Cheng (1987); Liang and Oset (2014) as shown in the
FIG.1(a). In the next step, the $c\bar{c}$ pair from decay of the $b$ quark
forms a $J/\psi$ and the $d\bar{d}$ hadronizes a pair of vector-pseudoscalar
mesons as depicted in FIG.1(b). Following Refs.Liang et al. (2018); Molina et
al. (2016); Miyahara et al. (2015), for the $d\bar{d}$ hadronization, we need
to introduce a $\bar{q}q$ pair with the quantum numbers of the vacuum,
$\bar{u}u+\bar{d}d+\bar{s}s$, and then two pairs of $q\bar{q}$(vector-
pseudoscalar mesons) arise. The hadronization process can be written

$\displaystyle
d(\bar{u}u+\bar{d}d+\bar{s}s)\bar{d}=\sum_{i=1}^{3}M_{2i}M_{i2}$ (1)

where $i$ denotes the quarks $u,d,s$, and $M$ is the $q\bar{q}$ matrix in
SU(3) group

$M=q\bar{q}=\begin{pmatrix}u\\\ d\\\
s\end{pmatrix}\begin{pmatrix}\bar{u}&\bar{d}&\bar{s}\end{pmatrix}=\begin{pmatrix}u\bar{u}&u\bar{d}&u\bar{s}\\\
d\bar{u}&d\bar{d}&d\bar{s}\\\ s\bar{u}&s\bar{d}&s\bar{s}\end{pmatrix}_{.}$ (2)

The matrix $M$ can also be rewritten in terms of pseudoscalar meson matrix
$M_{P}$ or vector meson matrix $M_{V}$,

$\displaystyle
M_{P}=\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}&K^{0}\\\
K^{-}&\bar{K}^{0}&-\frac{2}{\sqrt{6}}\eta_{8}\end{pmatrix}_{,}\qquad$ (3)
$\displaystyle
M_{V}=\begin{pmatrix}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{pmatrix}_{.}$

Eq.(1) can be rewritten with regard to $M_{V}M_{P}$ or $M_{P}M_{V}$

$\displaystyle M_{V}M_{P}\pm M_{P}M_{V},$ (4)

where the $\pm$ corresponds to $\mathcal{C}=\mp 1$($\mathcal{C}$ parity),
respectively Molina et al. (2016). Now, the hadronization process of
$d\bar{d}$ gives

$\displaystyle d(u\bar{u}$
$\displaystyle+d\bar{d}+s\bar{s})\bar{d}=(M_{V}M_{P}-M_{P}M_{V})_{22}$ (5)
$\displaystyle=K^{*0}\bar{K}^{0}-\bar{K}^{*0}K^{0}+\rho^{-}\pi^{+}-\rho^{+}\pi^{-}$
$\displaystyle+\left(\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}\right)\left(-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}\right)$
$\displaystyle-\left(-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}\right)\left(\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}\right).$

$\bar{B}^{0}$$V$$J/\psi$$P$ (a)
$\bar{B}^{0}$$J/\psi$$P^{\prime}$$V^{\prime}$$V$$P$ (b)

$\bar{B}^{0}$$J/\psi$$f_{1}(1285)$$V$$P$ (c)

Figure 2: Diagrammatic representation for the $\bar{B}^{0}\to J/\psi VP$
process. (a) The reaction at the tree level. (b) The rescattering between
pseudoscalar mesons and vector mesons. (c) Production process of the
$\bar{B}^{0}\to J/\psi f_{1}(1285)$ decay.

