11institutetext:  Department of Humanities, Social and Political Sciences, ETH
Zurich, Switzerland
22institutetext: Department of Computer Science, VU University Amsterdam,
Netherlands
<EMAIL_ADDRESS>
#  nanopub-java: A Java Library for Nanopublications

Tobias Kuhn 1122

###### Abstract

The concept of nanopublications was first proposed about six years ago, but it
lacked openly available implementations. The library presented here is the
first one that has become an official implementation of the nanopublication
community. Its core features are stable, but it also contains unofficial and
experimental extensions: for publishing to a decentralized server network, for
defining sets of nanopublications with indexes, for informal assertions, and
for digitally signing nanopublications. Most of the features of the library
can also be accessed via an online validator interface.

##  1 Introduction

This technical paper presents nanopub-java, which is a library for
nanopublications. Its source code can be found here:

> https://github.com/Nanopublication/nanopub-java

Nanopublications111http://nanopub.org [2, 8] are an approach to publish
scientific data and meta-data in RDF by subdividing them into small data
snippets. They are a concrete proposal to implement the visions of _semantic
publishing_ [9] and _linked science_ [3] by allowing for the publication and
sharing of formally represented scientific resources and data that are
semantically interlinked and provide provenance and context information for
their reliable integration and evaluation. Specifically, a nanopublication
consists of three named graphs of RDF triples (plus a fourth graph to keep
them together): the assertion graph contains the actual content of the
nanopublication (e.g. a scientific finding); the provenance graph contains
information about the provenance of the assertion (e.g. the scientific method
with which the assertion was derived); and the publication information graph
contains meta-data about the nanopublication itself (e.g. its creator and a
timestamp).

The library presented here can be useful in a number of scenarios:

  * •

To represent and share small chunks of scientific knowledge and metadata in
RDF in a provenance-aware manner (as nanopublications)

  * •

To make RDF content verifiable and immutable (with trusty URIs)

  * •

To define large or small datasets of RDF content where the data entries can be
individually addressed and recombined in new datasets (with nanopublication
indexes)

  * •

To quickly publish RDF snippets in a verifiable and permanent manner (relying
on an existing server network)

  * •

To retrieve existing nanopublications from the network (5 millions and
counting)

  * •

To digitally sign RDF snippets (though this is still experimental)

Below the details of the library and its web interface are explained.

##  2 Implementation

The library is built upon the Sesame library [1] to validate, represent, and
create RDF structures. The features of the nanopublication library are
centered around a Java interface representing a nanopublication, called
Nanopub, and the Java class NanopubImpl provides a reference implementation of
this interface. This implementation checks the well-formedness of a
nanopublication at the time of its creation based on the latest version of the
nanopublication guidelines,222http://nanopub.org/guidelines/working_draft/ and
raises an exception in the case of a violation of these rules.

Trusty URIs [6, 7] are the recommended way of how to make nanopublications
verifiable and immutable, and to give them identifiers based on cryptographic
hash values. The nanopublication library uses for that purpose the trustyuri-
java library.333https://github.com/trustyuri/trustyuri-java In a nutshell, a
trusty URI is a kind of URI reference that contains a cryptographic hash value
that is calculated on the digital artifact it represents. This allows one to
verify that a given content is really what the URI was supposed to represent
by its creator, and thereby to enforce the immutability of digital artifacts
such as nanopublications.

The features of the library are made available through the Java API as well as
via a command line interface using the command np. The following features are
part of the core of the library, which means that they deal with stable and
agreed-upon structures as defined by the community:

  * •

check / CheckNanopub reads a nanopublication or several of them and checks
whether any of the well-formedness criteria are violated. If a trusty URI or a
digital signature is found (see below), these are checked too.

  * •

mktrusty / MakeTrustyNanopub takes a nanopublication that does not yet have a
trusty URI and transforms it into one that is identified by a newly created
trusty URI.

  * •

fix / FixTrustyNanopub takes a nanopublication with a broken trusty URI and
fixes it, i.e. assigns it a new trusty URI. This is useful when a
nanopublication has to be changed, which invalidates the hash. Running this
command creates a _new_ nanopublication with a valid trusty URI.
(Nanopublications are immutable, so changing something necessarily leads to a
new nanopublication.)

In addition to these core features, the library also contains a number of
unofficial extensions (which may or may not become official at some point).
There is code to validate informal assertions specified as AIDA sentences [4];
code for creating index nanopublications to define small or large sets of
nanopublications [5]; code for publishing and retrieving nanopublications from
a decentralized server network [5]; and experimental code for digitally
signing nanopublications. These features are accessible via the following
commands and classes:

  * •

mkindex / MakeIndex takes a list of nanopublications and creates an index that
refers to them. A nanopublication index therefore represents a (possibly
large) set of nanopublications. Such indexes are themselves formatted as
nanopublications, and can therefore also be published to the server network
(see below).

  * •

publish / PublishNanopub uploads a given nanopublication that has a trusty
URIs to the server network. Such a nanopublication is then distributed among
the servers of the network (currently five) and made available even if some of
the servers should be inaccessible at a certain point in time. In this way,
the nanopublication is made permanent and its publication cannot be undone.

  * •

get / GetNanopub reliably retrieves a given nanopublication from the
decentralized server network. Nanopublications are verified according to their
trusty URI, and only verified nanopublications are returned by this command.
For nanopublication indexes, the whole set of nanopublications that is defined
by the index can be downloaded.

  * •

status / NanopubStatus checks whether and how often a given nanopublication
(identified by its trusty URI) is found on the server network.

  * •

server / GetServerInfo returns some information about a given server in the
network, such as the number of nanopublications it contains.

  * •

mkkeys / MakeKeys creates a new key-pair to be used to sign nanopublications.

  * •

sign / SignNanopub takes a nanopublication and signs it with a given private
key.

##  3 Examples

Below, some of the most important commands are explained by examples based on
the np command line tool. The same functionality is also available via the
Java API. For the sake of these examples, let us assume that we have a file
called nanopubfile.trig that starts with the following RDF prefixes:

⬇

{o}@prefix xsd: <http://www.w3.org/2001/XMLSchema#>.

{o}@prefix dc: <http://purl.org/dc/terms/>.

{o}@prefix pav: <http://purl.org/pav/>.

{o}@prefix prov: <http://www.w3.org/ns/prov#>.

{o}@prefix np: <http://www.nanopub.org/nschema#>.

{o}@prefix ex: <http://example.org/>.

{o}@prefix : <http://example.org/np1#>.{end}

The definition of the first nanopublication in this file starts with the head
graph that defines the structure of the nanopublication by linking to the
other graphs:

⬇

{o}:Head {

{o} : a np:Nanopublication; np:hasAssertion :assertion;

{o} np:hasProvenance :provenance; np:hasPublicationInfo :pubinfo.

{o}}{end}

The actual claim of the nanopublication is stored in the assertion graph:

⬇

{o}:assertion {

{o} ex:drugA ex:treats ex:diseaseB.

{o}}{end}

The provenance and publication info graphs provide meta-information about the
assertion and the entire nanopublication, respectively:

⬇

{o}:provenance {

{o} :assertion prov:wasDerivedFrom ex:some_publication.

{o}}

{o}:pubinfo {

{o} : pav:createdBy <http://orcid.org/0000-0002-1267-0234>.

{o} : dc:created "2015-08-18T15:36:22+01:00"^^xsd:dateTime.

{o}}{end}

The lines above constitute a very simple but complete nanopublication. To make
this example a bit more interesting, let us assume that our file contains two
more nanopublications that have different assertions but are otherwise
identical:

⬇

{o}@prefix : <http://example.org/np2#>.

{k}...

{o}:assertion {

{o} ex:Gene1 ex:isRelatedTo ex:diseaseB.

{o}}

{k}...

{o}

{o}@prefix : <http://example.org/np3#>.

{k}...

{o}:assertion {

{o} ex:Gene2 ex:isRelatedTo ex:diseaseB.

{o}}

{k}...{end}

To check and validate these three nanopublications, we can now use the
following command:

⬇

{c}$ np check nanopubfile.trig

{o}Summary: 3 valid (not trusty);{end}

These nanopublications can now be transformed into ones with trusty URIs using
the following command (resulting in a new file trusty.nanopubfile.trig):

⬇

{c}$ np mktrusty nanopubfile.trig{end}

Using the same command in verbose mode with the argument -v shows us the newly
generated trusty URIs for the three nanopublications:

⬇

{c}$ np mktrusty -v nanopubfile.trig

{o}Nanopub URI:
http://example.org/np1#RAHGB0WzgQijR88g_rIwtPCmzYgyO4wRMT7M91ouhojsQ

{o}Nanopub URI:
http://example.org/np2#RA4xTdhe2gPctqvAwdgTU4eRiR1aTQlTYJcF3Sohe5Cus

{o}Nanopub URI: http://example.org/np3#RAEjvXP0xTkeIa2mKmYT66i_PAJ-
u-k0uRBd6_sMe9qG0{end}

As they are tiny snippets of data, nanopublications are most useful when they
grouped and combined in small or large collections. We therefore need a simple
method to refer to collections or sets of nanopublications, which is achieved
by the experimental proposal of _nanopublication indexes_ [5], which are
themselves nanopublications. Such indexes can be used to group
nanopublications that have trusty URIs using the following command:

⬇

{c}$ np mkindex trusty.nanopubfile.trig

{o}Index URI:
http://np.inn.ac/RAFa_x4h0ng_NXtof35Ie9pQVsAY69Ab3ZQMir2NP8vGc{end}

The nanopublications of the new index are saved in a file called index.trig
unless specified otherwise with the argument -o.

