capbtabboxtable[][] ††institutetext: Université Paris-Saclay, CNRS/IN2P3,
IJCLab, 91405 Orsay, France

# Rigorous treatment of the $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold model
with brane-Higgs couplings

Ruifeng Leng  Grégory Moreau  Florian Nortier<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We build rigorously the attractive five-dimensional model where bulk fermions
propagate along the $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold and interact
with a Higgs boson localised at a fixed point of the extra dimension. The
analytical calculation of the fermion mass spectrum and effective Yukawa
couplings is shown to require the introduction of either Essential Boundary
Conditions (EBC) imposed by the model definition or certain Bilinear Brane
Terms (BBT) in the action, instead of the usual brane-Higgs regularisations.
The obtained fermion profiles along the extra dimension turn out to undergo
some discontinuities, in particular at the Higgs brane, which can be
mathematically consistent if the action is well written with improper
integrals. We also show that the $\mathbb{Z}_{2}$ parity transformations in
the bulk do not affect the fermion chiralities, masses and couplings, in
contrast with the EBC and the BBT, but when extended to the fixed points, they
can generate the chiral nature of the theory and even select the Standard
Model chirality set-up while fixing as well the fermion masses and couplings.
Thanks to the strict analysis developed, the duality with the interval model
is scrutinised.

##  1 Introduction

As it is well known since the 2000s, the paradigm of models with additional
spatial dimensions 111Together with the composite Higgs models which are dual
models via the AdS/CFT correspondance. constitutes an attractive alternative
to supersymmetry for addressing the Standard Model (SM) puzzle of the gauge
hierarchy. Furthermore, the warped dimension framework Randall:1999ee  with SM
fermions in the whole bulk Chang:1999nh  offers an elegant geometrical
principle of fermion profile overlap generating the SM fermion mass hierarchy
Gherghetta:2000qt  (see concrete application models e.g. in Ref. Huber:2001ug
; Casagrande:2008hr ; Chang:2005ya ; Moreau:2005kz ; Moreau:2006np ;
Bouchart:2008vp ). In order to realise these two hierarchical features, the
Brout-Englert-Higgs scalar field Englert:1964et ; Higgs:1964ia , which is at
the origin of the SM particle masses through the electroweak symmetry
breaking, must be stuck at the so-called TeV-brane 222Let us mention here
other possible phenomenological motivations, as from neutrino mass models, for
the Higgs boson to be stuck at the boundary of an interval Dienes:1998sb ;
Abada:2006yd ; Grossman:1999ra ; Huber:2002gp ; Moreau:2004qe  or for fermions
to propagate in the bulk Frere:2003hn ; Nortier:2020lbs . (or located in the
bulk with a wave function strongly peaked at this brane). The TeV-brane is a
3-brane (three spatial dimensions) possibly at a boundary of the finite warped
extra dimension 333See for instance Ref. Ledroit:2007ik ; Bouchart:2009vq ;
Djouadi:2006rk ; Djouadi:2007eg ; Djouadi:2007fm ; Djouadi:2009nb ;
Djouadi:2011aj ; Angelescu:2017jyj  for its phenomenology and Ref.
Bouchart:2011va  in a supersymmetric context.. More generally, a brane is an
hypersurface located in an higher-dimensional space. It can arise in the
context of string theories as D-branes which are dynamical objects with
quantum properties Polchinski:1996na ; Bachas:1998rg  (see also Ref.
Aharony:1999ti ; Duff:1996zn  for the supergravity limit of string theories)
444See Ref. Fichet:2019owx ; Nortier:2020vge  for brane-world effective field
theories..

In this paper, we will study the original version Randall:1999ee  of the
warped dimension scenario based on the ${\cal S}^{1}/\mathbb{Z}_{2}$ orbifold
Dixon:1985jw ; Dixon:1986jc  where the extra space is compactified on a circle
respecting a spatial parity of the Lagrangian 555An orbifold ${\cal O}$ being
defined as an extra compact manifold ${\cal C}$ with so-called fixed points
where the introduced spatial transformation (element from a discrete group
$G$) – letting the Lagrangian invariant – is just equivalent to the identity.
It is noted ${\cal O}={\cal C}/G$ and possesses thus singularities, not like a
smooth manifold Quiros:2003gg ; Choi:2020dws .. Focusing our attention on the
subtle bulk fermion interactions with the brane-Higgs field localised at a
fixed point, we will analyse the toy model with a flat extra dimension and the
minimal field content: the results obtained on the fermion-Higgs coupling
structure are directly applicable to the realistic warped model.

We will clarify the treatment of the bulk fermion couplings to the brane-
localised Higgs boson, within the ${\cal S}^{1}/\mathbb{Z}_{2}$ orbifold
background, by building rigorously the four-dimensional (4D) 666Including
time. effective Lagrangian of the minimal model, that is by calculating
consistently the Kaluza-Klein (KK) tower spectrum of fermion mass eigenvalues
and the 4D effective Yukawa couplings (via the fermion wave functions along
the extra dimension).

In particular, we will demonstrate that no brane-Higgs regularisation [like
smoothing the Higgs Dirac peak] should be applied (not necessary and no
theoretical argument for it) in contrast with the usual regularisation
procedure of literature (see Ref. Angelescu:2019viv  and references therein)
and that, instead, one must introduce either Essential Boundary Conditions
(EBC) on 5D fields 777Directly imposed by the model definition, in contrast
with the Natural Boundary Conditions (NBC) deduced from the action
minimisation condition., originating from the $\mathbb{Z}_{2}$ symmetry, or
equivalently some Bilinear Brane Terms (BBT) in the fundamental 5D Lagrangian.
The exact matching of the fermion mass spectra derived respectively through 4D
and 5D methods will be used in order to confirm our analytical results. All
those statements (except the 4D approach) hold as well for the free case i.e.
without Yukawa interactions.

This necessity of the presence of EBC or BBT (terms with the same form as in
Ref. Angelescu:2019viv ; Nortier:2020xms ), in the 4D or 5D approach, has been
found as well Angelescu:2019viv  in the finite interval scenario (the higher-
dimensional framework of the other warped model version) with identical brane-
Higgs couplings to bulk fermions: this conclusion confirms that a specific
treatment is required for point-like interactions between bulk fermions and
brane-Higgs bosons in higher-dimensional spaces.