In the last step from FIG.2(b) which will be explained in the following
section, we need pseudoscalar $P$ and vector mesons $V$ for rescattering to
form the molecular resonance $f_{1}(1285)$ with quantum numbers
$I^{G}(J^{PC})=0^{+}(1^{++})$ Zyla et al. (2020) since $\mathcal{C}V=-\bar{V}$
and $\mathcal{C}P=\bar{P}$ for $M_{V}$ and $M_{P}$, respectively. Thus, in
order to generate the resonance $f_{1}(1285)$ we must take the C-parity
positive combination between the $V$ and $P$, and the Eq.(5):

$\displaystyle|MM\rangle=|K^{*0}\bar{K}^{0}-\bar{K}^{*0}K^{0}+\rho^{-}\pi^{+}-\rho^{+}\pi^{-}\rangle.$
(6)

##  III The processes $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ and
$\bar{B}^{0}\to J/\psi f_{1}(1285)$

In this section, we will calculate the invariant mass distribution of
$\bar{K}^{*0}K^{0}$ in decay process $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$
and the decay width of $\bar{B}^{0}\to J/\psi f_{1}(1285)$. According to the
chiral unitary approach Roca et al. (2005), the molecular resonance
$f_{1}(1285)$ can be regarded as dynamically generated from the interaction of
$K^{*}\bar{K}-c.c.$. Meanwhile, the process of $VP\to VP$ rescattering is
shown in FIG.2(b). In the local hidden gauge approach Bando et al. (1985);
Meissner (1988); Birse (1996); Bando et al. (1988), the Lagrangian for the
interaction of two pseudoscalar and two vector mesons is

$\displaystyle\mathcal{L}_{VVPP}=-\frac{1}{4f_{\pi}^{2}}Tr\left(\left[V^{\mu},\partial^{\nu}V_{\mu}\right]\left[P,\partial_{\nu}P\right]\right)$
(7)

where $P$ and $V$ are given in Eq.(3), $f_{\pi}$ is the decay constant of pion
with $f_{\pi}=92.4$ MeV, $Tr$ represents the $SU(3)$ trace. The Lagrangian of
Eq.(7) will lead to the relevant the $s$-wave projection of the scattering
amplitude, or to say the $V$-matrix (Roca et al., 2005)

$\displaystyle V_{ij}(s)$ $\displaystyle=$
$\displaystyle-\frac{\epsilon\cdot\epsilon^{\prime}}{8f_{\pi}^{2}}C_{ij}\bigg{[}3s-(M^{2}+m^{2}+M^{\prime
2}+m^{\prime 2})$ (8) $\displaystyle-$
$\displaystyle\frac{1}{s}(M^{2}-m^{2})(M^{\prime 2}-m^{\prime 2})\bigg{]}$

where $\epsilon(\epsilon^{\prime})$ represents the polarization vector of the
incoming(outgoing) vector meson. For convenience,
$\epsilon\cdot\epsilon^{\prime}$ is calculated as a prefactor below. The
$M(M^{\prime})$, $m(m^{\prime})$ stand for the masses of initial(final) vector
mesons and initial(final) pseudoscalar mesons, respectively. The indices $i$
and $j$ correspond to the initial and final $VP$ states, respectively. There
are different $C_{ij}$ coefficients for different isospin basis $(S,I)$, the
results of $C_{ij}^{}$11footnotemark: 1 have been listed in Ref.(Roca et al.,
2005).

The rescattering amplitude $T$ for the decay of $\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}$ as shown in the FIG.2(b) was obtained by solving the
Bethe-Salpeter equation in coupled channels Roca et al. (2005)

$\displaystyle
T=[1+V\hat{G}]^{-1}(-V)\vec{\epsilon}\cdot\vec{\epsilon}^{\prime}$ (9)

where $\hat{G}=G\left(1+\frac{1}{3}\frac{q^{2}}{M^{2}}\right)$, and $G$ is the
two meson propagator loop function including a vector and a pseudoscalar:

$\displaystyle
G(\sqrt{s})=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{(P-q)^{2}-M^{2}+i\epsilon}\frac{1}{q^{2}-m^{2}+i\epsilon},~{}$
(10)

where $m$ and $M$ are the masses of the pseudoscalar and vector mesons,
respectively. Some amplitudes and $V$-matrices have been listed in Ref.Wang et
al. (2020); Toledo et al. (2020).