Moving to the part that involves the server network, nanopublications that
have trusty URIs (which includes nanopublication indexes) can be published to
the network with the following command:

⬇

{c}$ np publish trusty.nanopubfile.trig

{o}3 nanopubs published at http://np.inn.ac/{end}

The publication status of a given nanopublication can be checked like this:

⬇

{c}$ np status -a
http://example.org/np1#RAHGB0WzgQijR88g_rIwtPCmzYgyO4wRMT7M91ouhojsQ

{o}URL: http://np.inn.ac/RAHGB0WzgQijR88g_rIwtPCmzYgyO4wRMT7M91ouhojsQ

{o}URL: http://ristretto.med.yale.edu:8080/nanopub-
server/RAHGB0WzgQijR88g_rIwtPCmzYgyO{k}...

{o}URL: http://nanopub-
server.ops.labs.vu.nl/RAHGB0WzgQijR88g_rIwtPCmzYgyO4wRMT7M91ouhojsQ

{o}URL: http://nanopubs.stanford.edu/nanopub-
server/RAHGB0WzgQijR88g_rIwtPCmzYgyO4wRMT7{k}...

{o}URL:
http://nanopubs.semanticscience.org/RAHGB0WzgQijR88g_rIwtPCmzYgyO4wRMT7M91ouhojsQ

{o}Found on 5 nanopub servers.{end}

A given nanopublication that is published on the server network can be
retrieved via its URI:

⬇

{c}$ np get
http://www.tkuhn.ch/bel2nanopub/RAhV9IpiUEjbentzGivp1Lbx0BVegp5sgE3BwS0S2RAYM{end}

All the servers in the network are checked until the nanopublication is found
and successfully verified. This command is therefore reliable even if one or
several servers are down. Instead of the complete URI, it is also possible to
just specify the trusty URI artifact code:

⬇

{c}$ np get RAhV9IpiUEjbentzGivp1Lbx0BVegp5sgE3BwS0S2RAYM{end}

To get the content of a nanopublication index (and not just the top-most index
nanopublication), argument -c can be used:

⬇

{c}$ np get -c -o content.trig
RAtF0ivB9B8cb-u3K_zElgmRBxiDwfym1yVBRY6VAyWvE{end}

Argument -o specifies again the name of the output file. The remaining
commands as introduced above are equally intuitive to use. Just entering the
command without any arguments will output a list of all argument options.

##  4 Web Interface

Many of the features described above are made available through the
nanopublication validator Web interface,444https://github.com/tkuhn/nanopub-
validator an instance of which can be accessed at http://nanopub.inn.ac.
Figure 1 shows a screenshot. With this interface, the well-formedness of
nanopublications can be checked as well as the adherance to a number of
patterns. They can furthermore be transformed into different RDF
serializations, and published to the server network. Loading of
nanopublications is possible via form input, upload, fetching from a URL,
SPARQL endpoint access, and retrieval from the server network. In general,
this web interface and the underlying library are supposed to support the
development of best practices for the nanopublication community by providing a
solid basis for discussion, by allowing for experimental features to be tested
and discussed, and by facilitating the implementation of prototypes.

The current server network on which many of the unofficial features depend,
can be explored via a monitor interface at http://npmonitor.inn.ac.

Figure 1: This is a screenshot of the nanopublication validator interface at
http://nanopub.inn.ac.

##  5 Conclusions

The nanopub-java library provides a stable implementation of the
nanopublication concept, adhering to its specified guidelines. It is openly
licensed under the terms of the MIT license, is available on The Central
Repository,555https://search.maven.org/#artifactdetails|org.nanopub|nanopub|1.7|jar
and has so far been used in about a dozen open-source
codebases.666https://github.com/Nanopublication/nanopub-java#usage-tracking

In general, we believe that this library can be a valuable resource for tools
that use RDF data in the context of provenance recording, reproducibility,
data publishing, data reuse, and reliable retrieval of Linked Data.

## References

  * [1] J. Broekstra, A. Kampman, and F. Van Harmelen.  Sesame: A generic architecture for storing and querying rdf and rdf schema.  In The Semantic Web — ISWC 2002, pages 54–68. Springer, 2002\.
  * [2] P. Groth, A. Gibson, and J. Velterop.  The anatomy of a nano-publication.  Information Services and Use, 30(1):51–56, 2010.
  * [3] T. Kauppinen, A. Baglatzi, and C. Keßler.  Linked science: interconnecting scientific assets.  Data Intensive Science, 2012.
  * [4] T. Kuhn, P. E. Barbano, M. L. Nagy, and M. Krauthammer.  Broadening the scope of nanopublications.  In The Semantic Web: Semantics and Big Data — ESWC 2013, pages 487–501. Springer, 2013.
  * [5] T. Kuhn, C. Chichester, M. Krauthammer, and M. Dumontier.  Publishing without publishers: a decentralized approach to dissemination, retrieval, and archiving of data.  In The Semantic Web — ISWC 2015. Springer, 2015.
  * [6] T. Kuhn and M. Dumontier.  Trusty URIs: Verifiable, immutable, and permanent digital artifacts for linked data.  In The Semantic Web: Trends and Challenges — ESWC 2014, pages 395–410. Springer, 2014.
  * [7] T. Kuhn and M. Dumontier.  Making digital artifacts on the web verifiable and reliable.  IEEE Transactions on Knowledge and Data Engineering, 27(9), 2015\.
  * [8] B. Mons, H. van Haagen, C. Chichester, J. T. den Dunnen, G. van Ommen, E. van Mulligen, B. Singh, R. Hooft, M. Roos, J. Hammond, et al.  The value of data.  Nature genetics, 43(4):281–283, 2011.
  * [9] D. Shotton.  Semantic publishing: the coming revolution in scientific journal publishing.  Learned Publishing, 22(2):85–94, 2009.


KOBE-TH-13-1
HRI-P-13-01-001
RECAPP-HRI-2013-001

$CP$ phase from twisted Higgs
vacuum expectation value in extra dimension

Yukihiro Fujimoto, aaa E-mail<EMAIL_ADDRESS>Kenji Nishiwaki,
bbb E-mail<EMAIL_ADDRESS>and Makoto Sakamoto ccc E-mail:
<EMAIL_ADDRESS>
a,cDepartment of Physics, Kobe University, Kobe 657-8501, Japan
bRegional Centre for Accelerator-based Particle Physics,
Harish-Chandra Research Institute, Allahabad 211 019, India
2 September 2024

We propose a new mechanism for generating a $CP$ phase via Higgs a vacuum
expectation value originating from geometry of an extra dimension. A twisted
boundary condition is the key to produce an extra-dimension coordinate-
dependent vacuum expectation value, which contains a $CP$ phase degree of
freedom and can be a new source of a $CP$ phase in higher-dimensional gauge
theories. As an illustrative example, we apply our mechanism to a five-
dimensional gauge theory with point interactions and show that our mechanism
can dynamically produce a nontrivial $CP$-violating phase with electroweak
symmetry breaking, even though the five-dimensional model does not include any
$CP$-violating phases of Yukawa couplings in the five-dimensional Lagrangian
because of a single generation of five-dimensional fermions. We apply our
mechanism to a model with point interactions, which has no source of
$CP$-violating phases in the couplings of the higher-dimensional action, and
show that a nontrivial $CP$ phase dynamically appears.

##  1 Introduction

Pursuing the origin of the generations of the fermions is one of the important
themes in particle physics. The three generations (or more) are necessary to
produce the Kobayashi–Maskawa $CP$ phase, which causes $CP$-violating effects
and was proposed in Ref. [1]. If the number of generations were less than $3$,
any complex phases in the Cabbibo–Kobayashi–Maskawa (CKM) matrix would be
absorbed into phases of quark fields and then the Kobayashi-Maskawa
$CP$-violating mechanism would not work.