Besides, we will strictly describe and work out the entire known duality:
identical physical quantities, namely the mass eigenvalues and 4D effective
Yukawa couplings, are obtained in the different ${\cal S}^{1}/\mathbb{Z}_{2}$
orbifold scenario with the Higgs boson localised at a fixed point and finite
interval geometrical set-up with the Higgs field stuck at a boundary.

The EBC and BBT (forms including signs) choices, which should originate from
an Ultra-Violet (UV) completion of the theory, turn out to induce the chiral
nature of the low-energy effective theory as well as realising the specific SM
fermion chiralities. Indeed, all these chirality properties are in fact not
selected by the remaining sign choices for the 5D fields transformed via the
spatial $\mathbb{Z}_{2}$ group – as the solutions we find within this orbifold
configuration can exhibit twist transformations (sign modification here) of
the 5D fields, à la Scherk-Schwarz Scherk:1979zr ; Cremmer:1979uq , through
the extra space reflection. We will even show that the transformation sign
choices are just mathematical conventions without physical impacts on the SM
field chiralities, the fermion mass spectrum and the 4D effective Yukawa
couplings.

Nevertheless, in order to clarify the chirality aspects, we will also study a
different scenario – considered for example in Ref. Gherghetta:2000kr  – where
the $\mathbb{Z}_{2}$ transformation definitions on the fields cover as well
the fixed points themselves. It turns out that the associated transformation
sign choices precisely at these fixed points constitute here additional EBC,
noted EBC’, that have the capacity to select some of the previous EBC and
hence to fix the chirality set-up. Once more, the rôle of these EBC’ can be
played instead by certain of the above BBT. Interestingly, such an inclusive
$\mathbb{Z}_{2}$ symmetry definition can induce by itself the chiral nature of
the theory as well as the SM chirality distribution over the various fields.
This origin for the whole chirality configuration is not offered within the
simpler interval model for instance. In the presence of brane-localised Yukawa
couplings, such an inclusive $\mathbb{Z}_{2}$ scenario can only be treated
through the 4D method. The fermion masses and couplings are also affected by
this inclusive $\mathbb{Z}_{2}$ symmetry.

The action integral definition and integral domain end-points will be treated
carefully. In particular the decomposition of the action to introduce improper
integrals will appear to be required in the presence of orbifold fixed points
or point-like fermion-boson interactions (not located at the boundary of a
finite extra space like an interval). Within this new and appropriate approach
of the specific points along the extra dimension of the orbifold, we find for
the free or Yukawa case that some of the obtained consistent solutions exhibit
certain field jumps at these fixed points and localised-interaction point.
This interesting result of the possible existence of consistent profile jumps
stands against one’s first intuition Csaki:2003sh ; Azatov:2009na , but those
jumps are only induced by sign flipping and not by point-like changes of the
absolute value of the wave function amplitudes.

The analysis of the present orbifold background with brane-localised fermion-
scalar interactions, as well as the previous results Angelescu:2019viv  on the
interval background, show that generally speaking the action expression does
not systematically contain all the information allowing to fully define the
model: in particular some EBC may be used (in contrast, the BBT are terms in
the action) depending on the brane treatment adopted or on the UV completion
of the theory (which could introduce the BBT).

##  2 Minimal $\mathcal{S}^{1}/\mathbb{Z}_{2}$ consistent model

###  2.1 Geometry and symmetries: the proper action

We consider the 5D space-time model with the product geometry
$\mathcal{M}^{4}\times\mathcal{S}^{1}/\mathbb{Z}_{2}$ described just below.

  * •

$\mathcal{M}^{4}$ represents the usual 4D Minkowski space-time whose
coordinates are denoted by $x^{\mu}$ where $\mu\in\llbracket 0,3\rrbracket$ is
the Lorentz index of the covariant formalism. The metric conventions are given
in Appendix A.

  * •

$\mathcal{S}^{1}/\mathbb{Z}_{2}$ stands for the extra space orbifold obtained
from modding out the circle $\mathcal{S}^{1}$ by the discrete group 888Factor
element, $e^{\pm i\frac{2\pi}{2}}=-1$, and neutral element, $1$. symmetry
$\mathbb{Z}_{2}$.
This circle $\mathcal{S}^{1}$ is characterised by a radius $R$ and its
coordinate is $y\in(-\pi R,\pi R]$, not double-counting the point $y=\pi R$
since it is this point, by pure convention, that is chosen to be the junction
point geometrically identified with the point $y=-\pi R$ (which we note: $-\pi
R\equiv\pi R$) 999Another possible mathematical convention would have been,
for instance, $-\frac{3\pi R}{2}\equiv\frac{\pi R}{2}$. in order to implement
the circle periodicity. The circle could be constructed from the real axis by
imposing a periodicity, that is by identifying geometrically an infinite
number of translated regions of size $2\pi R$ and hence by limiting the 1D
space to the fundamental domain $(-\pi R,\pi R]$.
The (non-neutral) $\mathbb{Z}_{2}$ transformation on space, $y\to-y$ 101010The
convention above, of having taken the coordinate origin at the strict middle
of the circle domain (or fundamental domain), renders the $\mathbb{Z}_{2}$
parity with respect to the origin more explicit and convenient to study., has
a representation on a generic 5D field,

$\Phi(x^{\mu},-y)={\mathcal{T}}\Phi(x^{\mu},y)\,,\ \ \forall y\in(-\pi
R,0)\cup(0,\pi R)\,,$ (1)

which must let the Lagrangian density invariant, by definition of the
symmetry:

$\mathcal{L}\left[\Phi(x^{\mu},-y)\right]=\mathcal{L}\left[\Phi(x^{\mu},y)\right]\,,\
\ \forall\,y\in(-\pi R,\pi R]\,.$ (2)

We mention that this equation can define a class of equivalence of a given
coordinate $y_{0}$, defined as $[y_{0}]=\\{y\in\mathcal{S}^{1}\,|\,y\sim
y_{0}\\}$ with $y\sim\pm y$, as illustrated symbolically on Fig. 1.
Two fixed points arise: $(y=0)\to(-0=0)$ and $(y=\pi R)\to(-\pi R\equiv\pi
R)$. At these fixed points, the Lagrangian condition of Eq. (2) is
automatically satisfied:
$\mathcal{L}\left[\Phi(x^{\mu},-0)\right]=\mathcal{L}\left[\Phi(x^{\mu},0)\right]$,
and, $\mathcal{L}\left[\Phi(x^{\mu},-\pi
R)\right]=\mathcal{L}\left[\Phi(x^{\mu},\pi R)\right]$, so that
${\mathcal{T}}$ is naturally taken as the identity operator in Eq. (1) since
no transformation needs to apply on the fields there. Another scenario will be
analysed in Section 6.