The divergence of loop function Eq.(10) can be regularized by means of the
dimensional regularization Geng et al. (2015); Roca et al. (2005) or cut-off
scheme Aceti et al. (2015b), one has in the former scheme

$\displaystyle G^{D}(\sqrt{s})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\bigg{\\{}a(\mu)+\ln{\frac{M^{2}}{\mu^{2}}}+\frac{m^{2}-M^{2}+s}{2s}\ln{\frac{m^{2}}{M^{2}}}$
(11) $\displaystyle+$
$\displaystyle\frac{|\vec{q}|}{\sqrt{s}}[\ln{(s-(M^{2}-m^{2})+2|\vec{q}|\sqrt{s})}$
$\displaystyle+$ $\displaystyle\ln{(s+(M^{2}-m^{2})+2|\vec{q}|\sqrt{s})}$
$\displaystyle-$ $\displaystyle\ln{(-s+(M^{2}-m^{2})+2|\vec{q}|\sqrt{s})}$
$\displaystyle-$
$\displaystyle\ln{(-s-(M^{2}-m^{2})+2|\vec{q}|\sqrt{s})}]\bigg{\\}}.$

Also, in the cut-off scheme we have

$\displaystyle G(\sqrt{s})=$
$\displaystyle\int_{0}^{q_{max}}\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{\omega_{1}+\omega_{2}}{2\omega_{1}\omega_{2}}\frac{1}{s-(\omega_{1}+\omega_{2})^{2}+i\epsilon}$
(12) $\displaystyle\omega_{1}=$
$\displaystyle\sqrt{|\vec{q}|^{2}+m^{2}}\qquad\omega_{2}=\sqrt{|\vec{q}|^{2}+M^{2}}_{.}$

where the $\mu$ and $a(\mu)$ are the renormalization scale and subtraction
constant, respectively. And, the $\vec{q}$ stands for the three-momentum of
the vector or pseudoscalar meson in the center of mass frame and is given by

$|\vec{q}|=\frac{1}{2\sqrt{s}}\sqrt{[s-(M+m)^{2}][s-(M-m^{2})]}.$ (13)

The $s$ is the invariant mass of vector and pseudoscalar mesons in the loop of
FIG.2(b).

When the final state is the $\bar{K}^{*0}K^{0}$, the total amplitude
contributions correspond to FIG.3. Eq.(6) causes that subfigures (b) and (c)
have a minus sign comparing with (a) and (d). Meanwhile, we can write the
amplitude $\mathcal{U}_{j}$ for the transition $\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}$

$\displaystyle\mathcal{U}_{j}=$ $\displaystyle
V_{P}\left(h_{j}+\sum_{i}h_{i}\hat{G}_{i}(M_{inv})t_{ij}(M_{inv})\right)$ (14)

where the $\hat{G}$ is the loop function in Eq.(10), the scattering matrix
$t_{ij}$ is described in Eq.(9) and $M_{inv}$ is the invariant mass of the
vector-meson and pseudoscalar-meson in the final state. The $V_{P}$ is an
overall factor, which includes Cabibbo-Kobayashi-Maskawa(CKM) matrix elements
and kinematic prefactors. And the $V_{P}$ is a unknown quantity and it could
be cancelled below. Meanwhile, in the Refs. Kang et al. (2014); Xie and Geng
(2016); Daub et al. (2016); Xie et al. (2018); Wang et al. (2015) the authors
take $V_{P}$ as a constant. Also, considering the Eq.(8), for $h_{i(j)}$ we
have

$\displaystyle h_{K^{*0}\bar{K}^{0}}$
$\displaystyle=h_{\rho^{-}\pi^{+}}=1,\quad
h_{\bar{K}^{*0}K^{0}}=h_{\rho^{+}\pi^{-}}=-1,$ (15) $\displaystyle
h_{K^{*+}K^{-}}$
$\displaystyle=h_{K^{*-}K^{+}}=h_{\rho^{0}\pi^{0}}=h_{\rho^{0}\rho^{0}}=0.$

Take notice that the above amplitude only holds for an s-wave with every
intermediate particle being on mass-shell.