Extra-dimensional field theory is one of the appealing candidates beyond the
standard model (SM). Many studies have been done up to today based on many
ideas for pursuing the origin of fermion flavor [2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18]. Especially, we can find several attractive
models to solve the generation problem, in which the three generations of the
four-dimensional chiral fermions are dynamically realized from a single
generation of higher-dimensional fermions. However, such models have a common
problem: The number of higher-dimensional Yukawa couplings is not enough to
produce a $CP$-violating phase because of a single generation of fermions in
higher-dimensions, so we could not obtain a $CP$-violating phase à la
Kobayashi-Maskawa. Therefore, in models to solve the generation problem, we
need some new sources of $CP$-violating phases other than higher-dimensional
Yukawa couplings. Otherwise, those models without a $CP$-violating phase
should be discarded as phenomenological ones.111 In the gauge-Higgs
unification model, a similar problem arises because of lack of degree of
freedom in the Yukawa sector of an original five-dimensional action. The ways
to overcome this point have been studied [19, 20, 21].

In this paper, we propose a new mechanism to produce a $CP$ phase in the
context of five-dimensional gauge theories. Allowing a twisted boundary
condition (BC) for the Higgs doublet leads to a Higgs vacuum expectation value
(VEV) with an extra-dimension coordinate-dependent phase, which contains a
$CP$ phase degree of freedom. The properties of such kinds of scalar VEVs have
been studied in Refs. [22, 23, 24, 25, 26, 27, 28, 29, 30, 31].222 Point
interactions on $S^{1}$, which are additional boundary points (on $S^{1}$),
have been studied in Refs [32, 33, 34, 35, 36]. We can consider another
possibility that some terms are localized in boundary points at tree level
[37, 38, 39, 40].  We note that the electroweak symmetry is dynamically broken
at that time.

As a demonstration of our mechanism, we apply the mechanism to a five-
dimensional gauge theory with point interactions in which three generations in
four dimensions are produced from a single generation in five dimensions. We
show that a nontrivial CP phase dynamically appears in the CKM matrix in four
dimensions, even though any coupling constants in the five-dimensional
Lagrangian have no $CP$ phases. Our purpose of this paper is to show that our
mechanism does work as a new source of the $CP$ violation.

This paper is organized as follows. In Sec. 2, we discuss and verify a
possibility of the Higgs doublet with a twisted BC to explain the origin of
the $CP$ phase in the CKM matrix. In Sec. 3, we construct a model with point
interactions and a scalar singlets for which the VEV depends on the extra
coordinate exponentially. In Sec. 4, we check that the CP phase originating
from our mechanism can explain the CKM properties in the above model. Here, we
also discuss the properties of the realized quark masses and other mixings
briefly. In Sec. 5, we summarize our results and discuss some aspects of our
model. In the Appendix, details of choosing parameters is explored.

##  2 Position-dependent VEV (also as $CP$ phase) with twisted boundary
condition

In this section, we propose a new mechanism for generating $CP$ phase with
twisted boundary condition of a five-dimensional scalar $H$ on $S^{1}$.
Hereafter, we use a coordinate $y$ to indicate the position in the extra
space. A key aspect is that broken phase can be realized with the scalar, and
at the same time, the VEV profile itself turns out to be $y$-position
dependent and complex, which means that the scalar VEV possibly triggers the
$CP$ violation. Interestingly, the $y$-position dependence disappears in the
gauge boson masses, even though the VEV of $H$ depends on $y$. This is because
the $y$ dependence of the VEV of $H$ is cancelled out in the squared form
$H^{\dagger}H$. This property is very important and it works as an usual four-
dimensional Higgs mechanism without violating electroweak precision
measurements at the tree level. When we consider the situation that $H$ is the
$SU(2)_{W}$ Higgs doublet, we can dynamically generate both the suitable
electroweak symmetry breaking (EWSB) and the $CP$-violating phase
simultaneously. We note that its $SU(N)$ extension is possible and
straightforward.

The action we consider is

$\displaystyle S_{H}$ $\displaystyle=\int
d^{4}x\int_{0}^{L}dy\bigg{\\{}{H^{\dagger}(\partial_{M}\partial^{M}+{M}^{2})H-\frac{\lambda}{2}(H^{\dagger}H)^{2}}\bigg{\\}},$
(2.1)

where $M$ and $\lambda$ are the bulk mass and quartic coupling, respectively.
Since $S^{1}$ is a multiply connected space, we can impose the twisted
boundary condition on $H$ as [22, 23, 24, 25, 26]

$\displaystyle H(y+L)=e^{i\theta}H(y).$ (2.2)

Here, we take the range of $\theta$ as $-\pi<\theta\leq\pi$. $L$ shows the
circumference of $S^{1}$, and we choose the metric convention as
$\eta_{MN}=\eta^{MN}=\text{diag}(-1,1,1,1,1).$ The Latin indices run from $0$
to $3,\,5\ (\text{or\ }y)$, and Greek ones run from $0$ to $3$, respectively.

We note that the VEV of $\langle H(y)\rangle$ should be determined by
minimizing the functional

$\displaystyle\mathcal{E}[H]=\int_{0}^{L}dy\bigg{\\{}{\left|\partial_{y}H\right|^{2}-{M}^{2}\left|H\right|^{2}+\frac{\lambda}{2}\left|H\right|^{4}}\bigg{\\}}{,}$
(2.3)

because the VEV can possess the $y$ dependence to minimize the energy. Here,
we assume that the four-dimensional (4D) Lorentz invariance is unbroken.

After introducing $\mathcal{H}(y)$ by

$\displaystyle{H(y)=e^{i{\frac{\theta}{L}y}}\mathcal{H}(y),\quad\mathcal{H}(y+L)=\mathcal{H}(y),}$
(2.4)

the functional $\mathcal{E}$ can be rewritten as

$\displaystyle\mathcal{E}[H]$
$\displaystyle=\mathcal{E}_{1}[\mathcal{H}]+\mathcal{E}_{2}[\mathcal{H}],$
(2.5) $\displaystyle\mathcal{E}_{1}[\mathcal{H}]$
$\displaystyle=\int_{0}^{L}dy\bigg{\\{}{\left|\partial_{y}\mathcal{H}\right|^{2}}+i\frac{\theta}{L}\Big{(}(\partial_{y}\mathcal{H})^{\dagger}\mathcal{H}-\mathcal{H}^{\dagger}\partial_{y}\mathcal{H}\Big{)}\bigg{\\}},$
(2.6) $\displaystyle\mathcal{E}_{2}[\mathcal{H}]$
$\displaystyle=\int_{0}^{L}dy\bigg{\\{}{\frac{\lambda}{2}\left(\left|\mathcal{H}\right|^{2}-\frac{1}{\lambda}\left[{M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right]\right)^{2}-\frac{1}{2\lambda}\left[{M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right]^{2}}\bigg{\\}},$
(2.7)

where ${\cal E}_{1}$ corresponds to the contribution from the $y$-kinetic term
of ${\cal H}$.

Since $\mathcal{H}(y)$ satisfies the periodic boundary condition,
$\mathcal{H}(y)$ can be decomposed as

$\displaystyle{\mathcal{H}(y)=\sum_{n=-\infty}^{\infty}\frac{\vec{a}_{n}}{\sqrt{L}}e^{i2\pi
n\frac{y}{L}},}$ (2.8)

where $\vec{a}_{n}$ is a two-component $SU(2)_{W}$ constant vector.
Substituting Eq. (2.8) into $\mathcal{E}_{1}$, we obtain the expression

$\displaystyle\mathcal{E}_{1}=\sum_{n=-\infty}^{\infty}\left[\left(\frac{2\pi
n+\theta}{L}\right)^{2}-\left(\frac{\theta}{L}\right)^{2}\right]{\left|\vec{a}_{n}\right|^{2}\geq
0}$ (2.9)

and we can conclude that the minimum of $\mathcal{E}_{1}$ is given by
$\mathcal{E}_{1}=0$ when the values of $\theta$ and $\vec{a}_{n}$ satisfy one
of the conditions

(i) $\displaystyle{-\pi<\theta<\pi\text{\ and\ }{\vec{a}_{n}}=0\,(n\not=0):}$
$\displaystyle\mathcal{H}=\frac{{\vec{a}_{0}}}{\sqrt{L}},$ (2.10) (ii)
$\displaystyle\theta=\pi\text{\ and\ }{\vec{a}_{n}}=0\,(n\not=0,-1):$
$\displaystyle{\mathcal{H}=\frac{\vec{a}_{0}}{\sqrt{L}}\ \ \text{or}\ \
\mathcal{H}=\frac{{\vec{a}_{-1}}}{\sqrt{L}}e^{-i2\pi\frac{y}{L}},}$ (2.11)

where $\vec{a}_{0}$ in Eq. (2.10) and $\vec{a}_{0}$ or $\vec{a}_{-1}$ in Eq.
(2.11) are still undetermined. The functional $\mathcal{E}_{2}$ takes the
minimum value if the following condition is fulfilled:

$\displaystyle\left|\mathcal{H}\right|^{2}=\begin{cases}{\frac{1}{\lambda}\left({M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right)}&\text{for\
}{{M}^{2}}-\left(\frac{\theta}{L}\right)^{2}>0\\\ 0&\text{for\
}{{M}^{2}}-\left(\frac{\theta}{L}\right)^{2}\leq 0\end{cases}.$ (2.12)