Figure 1: $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold picture. The fixed points
at $y=0$ and $y=\pi R$ are indicated by the two black points. The two examples
of pairs of points with opposite coordinates, respectively indicated by the
double dashed arrows, correspond to an identical Lagrangian density (for each
pair).

In order to properly write down the initial action, we urge the importance of
taking care of possible field jumps along the extra dimension upon the reader.
We are going to show that the existence of a field jump in field theory can
make sense mathematically if the action integration domain is properly divided
at the jump location. Different discontinuity configurations must be
considered. First, the hypothesis of a possible jump at any point of the bulk
would lead to an infinite number of cuts in the action integration region
which would obviously not be treatable leading to unpredictable observables:
this assumption is thus excluded. Secondly, assuming an arbitrary finite
number of possible jumps and hence of mathematical separations in the action
domain, outside the fixed points, is not expected to affect the unique
physical results – like the fermion mass spectrum – since none of those jump
points exhibit some specific property: it is thus useless to explore this
direction. Thirdly, the case of possible profile jumps at the two specific
points that are the fixed points of the orbifold – one of those two, $y=\pi
R$, corresponding as well to the Yukawa coupling location (see Section 2.2.3)
– remains to be studied. The effective presence of such profile jumps in some
of the obtained solutions (see Figures 2 and 3 respectively for the free and
coupled fermion situations) confirms this possibility. For example, in case of
a profile jump at $y=0$ (an identical discussion holds for the other fixed
point at $y=\pi R$), regarding a well-defined Lagrangian integrand involving
5D fields over the whole action integration domain, we simply have to choose
between the mathematical definitions of the left or right continuity for a
generic profile function along the extra dimension:
$f(0)=f(0^{-})=\lim_{\epsilon\to 0}f(0-\epsilon)$ with $\epsilon>0$, or,
$f(0)=f(0^{+})$. This choice is conventional and hence cannot affect numerical
results, so let us choose conveniently

$\displaystyle f(0)=f(0^{+})\ \ \mbox{and}\ \ f(\pi R^{-})=f(\pi R)$ (3)

throughout this paper, in case of jumps at the fixed points. Then, the well-
defined global action of this model must be written as a sum of some brane
terms, an improper integral and a standard integration over different regions
covering the whole physical domain of the circle:

$\displaystyle S_{\rm 5D}=S_{bulk}+S_{branes}\,,\ \mbox{with,}\ \
S_{bulk}\,=\,\int d^{4}x\left(\int_{-\pi R^{+}}^{0^{-}}dy\
\mathcal{L}_{kin}+\int_{0}^{\pi R}dy\ \mathcal{L}_{kin}\right),\ \mbox{and,}$
$\displaystyle\int d^{4}x\int_{-\pi R^{+}}^{0^{-}}dy\ \mathcal{L}_{kin}\
\hat{=}\,\lim_{a\to 0^{-},\,b\to-\pi R^{+}}\,\int d^{4}x\int_{b}^{a}dy\
\mathcal{L}_{kin}\,=\,\lim_{\epsilon\to 0}\,\int d^{4}x\int_{-\pi
R+\epsilon}^{0-\epsilon}dy\ \mathcal{L}_{kin}\,,$ (4)

where $\epsilon>0$, $S_{branes}$ represents action terms located at the
orbifold fixed points and $\mathcal{L}_{kin}$ stands for the fermion kinetic
terms of the Lagrangian density (see next subsection). Indeed, all the
obtained fields will be well-defined at the two fixed points via Eq. (3).
Besides, for $\mathcal{L}_{kin}$ to be integrable over the entire region
$y\in[0,\pi R]$, this Lagrangian density, which will involve profile
derivatives $f^{\prime}(y)$, must be well-defined over this region. The
necessary (but not sufficient) condition for this last feature is that the
profiles $f(y)$ have to be continuous on $[0,\pi R]$ and Eq. (3) also
guarantees this continuity. For the consistency of the other integration term
in Eq. (4), the profile continuity along $y\in[-\pi R^{+},0^{-}]$ 111111To be
clear, the integration domain $[-\pi R^{+},0^{-}]$ corresponds to the spatial
region along the extra dimension $]-\pi R,0[\ \Leftrightarrow$ $(-\pi R,0)$ –
respectively the Francophone and Anglophone notations – which does not include
the fixed points at $y=0$ and $y=-\pi R$. obviously reads as:

$\lim_{\kappa\to 0}f(0^{-}-\kappa)=\lim_{\epsilon\to 0}\lim_{\kappa\to
0}f([0-\epsilon]-\kappa)=\lim_{\omega\to 0}f(0-\omega)=f(0^{-})\ ,$

with $\kappa>0$, $\epsilon>0$, $\omega=\epsilon+\kappa$ and similar equalities
hold at the other region boundary $y=-\pi R^{+}$. Furthermore, the worked out
solutions $f(y)$ will (well) be derivable over the two regions $[-\pi
R^{+},0^{-}]$ and $[0,\pi R]$ (see Sections 3.2 and 5.3 respectively for the
free and coupled fermion situations) so that $\mathcal{L}_{kin}$ will be well-
defined. For example, $f(y)$ is derivable in the region $[0,\pi R]$ at $y=0$
if and only if $f(y)$ is right-derivable at $y=0$, and the corresponding
right-derivative does not diverge thanks to the first equality of Eq. (3).
Notice that from the point of view of the integration by pieces of the action
in Eq. (4) precisely over the physical domain, the inclusion (or not) of the
single points at $y=0$ or $y=\pi R\equiv-\pi R$ does not affect the integral
results – given the continuous form of the even $\mathcal{L}_{kin}$ over the
two regions – so that only consistent action definition arguments were
considered here.