$\bar{B}^{0}$$J/\psi$$K^{0}$$\bar{K}^{*0}$$\ \ \bar{K}^{*0}$$\ \ K^{0}$ (a)
$\bar{B}^{0}$$J/\psi$$K^{0}$$\bar{K}^{*0}$$\ \ K^{*0}$$\ \ \bar{K}^{0}$ (b)

$\bar{B}^{0}$$J/\psi$$K^{0}$$\bar{K}^{*0}$$\ \ \rho^{-}$$\ \ \pi^{+}$ (c)
$\bar{B}^{0}$$J/\psi$$K^{0}$$\bar{K}^{*0}$$\ \ \rho^{+}$$\ \ \pi^{-}$ (d)

Figure 3: The total contributions for the $\bar{K}^{*0}K^{0}$ production.
Subfigures (a) and (d) have a minus sign corresponding with subfigures (b) and
(c).

The amplitude Eq.(14) for the $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ can be
written as

$\displaystyle\mathcal{U}=V_{P}\big{[}-1$ $\displaystyle-$ $\displaystyle
G_{\bar{K}^{*0}K^{0}}(M_{inv}(\bar{K}^{*0}K^{0}))t_{\bar{K}^{*0}K^{0}\to\bar{K}^{*0}K^{0}}(M_{inv}(\bar{K}^{*0}K^{0}))$
$\displaystyle+$ $\displaystyle
G_{K^{*0}\bar{K}^{0}}(M_{inv}(\bar{K}^{*0}K^{0}))t_{K^{*0}\bar{K}^{0}\to\bar{K}^{*0}K^{0}}(M_{inv}(\bar{K}^{*0}K^{0}))$
$\displaystyle+$ $\displaystyle
G_{\rho^{-}\pi^{+}}(M_{inv}(\bar{K}^{*0}K^{0}))t_{\rho^{-}\pi^{+}\to\bar{K}^{*0}K^{0}}(M_{inv}(\bar{K}^{*0}K^{0}))$
$\displaystyle-$ $\displaystyle
G_{\rho^{+}\pi^{-}}(M_{inv}(\bar{K}^{*0}K^{0}))t_{\rho^{+}\pi^{-}\to\bar{K}^{*0}K^{0}}(M_{inv}(\bar{K}^{*0}K^{0}))\big{]}.$

By means of the following isospin multiplets

$\displaystyle\begin{pmatrix}K^{+}\\\
K^{0}\end{pmatrix}_{,}\quad\begin{pmatrix}\bar{K}^{0}\\\
-K^{-}\end{pmatrix}_{,}\quad\begin{pmatrix}-\rho^{+}\\\ \rho^{0}\\\
\rho^{-}\end{pmatrix}_{,}\quad\begin{pmatrix}-\pi^{+}\\\ \pi^{0}\\\
\pi^{-}\end{pmatrix}_{,}$ (17)

and the CG-coefficients, we have amplitude relations

$\displaystyle t_{\bar{K}^{*0}K^{0}\to\bar{K}^{*0}K^{0}}=$
$\displaystyle+\frac{1}{2}t_{\bar{K}^{*}K\to\bar{K}^{*}K}^{I=0}$ (18)
$\displaystyle t_{K^{*0}\bar{K}^{0}\to\bar{K}^{*0}K^{0}}=$
$\displaystyle-\frac{1}{2}t_{K^{*}\bar{K}\to\bar{K}^{*}K}^{I=0}$
$\displaystyle t_{\rho^{-}\pi^{+}\to\bar{K}^{*0}K^{0}}=$
$\displaystyle-\frac{1}{\sqrt{6}}t_{\rho\pi\to\bar{K}^{*}K}^{I=0}$
$\displaystyle t_{\rho^{+}\pi^{-}\to\bar{K}^{*0}K^{0}}=$
$\displaystyle-\frac{1}{\sqrt{6}}t_{\rho\pi\to\bar{K}^{*}K.}^{I=0}$