Combining the above two results and using the $SU(2)_{W}$ global symmetry, we
can show that the VEV $\langle H(y)\rangle$ is given, without loss of
generality, as

(I)

${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$

$\displaystyle\left\langle
H(y)\right\rangle=\begin{cases}\frac{v}{\sqrt{2}}\,e^{i{\frac{\theta}{L}y}}\begin{pmatrix}0\\\
1\end{pmatrix}&\text{for\ }{-\pi<\theta<\pi},\\\
\frac{v}{\sqrt{2}}\,e^{i{\frac{{\pi}}{L}y}}\begin{pmatrix}0\\\
1\end{pmatrix}\text{\ \ or\ \
}\frac{v}{\sqrt{2}}\,e^{{-i\frac{{\pi}}{L}y}}\begin{pmatrix}0\\\
1\end{pmatrix}&\text{for\ }{\theta=\pi},\\\ \end{cases}$ (2.13)

(II)

${M}^{2}-\left(\frac{\theta}{L}\right)^{2}\leq 0$

$\displaystyle\left\langle H(y)\right\rangle=\begin{pmatrix}0\\\
0\end{pmatrix}{,}$ (2.14)

where $v$ is given by

$\displaystyle\left(\frac{v}{\sqrt{2}}\right)^{2}:={\left|\left\langle
H(y)\right\rangle\right|^{2}}={\frac{1}{\lambda}\left({M}^{2}-\left(\frac{\theta}{L}\right)^{2}\right)}.$
(2.15)

From now on, we will assume the case of (I)
${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$.

Now we discuss some properties of the derived VEV in Eq. (2.13). Differently
from the SM, the VEV possesses $y$-position dependence, and its broken phase
is realized only in the case of ${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$.
But like the SM, the squared VEV (2.15) is still constant even though $\langle
H(y)\rangle$ depends on $y$. This means that after $v\sqrt{L}$ is set as
$246\,\text{GeV}$, where the mass dimension of $v$ is $3/2$, the same
situation as the SM occurs in the EWSB sector. On the other hand, the $y$
dependence of the Higgs VEV in Eq. (2.13) is an important consequence for the
Yukawa sector. Since the VEV of the Higgs doublet appears linearly in each
Yukawa term, the overlap integrals which lead to effective 4D Yukawa couplings
will produce a nontrivial $CP$ phase in the CKM matrix.

In terms of the VEV and physical Higgs modes $h^{(n)}(x)$, $H$ can be expanded
as

$\displaystyle H(x,y)\rightarrow\sum_{n=-\infty}^{\infty}\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}\left(ve^{i\frac{\theta}{L}y}\delta_{n,0}+h^{(n)}(x)\frac{1}{\sqrt{L}}e^{i\left(\frac{2\pi
n+\theta}{L}\right)y}\right)\end{pmatrix},$ (2.16)

which obeys the boundary condition (2.2). The physical masses $\mu_{h^{(n)}}$
of the zero mode $(n=0)$ and the Kaluza–Klein (KK) modes $(n\not=0)$ are
easily calculated from Eq. (2.1) as

$\displaystyle\mu_{h^{(n)}}^{2}=\begin{cases}2\left(M^{2}-\left(\frac{\theta}{L}\right)^{2}\right)=\lambda
v^{2}&\text{for\ }n=0\\\ {2{M}^{2}+\frac{(\theta+2\pi
n)^{2}}{2L^{2}}+\frac{(\theta-2\pi
n)^{2}}{2L^{2}}-3\left(\frac{\theta}{L}\right)^{2}={\lambda v^{2}}+\frac{(2\pi
n)^{2}}{L^{2}}}&\text{for\ }{n\geq 1}\end{cases},$ (2.17)

with the hermiticity condition for a real field on $S^{1}$:
$h^{(n)\dagger}=h^{(-n)}$.

We mention that the relation between $\mu_{h^{(n)}}$ and $\lambda$ for $n=0$
in Eq. (2.17) is totally the same as that of the standard model. We also
comment on the Higgs-quarks couplings in our model. As shown in Eq. (2.16),
the profiles of the VEV and the Higgs physical zero mode are the same as
$e^{i\frac{\theta}{L}y}$ up to the coefficients. This means that the strengths
of the couplings are equivalent to those of the SM even though the mode
function gets to be $y$-position dependent. As a result, the decay branching
ratios of the Higgs boson are the same as those of the standard model.333Being
different from the universal extra dimension case [41, 42, 43, 44, 45], the
“low” KK mass less than a TeV scale is not allowed after considering the level
mixing in the top sector [46]. Then, the significant deviations do not occur
in the loop-induced single Higgs production via gluon fusion and Higgs decay
processes to a pair of photons and gluons in our model.

##  3 Model with point interactions on $S^{1}$

Figure 1:  The wave function profiles of the quarks and the VEV of $\Phi(y)$
are schematically depicted. Here we take $L_{0}^{(q)}=L_{0}^{(\Phi)}=0$. Note
that all the profiles have the periodicity along $y$ with the same period $L$.
Differently from the model on an interval in Ref. [46], we can find the
$(1,3)$ elements of the mass matrices due to the periodicity along the $y$
direction.  Figure 2:  This is an overview of our system with point
interactions. The red (blue) circular spots show the Dirichlet BC for left-
(right-)handed part at the corresponding boundary points, respectively. The
green, purple, and black circular spots represent the ordinary periodic in Eq.
(3.19), the Robin BCs in Eq. (3.16), and the twisted BCs in Eq. (2.2),
respectively. It is noted that we adopt the assumption in Eq. (4.3).

In the previous section, we introduced the twisted BC for the $SU(2)_{W}$
doublet $H$ and generated the EWSB by the $y$-position-dependent complex VEV
in Eq. (2.13) in the case of $M^{2}-(\theta/L)^{2}>0$. We expect that this VEV
also works as the source of the $CP$ phase of the CKM matrix, but here an
important issue, which we should think about carefully, exists.

If all the profiles of the three-generation quarks are flat, an effective
phase appears after integration over $y$ just as an overall factor, which can
be removed by $U(1)$ rephasing and never works as a physical $CP$ phase. To
circumvent this difficulty, profiles of the quarks are required to be
localized. On the other hand, field localization (in extra dimensions) is
known as an effective way of explaining the quark mass hierarchy and pattern
of flavor mixing. In this section, we consider a model with point interactions
as an illustrative example. Point interaction can be considered as zero-
thickness brane and we can arrange it anywhere in the bulk space of $S^{1}$.

At the location of a point interaction, we can consider five-dimensional (5D)
gauge-invariant boundary conditions, for which the variety is rich compared
with the case of $Z_{2}$ orbifolding. After we introduce three point
interactions for a 5D fermion, its zero-mode profile gets to be chiral, split
and localized. This situation is just what we want.444 Another interesting
idea for generating three-generation structure and field localization is
introducing magnetic flux on the torus [47, 48, 49].  We emphasize that flavor
mixing is naturally realized as overlapping of localized quark profiles. In
the model, an additional gauge singlet scalar is required for generating the
large mass hierarchy of the quarks. Its (almost) exponential shape in the VEV
is also generated by a suitable boundary condition at the corresponding point
interactions.

This basic idea is found in Ref. [46]. Nevertheless, there are two different
points between the models in this paper and in Ref. [46]:

  * •

In the previous model [46], the Higgs VEV cannot possess a nontrivial complex
phase, and a $CP$ phase in the CKM matrix has not been realized. On the other
hand, the VEV in our present model has a $y$-position-dependent complex phase,
which will produce a $CP$ phase of the CKM matrix.

  * •

In the previous model [46], the extra dimension has been taken to be an
interval, where the twisted BC in Eq. (2.2) cannot be realized. In the present
model, we set the extra dimension to be a circle $S^{1}$, for which the
geometry is compatible with the twisted BC (2.2).

In the following part, we briefly explain how to construct our model. The 5D
action for fermions is given by555 We adopt the representations of the gamma
matrices as $\Gamma_{\mu}=\gamma_{\mu},\
\Gamma_{y}=\Gamma^{y}=-i\gamma^{5}=\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$
and the Clifford algebra is defined as
$\left\\{\Gamma_{M},\Gamma_{N}\right\\}=-2\eta_{MN}.$

$\displaystyle S$ $\displaystyle=\int
d^{4}x\int_{0}^{L}dy\Bigg{\\{}\Big{[}\overline{Q}\left(i\partial_{M}\Gamma^{M}+M_{Q}\right)Q+\overline{\mathcal{U}}\left(i\partial_{M}\Gamma^{M}+M_{\mathcal{U}}\right)\mathcal{U}$
$\displaystyle\phantom{=\int d^{4}x\int_{0}^{L_{3}}\Bigg{\\{}\
\,}+\overline{\mathcal{D}}\left(i\partial_{M}\Gamma^{M}+M_{\mathcal{D}}\right)\mathcal{D}\Big{]}\Bigg{\\}},$
(3.1)

where we introduce an $SU(2)_{W}$ doublet ($Q$), an up-quark singlet
($\mathcal{U}$), and a down-type singlet ($\mathcal{D}$) with the
corresponding bulk masses ($M_{Q},M_{\mathcal{U}},M_{\mathcal{D}}$). We note
that our model contains only one generation for 5D quarks but each 5D quark
produces three generations of the 4D quarks, as we will see below.