Finally, the Lagrangians of the whole expression (4) will respect the
$\mathbb{Z}_{2}$ symmetry since the Lagrangian $\mathcal{L}_{kin}$ will
fulfill the condition (2) and the brane action will exclusively involve
Lagrangians taken at fixed points like for example [see Eq. (13)],
$S_{branes}\ni S_{Y}=\int d^{4}x\,\mathcal{L}_{Y}(x^{\mu},\pi R)$.

###  2.2 Field content and complete action

####  2.2.1 Bulk fermion fields

Let us introduce the minimal spin-1/2 field content which allows to write down
a SM Yukawa-like coupling between zero mode fermions (of different
chiralities) and a spin-0 field (see Section 2.2.3). It is constituted by a
pair of fermion fields called $Q$ and $D$. Those particles propagate along the
circle $\mathcal{S}^{1}$, as we have in mind an extension of this toy model to
a realistic scenario with bulk matter (c.f. Section 2.2.4) where $Q,D$ will
represent respectively the ${\rm SU(2)_{L}}$ gauge doublet down-component
quark and the singlet down-quark.

The 5D fields $Q(x^{\mu},y)$ and $D(x^{\mu},y)$ – of mass dimension 2 – have
the following kinetic terms [entering Eq. (4)] which allow to recover
canonical covariant kinetic terms for the associated fermions in the 4D
effective action (as imposed by the argument of decoupling limit 121212From
the theoretical consistency and phenomenological points of view, the SM must
be approximately recovered at low-energies in the limit of infinitely heavy KK
excitations.):

$\displaystyle\mathcal{L}_{kin}=\frac{i}{2}\
\left(\bar{Q}\Gamma^{M}\overleftrightarrow{\partial_{M}}Q+\bar{D}\Gamma^{M}\overleftrightarrow{\partial_{M}}D\right)\
,$ (5)

using the standard notations
$\overleftrightarrow{\partial_{M}}=\overrightarrow{\partial_{M}}-\overleftarrow{\partial_{M}}$,
$\partial_{M}=\partial/\partial x^{M}$, $x^{M}=(x^{\mu},y)$ with
$M\in\llbracket 0,4\rrbracket$ for the coordinates
$x^{M}\in\mathcal{M}^{4}\times\mathcal{S}^{1}/\mathbb{Z}_{2}$ and $\Gamma^{M}$
for the 5D Dirac matrices (c.f. Appendix A). In the used conventions, the 5D
Dirac spinor, being in the irreducible representation of the Lorentz group,
reads for example for $Q$ as,

$Q=\mathcal{Q}_{L}+\mathcal{Q}_{R}\ \ \text{with}\ \
\mathcal{Q}_{L}=\begin{pmatrix}Q_{L}\\\ 0\end{pmatrix}\ \ \text{,}\ \
\mathcal{Q}_{R}=\begin{pmatrix}0\\\ Q_{R}\end{pmatrix}\ ,$ (6)

in terms of the two two-component Weyl spinors $Q_{L}$, $Q_{R}$, $L/R$
standing for the Left/Right chirality, and as usually
$\bar{Q}=Q^{\dagger}\gamma^{0}$.

As stated at the end of previous section, the Lagrangian $\mathcal{L}_{kin}$
must obey the condition (2). For this purpose, the $\mathbb{Z}_{2}$
transformation (1) on the 5D fields $Q(x^{\mu},y)$ and $D(x^{\mu},y)$ can take
four different forms which constitute Essential Conditions (EC) issued from
the model definition:

$Type\,I\left\\{\begin{array}[]{c c
c}Q\left(x^{\mu},-y\right)=-\gamma^{5}\,Q\left(x^{\mu},y\right)\,\Longrightarrow\,Q_{L}\
\text{even},\ Q_{R}\ \text{odd},\\\ \vspace{-0.2cm}\hfil\\\
D\left(x^{\mu},-y\right)=\gamma^{5}\,D\left(x^{\mu},y\right)\,\Longrightarrow\,D_{L}\
\text{odd},\ D_{R}\ \text{even},\end{array}\right.$ (7)
$Type\,II\left\\{\begin{array}[]{c c
c}Q\left(x^{\mu},-y\right)=\gamma^{5}\,Q\left(x^{\mu},y\right)\,\Longrightarrow\,Q_{L}\
\text{odd},\ Q_{R}\ \text{even},\\\ \vspace{-0.2cm}\hfil\\\
D\left(x^{\mu},-y\right)=-\gamma^{5}\,D\left(x^{\mu},y\right)\,\Longrightarrow\,D_{L}\
\text{even},\ D_{R}\ \text{odd},\end{array}\right.$ (8)
$Type\,III\left\\{\begin{array}[]{c c
c}Q\left(x^{\mu},-y\right)=-\gamma^{5}\,Q\left(x^{\mu},y\right)\,\Longrightarrow\,Q_{L}\
\text{even},\ Q_{R}\ \text{odd},\\\ \vspace{-0.2cm}\hfil\\\
D\left(x^{\mu},-y\right)=-\gamma^{5}\,D\left(x^{\mu},y\right)\,\Longrightarrow\,D_{L}\
\text{even},\ D_{R}\ \text{odd},\end{array}\right.$ (9)
$Type\,IV\left\\{\begin{array}[]{c c
c}Q\left(x^{\mu},-y\right)=\gamma^{5}\,Q\left(x^{\mu},y\right)\,\Longrightarrow\,Q_{L}\
\text{odd},\ Q_{R}\ \text{even},\\\ \vspace{-0.2cm}\hfil\\\
D\left(x^{\mu},-y\right)=\gamma^{5}\,D\left(x^{\mu},y\right)\,\Longrightarrow\,D_{L}\
\text{odd},\ D_{R}\ \text{even},\end{array}\right.$ (10)

under which the Lagrangian (5) is indeed invariant, as appears by using the
properties of the $\gamma_{5}$ Dirac matrix and the odd parity of the fifth
partial derivative $\partial_{4}$. Notice that the $\mathbb{Z}_{2}$ parity
(second order cyclic group) does not allow complex phase factors in the
transformations:

$F|_{y}=e^{i\theta_{F}}\gamma^{5}F|_{-y}=e^{i\theta_{F}}\gamma^{5}e^{i\theta_{F}}\gamma^{5}F|_{y}=\left(e^{i\theta_{F}}\right)^{2}F|_{y}\,.$

Using the $\gamma_{5}$ definition of Appendix A together with Eq. (6), we
already deduce some information on the possible 5D chiral field parities with
respect to $y=0$, as indicated in Eq. (7)-(10).