And we can rewrite the Eq.(III) as

$\displaystyle\mathcal{U}=-V_{P}\big{[}1$ (19) $\displaystyle+$
$\displaystyle\frac{1}{2}G_{\bar{K}^{*}K}(M_{inv}(\bar{K}^{*0}K^{0}))t_{\bar{K}^{*}K\to\bar{K}^{*}K}^{I=0}(M_{inv}(\bar{K}^{*0}K^{0}))$
$\displaystyle+$
$\displaystyle\frac{1}{2}G_{K^{*}\bar{K}}(M_{inv}(\bar{K}^{*0}K^{0}))t_{K^{*}\bar{K}\to\bar{K}^{*}K}^{I=0}(M_{inv}(\bar{K}^{*0}K^{0}))\big{]}.~{}~{}~{}~{}~{}$

If the final state is chosen as $J/\psi K^{*0}\bar{K}^{0}$, the amplitude for
$\mathcal{U}_{J/\psi K^{*0}\bar{K}^{0}}$ has a minus sign compared with
Eq.(19).

With the above results, now the invariant mass distribution $d\Gamma/dM_{inv}$
can be written Zyla et al. (2020)

$\displaystyle\frac{d\Gamma}{dM_{inv}}$ $\displaystyle=$ $\displaystyle\sum
C_{s}^{2}\frac{1}{(2\pi)^{3}}\frac{M_{inv}}{8M_{\bar{B}^{0}}^{3}}$ (20)
$\displaystyle\times$
$\displaystyle\int_{M_{J/\psi,\bar{K}^{*0}}^{min}}^{M_{J/\psi,\bar{K}^{*0}}^{max}}|\mathcal{U}|^{2}M_{J/\psi,\bar{K}^{*0}}dM_{J/\psi,\bar{K}^{*0}},$

the $\sum C_{s}^{2}$ represents the total polarization structure, which can be
written as

$\displaystyle\sum
C_{s}^{2}=3+\frac{\vec{p}_{J/\psi}^{2}}{m_{J/\psi}^{2}}+\frac{\vec{p}_{\bar{K}^{*0}}^{2}}{m_{\bar{K}^{*0}}},$
(21)

where we make the approximationLiang et al. (2018); Sakai et al. (2018); Oset
and Ramos (2010) that the three-momenta of the vector mesons in the loop are
on-shell and small compared to their masses. And we have

$\displaystyle\vec{p}_{J/\psi}=$
$\displaystyle\frac{\lambda^{\frac{1}{2}}(M_{\bar{B}^{0}}^{2},M_{J/\psi}^{2},M_{inv}^{2})}{2M_{\bar{B}^{0}}},$
(22) $\displaystyle\vec{p}_{\bar{K}^{*0}}=$
$\displaystyle\frac{\lambda^{\frac{1}{2}}(M_{inv}^{2},M_{k^{0}}^{2},M_{\bar{k}^{*0}}^{2})}{2M_{inv}},$

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2xz$ is the Kählén function.

In the Eq.(20), $M_{\bar{K}^{*0}K^{0}}$ and $M_{J/\psi,\bar{K}^{*0}}$ are
interrelated, thus for specific $M_{\bar{K}^{*0}K^{0}}$, the range of
$M_{J/\psi,\bar{K}^{*0}}$ is defined as

$\displaystyle M_{J/\psi,\bar{K}^{*0}}^{max}=$
$\displaystyle\sqrt{(E_{\bar{K}^{*0}}+E_{J/\psi})^{2}-\left(\sqrt{E_{\bar{K}^{*0}}^{2}-M_{\bar{K}^{*0}}^{2}}-\sqrt{E_{J/\psi}^{2}-M_{J/\psi}^{2}}\right)^{2}},$
$\displaystyle M_{J/\psi,\bar{K}^{*0}}^{min}=$
$\displaystyle\sqrt{(E_{\bar{K}^{*0}}+E_{J/\psi})^{2}-\left(\sqrt{E_{\bar{K}^{*0}}^{2}-M_{\bar{K}^{*0}}^{2}}+\sqrt{E_{J/\psi}^{2}-M_{J/\psi}^{2}}\right)^{2}},$