We adopt the following BCs for $Q,\mathcal{U},\mathcal{D}$ with an
infinitesimal positive constant $\varepsilon$ [46]:

$\displaystyle Q_{R}$ $\displaystyle=0\qquad\text{at}\quad
y=L_{0}^{(q)}+\varepsilon,\,L_{1}^{(q)}\pm\varepsilon,\,L_{2}^{(q)}\pm\varepsilon,\,L_{3}^{(q)}-\varepsilon,$
(3.2) $\displaystyle\mathcal{U}_{L}$ $\displaystyle=0\qquad\text{at}\quad
y=L_{0}^{(u)}+\varepsilon,\,L_{1}^{(u)}\pm\varepsilon,\,L_{2}^{(u)}\pm\varepsilon,\,L_{3}^{(u)}-\varepsilon,$
(3.3) $\displaystyle\mathcal{D}_{L}$ $\displaystyle=0\qquad\text{at}\quad
y=L_{0}^{(d)}+\varepsilon,\,L_{1}^{(d)}\pm\varepsilon,\,L_{2}^{(d)}\pm\varepsilon,\,L_{3}^{(d)}-\varepsilon,$
(3.4)

where $\Psi_{R}$ and $\Psi_{L}$ denote the eigenstates of $\gamma^{5}$, i.e.,
$\Psi_{R}\equiv\frac{1+\gamma^{5}}{2}\Psi$ and
$\Psi_{L}\equiv\frac{1-\gamma^{5}}{2}\Psi$. Here $L^{(i)}_{j}$ for $i=q,u,d$
and $j=0,1,2,3$ means the positions of point interactions for the 5D fermions.
See Figs. 1 and 2 for details. A crucial consequence of the above BCs is that
there appear threefold degenerated left- (right-)handed zero modes in the mode
expansions of $Q$ ($\mathcal{U},\mathcal{D}$) and that they form the three
generations of the quarks. The details have been given in Ref. [46]. We will
not repeat the discussions here.

The fields $Q,\mathcal{U},\mathcal{D}$ with the BCs in Eqs (3.2)–(3.4) are KK
decomposed as follows:

$\displaystyle Q(x,y)=\begin{pmatrix}U(x,y)\\\ D(x,y)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\sum_{i=1}^{3}u^{(0)}_{iL}(x)f_{q^{(0)}_{iL}}(y)\\\
\sum_{i=1}^{3}d^{(0)}_{iL}(x)f_{q^{(0)}_{iL}}(y)\end{pmatrix}+(\text{KK
modes}),$ (3.5) $\displaystyle\mathcal{U}(x,y)$
$\displaystyle=\sum_{i=1}^{3}u^{(0)}_{iR}(x)f_{u^{(0)}_{iR}}(y)+(\text{KK
modes}),$ (3.6) $\displaystyle\mathcal{D}(x,y)$
$\displaystyle=\sum_{i=1}^{3}d^{(0)}_{iR}(x)f_{d^{(0)}_{iR}}(y)+(\text{KK
modes}).$ (3.7)

Here the zero-mode functions are obtained in the following forms:

$\displaystyle f_{q^{(0)}_{iL}}(y)$
$\displaystyle=\mathcal{N}_{i}^{(q)}e^{M_{Q}(y-L_{i-1}^{(q)})}\Big{[}\theta(y-L_{i-1}^{(q)})\theta(L_{i}^{(q)}-y)\Big{]}$
$\displaystyle\text{in\ }[L_{0}^{(q)},L_{3}^{(q)}],$ (3.8) $\displaystyle
f_{u^{(0)}_{iR}}(y)$
$\displaystyle=\mathcal{N}_{i}^{(u)}e^{-M_{\mathcal{U}}(y-L_{i-1}^{(u)})}\Big{[}\theta(y-L_{i-1}^{(u)})\theta(L_{i}^{(u)}-y)\Big{]}$
$\displaystyle\text{in\ }[L_{0}^{(u)},L_{3}^{(u)}],$ (3.9) $\displaystyle
f_{d^{(0)}_{iR}}(y)$
$\displaystyle=\mathcal{N}_{i}^{(d)}e^{-M_{\mathcal{D}}(y-L_{i-1}^{(d)})}\Big{[}\theta(y-L_{i-1}^{(d)})\theta(L_{i}^{(d)}-y)\Big{]}$
$\displaystyle\text{in\ }[L_{0}^{(d)},L_{3}^{(d)}],$ (3.10)

where

$\displaystyle\Delta L^{(l)}_{i}$
$\displaystyle=L^{(l)}_{i}-L^{(l)}_{i-1}\qquad(\text{for }i=1,2,3;\ l=q,u,d),$
(3.11) $\displaystyle\mathcal{N}_{i}^{(q)}=\sqrt{\frac{2M_{Q}}{e^{2M_{Q}\Delta
L_{i}^{(q)}}-1}},\quad\mathcal{N}_{i}^{(u)}=\sqrt{\frac{2M_{\mathcal{U}}}{1-e^{-2M_{\mathcal{U}}\Delta
L_{i}^{(u)}}}},\quad\mathcal{N}_{i}^{(d)}=\sqrt{\frac{2M_{\mathcal{D}}}{1-e^{-2M_{\mathcal{D}}\Delta
L_{i}^{(d)}}}}.$ (3.12)

$\mathcal{N}_{i}^{(q)},\mathcal{N}_{i}^{(u)},\mathcal{N}_{i}^{(d)}$ are the
wave function normalization factors for
$f_{q^{(0)}_{iL}},f_{u^{(0)}_{iL}},f_{d^{(0)}_{iL}}$, respectively.

Since the length of the total system is universal, $L^{(l)}_{3}-L^{(l)}_{0}\
(l=q,u,d)$ should be equal to the circumference of $S^{1}$, i.e.

$\displaystyle
L:=L^{(q)}_{3}-L^{(q)}_{0}=L^{(u)}_{3}-L^{(u)}_{0}=L^{(d)}_{3}-L^{(d)}_{0}.$
(3.13)

Note that all the mode functions in Eqs. (3.8)–(3.10) (and a form of a singlet
VEV in Eq. (3.17)) are periodic with the common period $L$, whereas we do not
indicate that thing explicitly in Eqs. (3.8)–(3.10).

In this model, the large mass hierarchy is naturally explained with the Yukawa
sector

$\displaystyle S_{\mathcal{Y}}=\int
d^{4}x\int_{0}^{{L}}dy\bigg{\\{}\Phi\Big{[}-\mathcal{Y}^{(u)}\overline{Q}(i\sigma_{2}H^{\ast})\mathcal{U}-\mathcal{Y}^{(d)}\overline{Q}H\mathcal{D}\Big{]}+\text{h.c.}\bigg{\\}},$
(3.14)

where $\mathcal{Y}^{(u)}/\mathcal{Y}^{(d)}$ is the Yukawa coupling for
up-/down-type quark; $H$ and $\Phi$ are an $SU(2)_{W}$ scalar doublet and a
singlet. It should be noted that although the Yukawa couplings
$\mathcal{Y}^{(u)}$ and $\mathcal{Y}^{(d)}$ can be complex, they cannot be an
origin of the $CP$ phase of the CKM matrix because our model contains only a
single quark generation so that the number of the 5D Yukawa couplings is not
enough to produce a $CP$ phase in the CKM matrix. An outline of our system is
depicted in Fig. 1. Note that the five terms of
$\overline{Q}(i\sigma_{2}H^{\ast})\mathcal{U},\overline{Q}H\mathcal{D},\Phi\overline{Q}Q,\Phi\overline{\mathcal{U}}\mathcal{U},\Phi\overline{\mathcal{D}}\mathcal{D}$
with the Pauli matrix $\sigma_{2}$ are excluded by introducing a discrete
symmetry $H\rightarrow-H,\Phi\rightarrow-\Phi$. $\Phi$ is a gauge singlet and
there is no problem with gauge universality violation.666 If there exists the
doublet-singlet mixing term $-CH^{\dagger}H\Phi^{\dagger}\Phi$ with a
coefficient $C$, which cannot be prohibited by the discrete symmetry
$H\to-H,\Phi\to-\Phi$ in our theory, gauge universality violation should be
revisited. A bound from the universality in $Z$ boson gauge couplings was
already calculated as $CL\lesssim 0.003$ (when a KK scale is around a few TeV)
in a model on an interval [46]. In this paper, we simply ignore this term.