Based on Eq. (6), we can rewrite the bulk Lagrangian of Eq. (5) in forms which
are convenient to see at a glance the Lagrangian even parity, simply by using
the occurence of fixed 5D field parities, different for the Left/Right
chiralities [c.f. Eq. (7)-(10)], and the $\partial_{4}$ odd parity:

$\displaystyle\mathcal{L}_{kin}$
$\displaystyle=\dfrac{1}{2}\left(iQ^{\dagger}_{R}\sigma^{\mu}\overleftrightarrow{\partial_{\mu}}Q_{R}+iQ^{\dagger}_{L}\bar{\sigma}^{\mu}\overleftrightarrow{\partial_{\mu}}Q_{L}-Q^{\dagger}_{R}\overleftrightarrow{\partial_{4}}Q_{L}+Q^{\dagger}_{L}\overleftrightarrow{\partial_{4}}Q_{R}\right)+\\{Q\leftrightarrow
D\\}\,,$
$\displaystyle=\dfrac{1}{2}\left(i\widebar{\mathcal{Q}}_{R}\gamma^{\mu}\overleftrightarrow{\partial_{\mu}}\mathcal{Q}_{R}+i\widebar{\mathcal{Q}}_{L}\gamma^{\mu}\overleftrightarrow{\partial_{\mu}}\mathcal{Q}_{L}-\widebar{\mathcal{Q}}_{R}\overleftrightarrow{\partial_{4}}\mathcal{Q}_{L}+\widebar{\mathcal{Q}}_{L}\overleftrightarrow{\partial_{4}}\mathcal{Q}_{R}\right)+\\{\mathcal{Q}\leftrightarrow\mathcal{D}\\}\,,$

where the low double arrows indicate a replacement of 5D fields in the
previous terms and the matrices $\sigma^{\mu},\bar{\sigma}^{\mu}$ are defined
in Appendix A.

####  2.2.2 Brane-localised scalar field

The questions about the mass calculation arise when the bulk fermions couple
to a single 4D real scalar field $H$ (mass dimension 1) which is confined at a
fixed point of the orbifold, as in the studied model (inspired by the warped
scenario addressing the gauge hierarchy problem). We simply choose this fixed
point to be at $y=\pi R$, rather than $y=0$, which is a purely mathematical
convention since these two points belong to a circle. The real scalar field
has an action of the generic form,

$\displaystyle S_{H}=\int
d^{4}x\left[\,\dfrac{1}{2}\,\partial_{\mu}H\partial^{\mu}H-V(H)\right]\,,$
(11)

with a potential $V$ possessing a minimum which generates a non-vanishing
Vacuum Expectation Value (VEV) for the field $H$ expanded as

$H(x^{\mu})=\frac{v+h(x^{\mu})}{\sqrt{2}}\,,$ (12)

in analogy with the SM Higgs field.

####  2.2.3 Yukawa interactions

We consider the following Yukawa interactions allowing to study the subtleties
induced by the coupling of the above brane-scalar field (at $y=\pi R$) to the
introduced bulk fermions,

$\displaystyle S_{Y}=\int d^{4}x\,\mathcal{L}_{Y}(x^{\mu},\pi
R)\,,\text{with,}\ \ \mathcal{L}_{Y}=-Y_{5}\
HQ^{\dagger}_{L}D_{R}-Y^{\prime}_{5}\ HQ^{\dagger}_{R}D_{L}+{\rm H.c.}\,.$
(13)

Notice that considering operators involving the fields $H$, $Q$, $D$ up to
dimension 5 allows to include such a Yukawa coupling. Let us recall here that
in case of profile jumps at the fixed point at $y=\pi R$, the 5D fields
$Q_{L/R}(x^{\mu},\pi R)$, $D_{L/R}(x^{\mu},\pi R)$ are defined through the
profile convention (3), as already described. The studied model with a Yukawa
coupling at a fixed point will turn out to be dual to the interval model
including a Yukawa coupling at a boundary (see Section 7).

The complex $Y_{5}=\text{e}^{i\alpha_{Y}}|Y_{5}|$ and
$Y_{5}^{\prime}=\text{e}^{i\alpha_{Y}^{\prime}}|Y_{5}^{\prime}|$ Yukawa
coupling constants, entering Eq. (13), are independent and a well-defined 4D
chirality holds for the fermion fields on the 3-brane strictly at $y=\pi R$
Angelescu:2019viv ; Azatov:2009na . To avoid the introduction of a new energy
scale, in the spirit of the warped model, we can define the 5D Yukawa coupling
constants as

$\displaystyle Y_{5}=y_{4}\times 2\pi R\,,\ {\rm and,}\
Y^{\prime}_{5}=y_{4}^{\prime}\times 2\pi R\,,$ (14)

where $y_{4}$, $y_{4}^{\prime}$ are dimensionless coupling constants of ${\cal
O}(1)$. Then, $y_{4}$ can be approximately identified with the SM Yukawa
coupling constant within the decoupling limit, as will be described in Eq.
(91)-(92).