where $E_{\bar{K}^{*0}}=(M_{inv}^{2}-M_{K}^{2}+M_{\bar{K}^{*0}}^{2})/2M_{inv}$
and $E_{J/\psi}=(M_{\bar{B}^{0}}^{2}-M_{inv}^{2}-M_{J/\psi}^{2})/2M_{inv}$ are
the energies of $\bar{K}^{*0}$ and $J/\psi$ in the $\bar{K}^{*0}K^{0}$ rest
frame. The Dalitz plot for $M_{\bar{K}^{*0}K^{0}}$ and
$M_{J/\psi\bar{K}^{*0}}$ invariant masses in the $\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}$ decay is shown in FIG.4.

Figure 4: Dalitz plot for $\bar{K}^{*0}K^{0}$ and $J/\psi\bar{K}^{*0}$
invariant masses in the $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ decay.

On the other hand, notice that Eq.(20) contains an unknown quantity $V_{P}$.
In order to eliminate $V_{P}$, we could consider the production of the
$f_{1}(1285)$ resonance. The relevant mechanism is depicted in FIG.2(c) and we
have

$\displaystyle T_{\bar{B}^{0}\to J/\psi
f_{1}(1285)}=V_{P}C_{s}^{\prime}G(M_{f_{1}})g_{f_{1}}$ (24)

where the polarization factor $C_{s}^{\prime}$ is obtained analogy to Eq.(21).
The coupling of $f_{1}(1285)$ to $K^{*}\bar{K}$ channel can be expressed as
$g_{f_{1}}\epsilon_{i}(f_{1})\epsilon_{i}(K^{*})$, by contracting the two
polarization vectors $\epsilon_{i}(K^{*})\epsilon_{j}(K^{*})=\delta_{ij}$ in
the loop of FIG.2.(c), and give

$\displaystyle\sum C_{s}^{\prime
2}=3+\frac{\vec{p}_{J/\psi}^{2}}{m_{J/\psi}^{2}}+\frac{\vec{p}_{f_{1}}^{2}}{m_{f_{1}}^{2}}.$
(25)

where
$\vec{p}_{f_{1}}=\vec{p}_{J/\psi}=\lambda^{\frac{1}{2}}(M_{\bar{B}^{0}}^{2},M_{J/\psi}^{2},M_{f_{1}}^{2})/2M_{\bar{B}^{0}}$
is the three momentum of $f_{1}$ and $J/\psi$ in the $\bar{B}^{0}$ rest frame.

Then, the partial decay width of $\bar{B}^{0}\to J/\psi f_{1}(1285)$ can be
written

$\displaystyle\Gamma_{\bar{B}^{0}\to J/\psi f_{1}}$ $\displaystyle=$
$\displaystyle\frac{V_{P}^{2}}{8\pi}\frac{g_{f_{1}}^{2}G^{2}(M^{2}_{f_{1}})|\vec{p}_{J/\psi}|}{M_{\bar{B}^{0}}^{2}}$
(26) $\displaystyle\times$
$\displaystyle\left(2+\frac{\left(M_{\bar{B}^{0}}^{2}-M_{J/\psi}^{2}-M_{f_{1}}^{2}\right)^{2}}{4M_{J/\psi}^{2}M_{f_{1}}^{2}}\right)$

By considering $\bar{B}^{0}\to J/\psi f_{1}(1285)$, we define

$\displaystyle R_{\Gamma}=\frac{d\Gamma_{\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}}/dM_{inv}(\bar{K}^{*0}K^{0})}{\Gamma_{\bar{B}^{0}\to
J/\psi f_{1}(1285)}}$ (27)

where $V_{P}$ has been cancelled so that $R_{\Gamma}$ has no other free
parameters except the energy scale. It can be a function of invariant mass
$M_{inv}$ and compared with the experiments at a certain energy scale.
Although there are different $\mathcal{U}$ for different final states
$J/\psi\bar{K}^{*0}K^{0}$ and $J/\psi K^{*0}\bar{K}^{0}$, they have an equal
$R_{\Gamma}$.