The 5D action and the BCs for $\Phi$ are assumed to be of the form [46, 50]

$\displaystyle S_{\Phi}=\int
d^{4}x\int_{0}^{{L}}dy\bigg{\\{}{\Phi^{\dagger}\left(\partial_{M}\partial^{M}-{M_{\Phi}}^{2}\right)\Phi-\frac{\lambda_{\Phi}}{2}\left(\Phi^{\dagger}\Phi\right)^{2}}\bigg{\\}},$
(3.15) $\displaystyle\Phi+L_{+}\partial_{y}\Phi$
$\displaystyle=0\qquad\text{at}\quad y=L_{0}^{(\Phi)}+\varepsilon,$
$\displaystyle\Phi-L_{-}\partial_{y}\Phi$ $\displaystyle=0\qquad\text{at}\quad
y=L_{3}^{(\Phi)}-\varepsilon,$ (3.16)

where $M_{\Phi}$ ($\lambda_{\Phi}$) is the bulk mass (quartic coupling) of the
scalar singlet $\Phi$ and $L_{\pm}$ can take values in the range of
$-\infty\leq L_{\pm}\leq\infty$ and $L_{0}^{(\Phi)}$ and $L_{3}^{(\Phi)}$
indicate the locations of the two “end points” of the singlet.

The VEV of $\Phi$ with the BCs, named Robin BCs, in Eq. (3.16) is expressed in
terms of Jacobi’s elliptic functions in general and its phase structure has
been discussed in Ref [50]. We adopt a specific form in the region
$[L_{0}^{(\Phi)}+\varepsilon,L_{3}^{(\Phi)}-\varepsilon]$ [46],

$\displaystyle\left\langle\Phi(y)\right\rangle=\left[{\frac{M_{\Phi}}{\sqrt{\lambda_{\Phi}}}}\left\\{\sqrt{1+X}-1\right\\}^{1/2}\right]\times\frac{1}{\text{cn}\left({M_{\Phi}}\left\\{1+X\right\\}^{1/4}(y-y_{0}),\sqrt{\frac{1}{2}\left(1+\frac{1}{\sqrt{1+X}}\right)}\right)},$
(3.17)

with

$\displaystyle X:={\frac{4\lambda_{\Phi}|Q|}{M^{4}_{\Phi}}}.$ (3.18)

Here $y_{0}$ and $Q$ are parameters which appear after integration on $y$ and
we focus on the choice of $Q<0$. We note that the values of $y_{0}$ and $Q$
are automatically determined after choosing those of $L_{\pm}$. As shown in
Ref. [46], we get the form of $\left\langle\Phi(y)\right\rangle$ to be an
(almost) exponential function of $y$ by choosing suitable parameter
configurations. Although there is a discontinuity in the wave function profile
of $\langle\Phi\rangle$ between $y=L_{0}^{(\Phi)}+\varepsilon$ and
$y=L_{3}^{(\Phi)}-\varepsilon$ in Eqs. (3.16), this type of BC is derived from
the variational principle on $S^{1}$ and leads to no inconsistency [50].

The BCs for the 5D $SU(3)_{C},SU(2)_{W},U(1)_{Y}$ gauge bosons
$G_{M},W_{M},B_{M}$ are selected as

$\displaystyle{G_{M}|_{y=0}=G_{M}|_{y=L},\quad\partial_{y}G_{M}|_{y=0}=\partial_{y}G_{M}|_{y=L},}$
(3.19)

where we only show the $G_{M}$’s case. In this configuration, we obtain the
standard model gauge bosons in zero modes. Based on the discussion in Sec. 2,
we conclude that the W and Z bosons become massive and their masses are
suitably created through “our” Higgs mechanism as $m_{W}\simeq
81\,\text{GeV},m_{Z}\simeq 90\,\text{GeV}$. The overview of the BCs is
summarized in Fig. 2. We mention that, on $S^{1}$ geometry, $G^{(0)}_{y}$,
$W^{(0)}_{y}$, and $B^{(0)}_{y}$ would exist as massless 4D scalars at the
tree level, but they will become massive via quantum corrections and are
expected to be uplifted to near KK states. We will discuss those modes in
another paper. We should note that in our model on $S^{1}$ with point
interactions, the 5D gauge symmetries are intact under the BCs summarized in
Fig. 2.777 In Refs. [51, 52, 53], the 5D gauge invariance has been discussed
from a quantum mechanical supersymmetry point of view.  Hence the unitarity in
the scattering processes of massive particles are ensured in our model.888
Some related works are found in Refs. [54, 55, 56, 57, 58, 59, 60, 61].

##  4 $CP$ phase in the CKM matrix

In this section, we verify that our mechanism can actually produce a
nontrivial $CP$ phase in the CKM matrix. We further would like to find a set
of parameter configurations in which the quark mass hierarchy and the
structure of the CKM matrix are derived naturally. In the following analysis,
we rescale all the dimensional valuables by the $S^{1}$ circumference $L$ to
make them dimensionless and the rescaled valuables are indicated with the
tilde $\tilde{{}}$ .

We set the parameters concerning the scalar singlet $\Phi$ as

$\displaystyle{\tilde{M}_{\Phi}}=8.67,\quad\tilde{y}_{0}=-0.1,\quad{\tilde{\lambda}_{\Phi}}=0.001,\quad|\tilde{Q}|=0.001,$
(4.1)

where the VEV profile becomes an (almost) exponential function of $y$, which
is suitable for generating the large mass hierarchy.999 The smallness of $Q$
is not an unnatural thing because they are resultant values derived from the
two input parameters $L_{\pm}$, for which the dimensionless values are within
$\mathcal{O}(10)$ as in Eq. (4.2). We note that $\lambda_{\Phi}$ always
appears in the form of the singlet VEV in Eq. (3.17) as the combination
$|Q|\lambda_{\Phi}$. $\lambda_{\Phi}$ in itself only affects the overall
normalization. Therefore some room might remain for a more “natural” choice of
$\lambda_{\Phi}$.  In this case, the values of $L_{\pm}$ in Eq. (3.16)
correspond to

$\displaystyle\frac{1}{\tilde{L}_{+}}=-6.07,\quad\frac{1}{\tilde{L}_{-}}=8.69,$
(4.2)

where the broken phase is realized [46].

$M_{ij}^{(u)}$ | $a$ | $b$ | $M_{ij}^{(d)}$ | $a$ | $b$
---|---|---|---|---|---
$M_{11}^{(u)}$ | $L_{0}^{(u)}$ | $L_{1}^{(u)}$ | $M_{11}^{(d)}$ | $L_{0}^{(d)}$ | $L_{1}^{(d)}$
$M_{22}^{(u)}$ | $L_{1}^{(q)}$ | $L_{2}^{(u)}$ | $M_{22}^{(d)}$ | $L_{1}^{(q)}$ | $L_{2}^{(d)}$
$M_{33}^{(u)}$ | $L_{2}^{(q)}$ | $L$ | $M_{33}^{(d)}$ | $L_{2}^{(q)}$ | $L$
$M_{12}^{(u)}$ | $L_{1}^{(u)}$ | $L_{1}^{(q)}$ | $M_{12}^{(d)}$ | $L_{1}^{(d)}$ | $L_{1}^{(q)}$
$M_{23}^{(u)}$ | $L_{2}^{(u)}$ | $L_{2}^{(q)}$ | $M_{23}^{(d)}$ | $L_{2}^{(d)}$ | $L_{2}^{(q)}$
$M_{31}^{(u)}$ | $L$ | $L+L_{0}^{(u)}$ | $M_{31}^{(d)}$ | $L$ | $L+L_{0}^{(d)}$
Table 1: The summary table for the overlap integrals in Eq. (4.6).