When calculating the tower of excited fermion masses, we restrict our
considerations to the VEV of $H$ and concentrate our attention on the
following part of the action (13),

$\displaystyle S_{X}=\int d^{4}x\,\mathcal{L}_{X}(x^{\mu},\pi
R)\,,\text{with,}\ \ \mathcal{L}_{X}=-X\ Q^{\dagger}_{L}D_{R}-X^{\prime}\
Q^{\dagger}_{R}D_{L}+{\rm H.c.}\,,$ (15)

with the compact notations $X=vY_{5}/\sqrt{2}$ and
$X^{\prime}=vY_{5}^{\prime}/\sqrt{2}$. Based on Eq. (12), the complete action
reads as, $S_{Y}=S_{X}+S_{int}$, with the localised fermion-scalar interaction
terms:

$\displaystyle S_{int}=\int d^{4}x\,\mathcal{L}_{int}(x^{\mu},\pi
R)\,,\text{with,}\ \ \mathcal{L}_{int}=-\dfrac{Y_{5}}{\sqrt{2}}\
hQ^{\dagger}_{L}D_{R}-\dfrac{Y_{5}^{\prime}}{\sqrt{2}}\
hQ^{\dagger}_{R}D_{L}+{\rm H.c.}\,,$ (16)

that allow to work out the 4D effective Yukawa coupling constants.

####  2.2.4 Bilinear brane terms

Introducing all the covariant operators up to mass dimension 5 [like for the
Yukawa couplings (13)] in this model, one should consider as well the
dimension 4 operators given just below, that we call the BBT like in Ref.
Angelescu:2019viv . Furthermore, the presence of the BBT has several
justifications: (i) they allow to avoid physical consistency problems both in
the free case (see Sections 3.1 and 3.3) and with Yukawa couplings (Sections
5.1 and 5.3); (ii) they play the rôle of defining well the model at the two
orbifold fixed points both in the free case (see Sections 3.2 and 3.3) and
with Yukawa couplings (Section 5.3); (iii) they induce the expected matching
of the analytical results on the spectrum derived through the 4D and 5D
approaches (see Sections 4 and 5.3).

The following BBT lead to the SM chirality configuration,

$\displaystyle S_{B}=\int d^{4}x\
\left(\sigma^{Q}_{0}\,\left.\bar{Q}Q\right|_{0}+\sigma^{Q}_{\pi
R}\,\left.\bar{Q}Q\right|_{\pi
R}+\sigma^{D}_{0}\,\left.\bar{D}D\right|_{0}+\sigma^{D}_{\pi
R}\,\left.\bar{D}D\right|_{\pi R}\,\right)\,,$ (17)

where $\sigma^{Q}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}$,
$\sigma^{D}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}$
and for example $\left.\bar{Q}Q\right|_{0}=\bar{Q}(x^{\mu},0)Q(x^{\mu},0)$.
Indeed, without Yukawa couplings, these terms will induce only a non-vanishing
profile $q_{L}^{0}(y)$ [see line 2 of Eq. (LABEL:completeEBCsm0) and Table 1
in case of the zero-mode with mass $m_{0}=0$] in the 5D field
$Q_{L}(x^{\mu},y)$ so that only the Left-handed 4D field $Q^{0}_{L}(x^{\mu})$
will exist. This zero-mode $Q^{0}_{L}(x^{\mu})$, without KK mass contribution,
constitutes the lightest mode of the KK tower and also the SM state. Hence, we
can well recover the SM configuration: a chiral field content and a Left-
handed 4D field potentially representing the ${\rm SU(2)_{L}}$ quark doublet
in the direct extension to gauge symmetries (and three flavours). Given that,
similarly, the BBT (17) will exclusively lead to a Right-handed 4D field
$D^{0}_{R}(x^{\mu})$ [line 1 of Eq. (LABEL:completeEBCsm0)] potentially
representing the SM down quark type (gauge singlet). When adding the Yukawa
couplings (13), this SM chirality set-up remains though it is no more explicit
due to the $Q^{n}(x^{\mu})$-$D^{n}(x^{\mu})$ mixing, via vector-like KK state
mixings, which induces some vector-like mass eigenstates
$\psi^{0}_{L/R}(x^{\mu})$ for the lightest modes of the tower (see Sections 4
and 5.3). In the decoupling limit where heavy KK state mixings tend to vanish,
the SM chirality configuration is recovered as expected.

For completeness, let us underline that in the free case, the opposite BBT
signs, $\sigma^{Q}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}$,
$\sigma^{D}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}$,
would lead to a chiral set-up for the zero-modes but different from the
potential SM chirality configuration, namely:
$Q_{R}^{0}(x^{\mu}),D_{L}^{0}(x^{\mu})$. Similarly, the BBT signs,
$\sigma^{Q}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}$,
$\sigma^{D}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}$,
would lead to the set-up, $Q_{L}^{0}(x^{\mu}),D_{L}^{0}(x^{\mu})$, and,
$\sigma^{Q}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}$,
$\sigma^{D}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}$,
to, $Q_{R}^{0}(x^{\mu}),D_{R}^{0}(x^{\mu})$.

Finally, as will be described in the Sections 3.2 and 3.3, the possible signs,
$\sigma^{Q}_{0(\pi R)}=\pm$ (same sign for $0$ and $\pi R$), would instead
lead to the profile solutions (LABEL:completeEBCcusto0) with two non-vanishing
profiles for the lightest modes (as $m_{0}\neq 0$) and hence to vector-like
states: $Q_{L/R}^{0}(x^{\mu})$. The same statement holds for
$\sigma^{D}_{0(\pi R)}=\pm$ and thus $D_{L/R}^{0}(x^{\mu})$. Such massive
vector-like states 131313Extensive phenomenology at colliders has been
developed about such vector-like particles Azatov:2012rj ; Bonne:2012im ;
Moreau:2012da ; Gopalakrishna:2013hua ; Angelescu:2015kga . can be used to
build custodially protected warped models Agashe:2003zs  and are then called
custodians (see for instance Ref. Bouchart:2008vp ). Of course there exist 8
remaining cases combining the above Lagrangian sign configurations:
$\sigma^{Q}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}},\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}$,
$\sigma^{D}_{0(\pi R)}=\pm$, and, $\sigma^{Q}_{0(\pi R)}=\pm$,
$\sigma^{D}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}},\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}$.

Therefore, it appears clearly that the BBT control the chiral configurations
of the model. The UV completion of the theory can be at the origin of the BBT
and hence of the chirality set-up: chiral nature of the theory and specific
chiralities of the various fields.