##  IV NUMERICAL RESULTS AND DISCUSSION

With the former formulae, we present, in FIG.5, the predictions for the ratio
$R_{\Gamma}$ as a function of the invariant mass $M_{inv}$ of the
$\bar{K}^{*0}K^{0}$ state. As mentioned above, the unknown parameter $V_{P}$
cancels, therefore it does not appear in the final result of the ratio
$R_{\Gamma}$. Meanwhile, we consider the following renormalization scales
$\mu=1100\text{MeV},\ 1200\text{MeV},\ 1300\text{MeV}$ and corresponding
subtraction constants $a(\mu)=-1.66,\ -1.62,\ -1.59$. A straightforward
relation between $a(\mu)$ and $\mu$ is given by Oller and Meissner (2001)

$a_{i}(\mu)=-2\log{\left(1+\sqrt{1+\frac{M_{i}^{2}}{\mu^{2}}}\right)}+\mathcal{O}\left(\frac{m_{i}}{M_{i}}\right).$
(28)

Figure 5:  The distribution of $R_{\Gamma}$ as a function of the invariant
mass $M_{inv}$ of $\bar{K}^{*0}K^{0}$ at $\mu=1100\text{MeV},\ 1200\text{MeV}$
and $1300\text{MeV}$, respectively. The solid (dashed) curve represents the
dynamically generated prediction with (without) the polarization structure
factor. The dot-dashed curve stands for the phase space distribution which has
been normalized to make it have the same area with the solid curve in the
$M_{inv}$ range.

As shown in FIG.5, the solid curve gives a good description for the invariant
mass distribution of $M_{inv}(\bar{K}^{*0}K^{0})$, which produces the broad
peak of the mass distribution. The broad peak is caused by the $f_{1}(1285)$
which is dynamically generated by the $K^{*}\bar{K}-c.c.$ interaction. The
range of the invariant mass $M_{inv}$ of $\bar{K}^{*0}K^{0}$ has been
considered as the $1390-1900$ MeV. In principle, the range of the invariant
mass $M_{inv}$ of $K^{*}\bar{K}$ can reach the maximum value $2182$
MeV($M_{\bar{B}^{0}}-M_{J/\psi})$ in the process of $\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}$ decay. Nevertheless, the results of the previous
sections are based on the chiral unitary theory, thus the range of the
invariant mass should be $200-300$ MeV smaller than the maximum value $2182$
MeV to make sure the theory works better Xie and Oset (2016). At the same
time, by comparing three subgraphs in FIG.5, we can see that as the
renormalization scale $\mu$ increases, the strength of the $R_{\Gamma}$
decreases and the position of the broad peak of $R_{\Gamma}$ almost does not
move. In addition, in order to compare, the result without considering the
polarization structure factor of the $R_{\Gamma}$ is also shown by the dashed
curve. We can see that the dashed curve also gives a clear broad peak and is
larger than the solid one. Finally, the phase space as a function of the
invariant mass $M_{inv}$ has been depicted by the dot-dashed curve. Actually,
the value from the phase space distribution is much smaller than the values of
the other two curves, and we normalize the phase space distribution to make it
have the same area with the solid curve.

We define the parameter $R_{t}$ by

$\displaystyle R_{t}=\frac{\Gamma_{\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}}}{\Gamma_{\bar{B}^{0}\to J/\psi
f_{1}(1285)}}=\int_{m_{\bar{K}^{*0}}+m_{K^{0}}}^{m_{\bar{B}^{0}}-m_{J/\psi}}R_{\Gamma}\
dM_{inv},$ (29)

and obtain a theoretical value of $R_{t}=0.451$ for $\mu=1200$ MeV. When we
choose the values of $\mu$ in the range of $1100\sim 1300$ MeV, the results of
$R_{\Gamma}$ are located around the range of $0.301\sim 0.734$. Thus the main
uncertainty of the theoretical ratio comes from the renormalization scale
$\mu$.