As in the previous analysis [46], the signs of the fermion bulk masses are
assigned as $M_{Q}>0,M_{\mathcal{U}}<0,M_{\mathcal{D}}>0$ to make much larger
overlapping in the up-quark sector than in down ones for top mass. Here we
assume the positions of the two end points of both the quark doublet and the
scalar singlet are the same

$\displaystyle L_{0}^{(q)}=L_{0}^{(\Phi)}=0,\quad
L_{3}^{(q)}=L_{3}^{(\Phi)}=L,$ (4.3)

where we set $L_{0}^{(q)}$ and $L_{0}^{(\Phi)}$ as zero. In addition, we also
assume that the orders of the positions of point interactions are settled as

$\displaystyle
0<L_{0}^{(u)}<L_{1}^{(u)}<L_{1}^{(q)}<L_{2}^{(u)}<L_{2}^{(q)}<L<L_{3}^{(u)},$
$\displaystyle
0<L_{0}^{(d)}<L_{1}^{(d)}<L_{1}^{(q)}<L_{2}^{(d)}<L_{2}^{(q)}<L<L_{3}^{(d)}.$
(4.4)

Here our up-quark mass matrix ${\mathcal{M}}^{(u)}$ and that of down ones
${\mathcal{M}}^{(d)}$ take the forms

$\displaystyle\mathcal{M}^{(u)}=\begin{bmatrix}M^{(u)}_{11}&M^{(u)}_{12}&M^{(u)}_{13}\\\
0&M^{(u)}_{22}&M^{(u)}_{21}\\\
0&0&M^{(u)}_{33}\end{bmatrix},\quad\mathcal{M}^{(d)}=\begin{bmatrix}M^{(d)}_{11}&M^{(d)}_{12}&M^{(d)}_{13}\\\
0&M^{(d)}_{22}&M^{(d)}_{21}\\\ 0&0&M^{(d)}_{33}\end{bmatrix},$ (4.5)

where the row (column) index of the mass matrices shows the generations of the
left- (right-)handed fermions, respectively. Differently from the model on an
interval in Ref. [46], the $(1,3)$ elements of the mass matrices are allowed
geometrically due to the periodicity along the $y$ direction. The general form
of the nonzero matrix elements of ${\mathcal{M}}^{(u)}$ and
${\mathcal{M}}^{(d)}$ can be expressed as

$\displaystyle
M^{(\kappa)}_{ij}={\mathcal{Y}}^{(\kappa)}\int_{a}^{b}\\!\\!dyf_{q^{(0)}_{iL}}(y)f_{\kappa^{(0)}_{jR}}(y)\langle\Phi(y)\rangle{\langle
H(y)\rangle},$ (4.6)

where $\kappa$ indicates the up/down type of quark and the concrete
information is stored in Table 1.

###  4.1 Quark masses and mixing parameters

The parameters which we use for calculation are

$\begin{array}[]{llll}\tilde{L}_{0}^{(q)}=0,&\tilde{L}_{1}^{(q)}={0.30},&\tilde{L}_{2}^{(q)}={0.660},&\tilde{L}_{3}^{(q)}=1,\\\
\tilde{L}_{0}^{(u)}={0.024},&\tilde{L}_{1}^{(u)}={0.026},&\tilde{L}_{2}^{(u)}={0.52},&\tilde{L}_{3}^{(u)}={1.024},\\\
\tilde{L}_{0}^{(d)}={0.07},&\tilde{L}_{1}^{(d)}={0.18},&\tilde{L}_{2}^{(d)}={0.646},&\tilde{L}_{3}^{(d)}={1.07},\\\
\tilde{M}_{Q}={6},&\tilde{M}_{\mathcal{U}}={-6},&\tilde{M}_{\mathcal{D}}={5},&\theta={3},\end{array}$
(4.7)

where the twist angle $\theta$ is a dimensionless value and should be within
the range $-\pi<\theta\leq\pi$. We note that $\tilde{L}_{3}^{(q)}$,
$\tilde{L}_{3}^{(u)}$, and $\tilde{L}_{3}^{(d)}$ are considered to be not
independent degree of freedom, for which the values are automatically
determined after we choose the other positions of the point interactions. In
Appendix A, we will comment on the orders of significant digits of the input
parameters in Eq. (4.7). We should note that in our system, the EWSB is only
realized on the condition of ${M}^{2}-\left(\frac{\theta}{L}\right)^{2}>0$ as
in Eqs. (2.13). Recently, the ATLAS and CMS experiments have announced that
the physical Higgs mass is around $126\,\text{GeV}$ over $5\sigma$ confidence
level [62, 63]. $\tilde{\lambda}$ is $0.262$ irrespective of the value of $L$,
while $\tilde{M}$ is slightly dependent on the value of $L$ as $3.01303$
$(3.00052)$ in the case of $M_{\text{KK}}=2\,\text{TeV}$
($M_{\text{KK}}=10\,\text{TeV}$), where $M_{\text{KK}}$ is a typical scale of
the KK mode and defined as $2\pi/L$. Here some tuning is required to obtain
the suitable values realizing the EWSB.

After the diagonalization of the two mass matrices, the quark masses are
evaluated as

$\begin{array}[]{lll}m_{\text{up}}={2.5}\,\text{MeV},&m_{\text{charm}}={1.339}\,\text{GeV},&m_{\text{top}}={173.3}\,\text{GeV},\\\
m_{\text{down}}={4.8}\,\text{MeV},&m_{\text{strange}}={104}\,\text{MeV},&m_{\text{bottom}}={4.183}\,\text{GeV},\\\
\vspace{-4mm}\\\
\displaystyle\frac{m_{\text{up}}}{m_{\text{up}}|_{\text{exp.}}}={1.07},&\displaystyle\frac{m_{\text{charm}}}{m_{\text{charm}}|_{\text{exp.}}}={1.05},&\displaystyle\frac{m_{\text{top}}}{m_{\text{top}}|_{\text{exp.}}}=1.00,\\\
\vspace{-4mm}\\\
\displaystyle\frac{m_{\text{down}}}{m_{\text{down}}|_{\text{exp.}}}={0.993},&\displaystyle\frac{m_{\text{strange}}}{m_{\text{strange}}|_{\text{exp.}}}={1.10},&\displaystyle\frac{m_{\text{bottom}}}{m_{\text{bottom}}|_{\text{exp.}}}=1.00,\end{array}$
(4.8)

and the absolute values of the CKM matrix elements are given as101010 The
values of $\tilde{\mathcal{Y}}^{(u)}$ and $\tilde{\mathcal{Y}}^{(d)}$ are also
chosen as $\tilde{\mathcal{Y}}^{(u)}=-0.0532+0.0156\,i$ and
$\tilde{\mathcal{Y}}^{(d)}=-0.00335-0.00146\,i$ by setting the initial
conditions $M_{33}^{(u)}=m_{t}$ and $M_{33}^{(d)}=m_{b}$ with the top mass
$m_{t}$ and the bottom mass $m_{b}$.

$\displaystyle|V_{\text{CKM}}|=\begin{bmatrix}{0.971}&{0.238}&{0.00377}\\\
{0.237}&{0.971}&{0.0403}\\\
{0.00887}&{0.0395}&0.999\end{bmatrix},\quad\left|\frac{V_{\text{CKM}}}{V_{\text{CKM}}|_{\text{exp.}}}\right|=\begin{bmatrix}0.997&1.06&{1.07}\\\
1.06&0.998&{0.978}\\\ {1.02}&{0.978}&1.00\end{bmatrix}.$ (4.9)

###  4.2 $CP$ phase

The Jarlskog parameter $J$ containing information about the $CP$ phase is
defined by

$\displaystyle\text{Im}\left[(V_{\text{CKM}})_{ij}(V_{\text{CKM}})_{kl}(V_{\text{CKM}}^{\ast})_{il}(V_{\text{CKM}}^{\ast})_{kj}\right]=J\sum_{m,n=1}^{3}\epsilon_{ikm}\epsilon_{jln}$
(4.10)

with the completely antisymmetric tensor $\epsilon$, and is invariant under
the $U(1)$ unphysical rephasing operations of six types of quarks [64, 65].
This value is easily estimated as

$\displaystyle J={3.23}\times 10^{-5},\quad\frac{J}{J|_{\text{exp.}}}={1.09},$
(4.11)

where we also provide the differences from the latest experimental values in
Ref. [66]. All the deviations from the latest experimental values are within
about $10\%$, and we can conclude that the situation of the SM is suitably
generated. In Appendix A, we discuss distribution patterns of quark mass-
matrix elements and required orders in tuning the input parameters with the
results for realizing the accuracy.

##  5 Summary and discussion

In this paper, we have proposed a new mechanism for generating a $CP$ phase
via a Higgs VEV originating from the geometry of an extra dimension. A twisted
BC for the Higgs doublet has been found to lead to an extra-dimension
coordinate-dependent phase in the Higgs VEV, which contains a nontrivial $CP$
phase degree of freedom. This mechanism is useful for generating a $CP$ phase
to a single generation extra-dimensional field theory incorporating with a
generation production mechanism. The electroweak symmetry breaking is also
generated dynamically due to the twisted boundary condition with the suitable
W- and Z-boson masses.

As an illustrative example, we applied our mechanism to a five-dimensional
gauge theory on a circle with point interactions [46]. Point interactions,
which are additional boundary points with respect to the extra dimension, are
responsible for producing the three generations while the model consists of a
single-generation fermion and make the quark profiles be localized. Since each
element of the mass matrices picks up a different phase through the overlap
integrals, there is some possibility of realizing a nontrivial $CP$ phase.
After numerical calculations, we found that a nontrivial $CP$ phase appears
with good precision, maintaining the property of the original model in which
the generations, the quark mass hierarchy and the CKM matrix appear from the
geometry of the extra dimension. Certainly, our new mechanism for generating a
$CP$ phase via Higgs VEV works.