To end up this section, we note that the complete toy model studied is
characterised by the action,

$\displaystyle S_{5D}=S_{bulk}+S_{branes}=S_{bulk}+S_{H}+S_{X}+S_{int}+S_{B}\
.$ (18)

The conclusions that will be derived in the present work can be directly
extended to the realistic warped model with SM bulk matter addressing the
fermion mass and gauge hierarchies, along the same lines as the flavour and
gauge symmetry generalisations described in details in the Section 2.6 of Ref.
Angelescu:2019viv .

##  3 Free bulk fermions on the orbifold

In this section, we calculate the fermionic mass spectrum for the free case
where $Y_{5}=Y_{5}^{\prime}=0$ in the action piece $S_{Y}$ given by Eq. (13).

###  3.1 Applying the NBC

We start by considering the bulk action part,

$S_{bulk}\,,$

of Eq. (4), from the considered action, $S_{5D}$, of Eq. (18). We apply the
least action principle to it which leads to two relations of the kind,
$\delta_{\bar{F}}S_{bulk}=0$, one for each of the unknown 5D fields $F=Q,D$,
and two corresponding ones, $\delta_{F}S_{bulk}=0$, involving the complex
conjugate fields 141414The equations of motion and boundary conditions derived
from the least action principle for the fields and their conjugates are
trivially related through the Hermitian conjugation., since the elementary
field variations $\delta Q_{\alpha}$, $\delta\bar{Q}_{\alpha}$, $\delta
D_{\alpha}$ and $\delta\bar{D}_{\alpha}$ (see Appendix B.1) are generic and
hence independent from each other. Using compact notations, like for example,

$\sum_{\alpha=1}^{4}\delta\bar{F}_{\alpha}\dfrac{\partial\mathcal{L}_{kin}}{\partial\bar{F}_{\alpha}}\,\hat{=}\,\delta{\bar{F}}\dfrac{\partial\mathcal{L}_{kin}}{\partial\bar{F}}\,,$

we can write in particular 151515We omit the global 4-divergence which
vanishes in the action integration due to vanishing fields at the boundaries
at infinities. Indeed, when minimising the action, the varied terms must
vanish separately at infinite boundaries, since the non-vanishing field
variations at boundaries are independent from each other and from the bulk
ones (see also Ref. Peskin:1995ev ). This is realised by the local physics
statement which induces vanishing fields at infinities due to the wave
function normalisation conditions (see also Ref. Schwartz:2013pla ).,

$\displaystyle\delta_{\bar{F}}S_{bulk}$ $\displaystyle=\displaystyle{\int
d^{4}x\,\left(\int_{-\pi R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\left\\{\delta{\bar{F}}\dfrac{\partial\mathcal{L}_{kin}}{\partial\bar{F}}+\delta\left(\partial_{M}\bar{F}\right)\dfrac{\partial\mathcal{L}_{kin}}{\partial\,\partial_{M}\bar{F}}\right\\}}$
$\displaystyle=\displaystyle{\int d^{4}x\,\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\left\\{\delta{\bar{F}}\dfrac{\partial\mathcal{L}_{kin}}{\partial\bar{F}}+\partial_{M}\left[\delta\bar{F}\dfrac{\partial\mathcal{L}_{kin}}{\partial\,\partial_{M}\bar{F}}\right]-\delta\bar{F}\,\partial_{M}\dfrac{\partial\mathcal{L}_{kin}}{\partial\,\partial_{M}\bar{F}}\right\\}}$
$\displaystyle=\displaystyle{\int d^{4}x\,\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\left\\{\delta{\bar{F}}\left[\dfrac{\partial\mathcal{L}_{kin}}{\partial\bar{F}}-\partial_{M}\dfrac{\partial\mathcal{L}_{kin}}{\partial\,\partial_{M}\bar{F}}\right]\right\\}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\displaystyle{\int d^{4}x\
\left(\left.\delta{\bar{F}}\dfrac{\partial\mathcal{L}_{kin}}{\partial\,\partial_{4}\bar{F}}\right|_{-\pi
R^{+}}^{0^{-}}+\left.\delta{\bar{F}}\dfrac{\partial\mathcal{L}_{kin}}{\partial\,\partial_{4}\bar{F}}\right|_{0}^{\pi
R}\right)}\ .$ (19)

Based on the Lagrangian $\mathcal{L}_{kin}$ of Eq. (5), these two bulk terms
take the same form (the first one being calculated explicitly in Eq. (113) to
clarify the spinor component treatment) and the two remaining brane terms can
be calculated as well:

$\displaystyle\delta_{\bar{F}}S_{bulk}$ $\displaystyle=\displaystyle{\int
d^{4}x\,\left(\int_{-\pi R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\left\\{\delta{\bar{F}}\left[i\Gamma^{M}\partial_{M}F\right]\right\\}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\displaystyle{\int d^{4}x\
\left(\left.\delta{\bar{F}}\left[-\dfrac{\gamma^{5}}{2}F\right]\right|_{0^{+}}^{\pi
R^{-}}+\left.\delta{\bar{F}}\left[-\dfrac{\gamma^{5}}{2}F\right]\right|_{0}^{\pi
R}\right)}\ ,$ (20)

where we have further invoked the $\mathbb{Z}_{2}$ transformations (7)-(10)
for the generic 5D field, Eq. (114) for its variation and $\gamma^{5}$
properties:

$\displaystyle\left.\delta{\bar{F}}\left[-\dfrac{\gamma^{5}}{2}F\right]\right|_{0^{-},-\pi
R^{+}}$
$\displaystyle=\left.\left(\mp\delta\bar{F}\gamma^{5}\right)\left[-\dfrac{\gamma^{5}}{2}\left(\pm\gamma^{5}F\right)\right]\right|_{0^{+},\pi
R^{-}}=-\left.\delta{\bar{F}}\left[-\dfrac{\gamma^{5}}{2}F\right]\right|_{0^{+},\pi
R^{-}}\ .$