On the other side, the processes $K^{*0}\to K^{+}\pi^{-},K^{0}\pi^{0}$ have
strengths $\frac{2}{3}$, $\frac{1}{3}$, respectively. The strength of the
process $\bar{B}^{0}\to J/\psi K^{*}\bar{K}$ can be obtained by using the
experimental branching ratio of $\bar{B}^{0}\to J/\psi K^{0}K^{-}\pi^{+}+c.c$,
thus similar to Ref.Liang et al. (2018), we have

$\displaystyle Br(\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0})=\frac{1}{8}\frac{3}{2}\times 2.1\times 10^{-5},$
$\displaystyle Br(\bar{B}^{0}\to J/\psi f_{1}(1285))=(8.4\pm 2.1)\times
10^{-6},$ (30)

where the branching ratio for $\bar{B}^{0}\to J/\psi f_{1}(1285)$ can be
obtained directly from PDG. The Eq.(IV) gives an experiment result

$R_{e}=\frac{Br(\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0})}{Br(\bar{B}^{0}\to
J/\psi f_{1}(1285))}=0.469\pm 0.117.$ (31)

In principle, the $R_{e}$ should be related with $R_{t}$ by $R_{e}=R_{t}$.
However, the $Br(\bar{B}^{0}\to J/\psi K^{0}K^{-}\pi^{+}+c.c.)=2.1\times
10^{-5}$ is not accurate within the corresponding experimental condition.
Furthermore, as mentioned, in order that the chiral unitary approach can work
well, the upper limits of the $M_{inv}$ should be $200-300$ MeV far from the
boundary($m_{\bar{B}^{0}}-m_{J/\psi}$). The above two reasons lead to the main
discrepancy between $R_{e}$ and $R_{t}$. There are also some other reasons,
for examples, there are higher mass states with spin-parity $J^{P}=1^{+}$ and
$2^{+}$ at higher invariant mass range of $\bar{K}^{*0}K^{0}$ (Xie and Oset,
2016). As for the relation between $\mu$ and $a(\mu)$, we ignore higher-order
effects. These contributions are small compared with the previous two main
reasons. We expect that future experimental observation of the invariant mass
distribution $R_{\Gamma}$ on the LHCb would provide valuable information on
the mechanism of the $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ decay.

##  V SUMMARY

In the present work, we have shown the distribution of the invariant mass
$\bar{K}^{*0}K^{0}$ in the decay $\bar{B}^{0}\to J/\psi\bar{K}^{*0}K^{0}$ to
investigate the basic nature of the $f_{1}(1285)$ resonance. In the frame of
chiral unitary theory, the $f_{1}(1285)$ resonance is dynamically generated
from the $K^{*}\bar{K}-c.c.$ interaction, further we get the theoretical
prediction $R_{\Gamma}=\frac{d\Gamma_{\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}}/dM_{inv}(\bar{K}^{*0}K^{0})}{\Gamma_{\bar{B}^{0}\to
J/\psi f_{1}(1285)}}$ as a function of invariant mass
$M_{inv}(\bar{K}^{*0}K^{0})$. The result of $R_{\Gamma}$ reproduces the peak
of the mass distribution of $M_{inv}(\bar{K}^{*0}K^{0})$, which is caused by
the production of the $f_{1}(1285)$ state. Finally, we can obtain a
theoretical result $R_{t}=\Gamma_{\bar{B}^{0}\to
J/\psi\bar{K}^{*0}K^{0}}/\Gamma_{\bar{B}^{0}\to J/\psi f_{1}(1285)}$ which is
possible to be compared with the experimental result $R_{e}$. We expect that
there will be more valuable data in future experimental observations, which
could advance our understanding of $f_{1}(1285)$.

###### Acknowledgements.

The authors thank Eulogio Oset for his valuable comments. Hao Sun is supported
by the National Natural Science Foundation of China (Grant No.12075043).

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