A key point of our mechanism is that we can generate both the EWSB and a $CP$
phase simultaneously as a complex Higgs VEV. To make our $CP$-violation
mechanism work correctly, quark profiles should be split and localized. In
this situation, flavor mixing and mass hierarchy of the quarks are also
naturally activated. We would like to emphasize that in the model adopting our
mechanism, all the concepts of quark flavor in the SM, namely EWSB, the number
of generations, flavor mixing, mass hierarchy, and $CP$ violation, are
interlinked closely.

One of the most important remaining tasks is to construct a model which brings
both the quarks and the leptons into perspective. Using our mechanism, not
only the quark sector but also the lepton sector can acquire a nontrivial $CP$
phase. Since the origin of the $CP$ phase is common, we can predict the value
of the $CP$ phase of the lepton sector after fitting the value of the $CP$
phase in the quark sector. The result will be reported elsewhere.
Accommodation of our mechanism to another single generation model is also an
important task.

Another crucial topic is the stability of the system. Our system is possibly
threatened with instability. Some mechanisms will be required to stabilize the
moduli representing the positions of point interactions (branes).111111 Moduli
stabilization via Casimir energy in the system where a scalar takes the Robin
BCs (but no point interaction in the bulk) has been studied in Refs.[67, 68,
69]. In a multiply connected space of $S^{1}$, there is another origin of
gauge symmetry breaking, i.e., the Hosotani mechanism [70, 71]. Since further
gauge symmetry breaking causes a problem in the model, we need to insure that
the Hosotani mechanism does not occur. To this end, we might introduce
additional 5D matter to prevent zero modes of $y$ components of gauge fields
from acquiring nonvanishing VEVs. We will leave those issues in future work.

## Acknowledgments

The authors would like to thank T.Kugo for valuable discussions. K.N. is
partially supported by funding available from the Department of Atomic Energy,
Government of India for the Regional Centre for Accelerator-based Particle
Physics (RECAPP), Harish-Chandra Research Institute. This work is supported in
part by a Grant-in-Aid for Scientific Research [Grants No. 22540281 and No.
20540274 (M.S.)] from the Japanese Ministry of Education, Science, Sports and
Culture.

## Appendix

##  Appendix A Considering input-parameter dependence

Figure 3:  The left scatter plot shows the distribution of
$M_{33}^{(u)}-M_{13}^{(u)}$ when we choose 100,000 points randomly around the
configuration in Eq.(4.7) within $\pm 10\%$ being consistent with the order in
Eq. (4.4). The right one represents the same thing when we pick up 100,000
points randomly only with following the order in Eq. (4.4).

Figure 4:  From left to right, distributions of
$m_{\text{up}}-m_{\text{down}}$, $m_{\text{charm}}-m_{\text{strange}}$ and
$m_{\text{top}}-m_{\text{bottom}}$ with 10,000 random points within $\pm 20\%$
parameter deviations from the central values in Eq. (4.7 following the order
in Eq. (4.4). The green (yellow) band in the left and center plots represents
$\pm 30\%$ range from the central values in Eq. (4.7). In the right plot, we
skip depicting the bands because all the shown ranges are covered by them.

Figure 5:  From left to right, distributions of
$\sin^{2}{\theta_{12}}-\sin^{2}{\theta_{23}}$ and $\sin^{2}{\theta_{13}}-J$
with 10,000 random points within $\pm 20\%$ parameter deviations from the
central values in Eq. (4.7) following the order in Eq. (4.4). The allowed
region is just near the point $(0,0)$ in both the plots.

In this appendix, we discuss distribution patterns of quark mass-matrix
elements and required orders in tuning the input parameters with the results
given in Sec. 4.1 and 4.2.

We first focus on the matrices in Eq. (4.5). In our model, the geometry of the
extra dimension strongly restricts the form of the matrices. In fact, we
cannot fill all the elements of the mass matrices and at least three of the
nine elements for each mass matrix have to be zero, as shown in Eq. (4.5).
This property is contrasted with that of the standard model, where all the
mass matrix elements are free parameters. This fact means that possible
patterns of mass matrices are constrained by the shape of the geometry of our
model.

Furthermore, it turns out that the values of the nonzero elements in the mass
matrices (4.5) cannot be controlled freely. To see this, we investigated
correlations of matrix elements. In the left figure of Fig. 3, we chose
100,000 points randomly around the configuration in Eq.(4.7) within $\pm 10\%$
being consistent with the order in Eq. (4.4) and depicted the resultant values
as two scatter plots ($M_{33}^{(u)}-M_{13}^{(u)}$). The right figure of Fig. 3
shows the same thing when we pick up 100,000 points randomly only with
following the order in Eq. (4.4).

It follows from Fig. 3 that we find no random distribution in the
$M^{(u)}_{33}-M^{(u)}_{13}$ plane and a strong correlation between
$M^{(u)}_{13}$ and $M^{(u)}_{33}$. We further see the property that the
typical value of $M^{(u)}_{13}$ is much smaller than that of $M^{(u)}_{33}$.
In the quark sector, the mixing angles are known to be small, so that off-
diagonal elements of the mass matrices will be preferred to be subleading
compared with the diagonal ones with suitable magnitudes. Our geometry
realizes this point naturally via its geometry.

From the above observations, we may conclude that the quark mass matrices in
our model are considerably restricted from the geometry of the extra
dimension, and hence that it is nontrivial to reproduce the quark-related
properties of the standard model, although we have 16 input parameters for
quark profiles, where 13 parameters are independent, to explain the 10
standard model parameters (6 quarks masses and 4 CKM parameters).

As another consideration, we investigate the resultant quark masses and CKM
matrix elements when we change the input parameters around the central values
in Eq. (4.7) and then find that the quark masses and CKM matrix elements are
very sensitive to some of the input parameters.

To show the sensitivity of the input parameters, we first alter all the
parameters randomly within $10\%$, $1\%$, $0.1\%$, and $0.01\%$, respectively,
obeying the order in Eq. (4.4), and calculate the masses and the elements.
When we proceed with the above procedure 100,000 times in each case, 11,
11,039, 81,955, and 100,000 points survive after putting the cut where all the
resultants are within $15\%$. These results indicate that parameter tuning
less than $1\%$ is required as our inputs are so in Eq. (4.7). The scatter
plots in Figs. 4 and 5 represent the distributions of the physical parameters
with 10,000 random points within $\pm 20\%$ around the central values, where
the CKM angles and the Jarlskog parameter are apt to getting away from the
required range easily, for which the experimental central values are
$\sin^{2}{\theta_{12}}\sim 0.05$, $\sin^{2}{\theta_{23}}\sim 0.002$,
$\sin^{2}{\theta_{13}}\sim 0.00001$, and $J\sim 0.00003$, respectively. This
issue is explained by the fact that off-diagonal elements of the CKM matrix
are closely related to those of the up- and down-quark mass matrices, which at
least parts of them are, sensitive to perturbation of the input parameters. We
can also find the tendency that bottom and top masses are not away from the
central values by the perturbation.

On the other hand, we try to alter an input parameter separately. In each
case, points of the numbers in Table 2 pass the cut which rejects the
possibilities that at least one resultant value is out of the $\pm 15\%$
deviation range from the central value in Eq. (4.7). According to the result,
we understand that quark masses and mixing angles are sensitive to the
positions of point interactions, while those are insensitive to (absolute
values of) the bulk masses and the twisted angle. Then we can conclude that
parameter tuning in $M_{Q}$, $M_{\mathcal{U}}$, $M_{\mathcal{D}}$, and
$\theta$ in Eq. (4.7) is not always necessary.

Finally, we briefly comment on the required orders of significant digits in
the input parameters. As we have discussed before based on Table 2, the system
is insensitive to $M_{Q}$, $M_{\mathcal{U}}$, $M_{\mathcal{D}}$, and $\theta$
around the central region of the parameters, and then single digits are
sufficient for them. On the other hand, for $L_{2}^{(q)}$ and $L_{2}^{(d)}$,
as also expressed in Table 2, triple digits are required because of their
great sensitivity. For the other values, tuning up to double digits is enough
for our purpose since they are less sensitive than $L_{2}^{(q)}$ and
$L_{2}^{(d)}$ as shown in Table 2.

parameter | $L_{1}^{(q)}$ | $L_{2}^{(q)}$ | $L_{0}^{(u)}$ | $L_{1}^{(u)}$ | $L_{2}^{(u)}$ | $L_{0}^{(d)}$ | $L_{1}^{(d)}$ | $L_{2}^{(d)}$ | $M_{Q}$ | $M_{\mathcal{U}}$ | $M_{\mathcal{D}}$ | $\theta$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\\#$ of surviving points | 105 | 11 | 46 | 85 | 42 | 270 | 172 | 10 | 679 | 730 | 372 | 971
Table 2:  Numbers of surviving points out of 1,000 ones after the cut where
all the physical resultants are within $\pm 15\%$ with changing an input
parameter within $\pm 10\%$ from the central values in Eq. (4.7) individually.

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