Then thanks to Eq. (3) 161616Those continuity relations lead to
$\bar{F}|_{0}=\bar{F}|_{0^{+}}$, i.e.
$\bar{F}_{\alpha}|_{0}=\bar{F}_{\alpha}|_{0^{+}}$ [c.f. Eq. (109)], and in
turn to $\delta\bar{F}_{\alpha}|_{0}=\delta\bar{F}_{\alpha}|_{0^{+}}$ which
can be written as $\delta\bar{F}|_{0}=\delta\bar{F}|_{0^{+}}$ via Eq. (110).
Similarly we get $\delta\bar{F}|_{\pi R}=\delta\bar{F}|_{\pi R^{-}}$. and Eq.
(111)-(112), respectively, the expression (20) simplifies to,

$\displaystyle\delta_{\bar{F}}S_{bulk}$ $\displaystyle=\displaystyle{\int
d^{4}x\,\left(\int_{-\pi R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\left\\{\delta{\bar{F}}\left[i\Gamma^{M}\partial_{M}F\right]\right\\}}+\displaystyle{\int
d^{4}x\
2\left.\delta{\bar{F}}\left[-\dfrac{\gamma^{5}}{2}F\right]\right|_{0}^{\pi
R}}$ $\displaystyle=$ $\displaystyle\displaystyle{\int
d^{4}x\,\left(\int_{-\pi R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\left\\{\delta{\bar{F}}\left[i\Gamma^{M}\partial_{M}F\right]\right\\}}+\displaystyle{\int
d^{4}x\left.\left[\delta{F^{\dagger}_{R}}F_{L}-\delta{F^{\dagger}_{L}}F_{R}\right]\right|_{0}^{\pi
R}}.$ (21)

In this expression, the bulk and brane variations – respectively the volume
and surface terms – must vanish separately due to independent field variations
(no reason to be linked). Besides all those field variations are not vanishing
(unknown fields) so that we get the bulk Equations Of Motion (EOM),

$i\,\Gamma^{M}\partial_{M}F=0\,,\ \ \ \ \forall\,x^{\mu},\ \forall\,y\in[-\pi
R^{+},0^{-}]\cup[0,\pi R]\ ,$ (22)

and the Natural Boundary Conditions (NBC),

$F_{L}|_{0}=F_{R}|_{0}=F_{L}|_{\pi R}=F_{R}|_{\pi R}=0\,.$ (23)

At this level, we can first solve Eq. (22) together with Eq. (23) to find out
the $F$ fields over the domain, $y\in[0,\pi R]$. This is precisely what has
been done in the preliminary Ref. Angelescu:2019viv  where the two exactly
identical Eq. (3.3) and (3.4) [there] have been solved over the interval,
$y\in[0,L]$. Since the fields are continuous over $y\in[0,\pi R]$ [c.f. Eq.
(3)] like there over $y\in[0,L]$, we can thus apply here the results obtained
in this reference: the solutions found for Eq. (22)-(23) are expressed through
the KK decomposition (with a similar choice of global factor),

$F_{L/R}\left(x^{\mu},y\right)=\dfrac{1}{\sqrt{2\pi
R}}\displaystyle{\sum_{n=0}^{+\infty}f^{n}_{L/R}(y)\,F^{n}_{L/R}\left(x^{\mu}\right)}\
,$ (24)

where the 4D fields $F^{n}_{L/R}=Q^{n}_{L/R},D^{n}_{L/R}$ represent the KK
states and satisfy the Dirac-Weyl equations,

$\forall\,n\in\mathbb{N},\ \left\\{\begin{array}[]{r c
l}i\bar{\sigma}^{\mu}\partial_{\mu}F_{L}^{n}\left(x^{\mu}\right)-m_{n}\,F_{R}^{n}\left(x^{\mu}\right)&=&0\,,\\\
\vspace{-0.2cm}\hfil\\\
i\sigma^{\mu}\partial_{\mu}F_{R}^{n}\left(x^{\mu}\right)-m_{n}\,F_{L}^{n}\left(x^{\mu}\right)&=&0\,,\end{array}\right.$
(25)

involving the KK mass eigenvalues $m_{n}$, while the only resulting profiles
$f^{n}_{L/R}(y)=q^{n}_{L/R}(y)$, $d^{n}_{L/R}(y)$, included respectively into
$F=Q,D$, are vanishing over $y\in[0,\pi R]$. The opposite signs in front of
each mass term of the bulk profile EOM induced by Eq. (22), with respect to
the calculations of Ref. Angelescu:2019viv , just originate from a different
sign convention for the $\Gamma^{4}$ matrix [see Eq. (107)] and hence do not
modify the (un)physical result of vanishing profiles. Now let us study the
profile solutions in the complementary region, $y\in[-\pi R^{+},0^{-}]$.
Inserting the KK decomposition (24) into the first type of $\mathbb{Z}_{2}$
transformation (7), one obtains the $\mathbb{Z}_{2}$ transformations directly
on the $f^{n}_{L/R}(y)$ profiles ($\forall\,n\in\mathbb{N}$):

$Type\,I\left\\{\begin{array}[]{c c
c}\displaystyle{\sum_{n=0}^{+\infty}\left[q^{n}_{L(R)}(-y)\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}q^{n}_{L(R)}(y)\right]\,Q^{n}_{L(R)}\left(x^{\mu}\right)}=0\,\Rightarrow\,q^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}q^{n}_{L(R)}(y)\\\
\vspace{-0.2cm}\hfil\\\
\displaystyle{\sum_{n=0}^{+\infty}\left[d^{n}_{L(R)}(-y)\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}d^{n}_{L(R)}(y)\right]\,D^{n}_{L(R)}\left(x^{\mu}\right)}=0\,\Rightarrow\,d^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}d^{n}_{L(R)}(y)\end{array}\right.$
(26)

where the implications come from the linear independence of mass eigenstates
$F^{n}_{L/R}\left(x^{\mu}\right)$. Similarly, for the three other types of
$\mathbb{Z}_{2}$ transformations (8)-(10), we have the following profile
parities:

$\displaystyle II\left\\{\begin{array}[]{c c
c}q^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}q^{n}_{L(R)}(y)\\\
\vspace{-0.2cm}\hfil\\\
d^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}d^{n}_{L(R)}(y)\end{array}\right.III\left\\{\begin{array}[]{c
c
c}q^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}q^{n}_{L(R)}(y)\\\
\vspace{-0.2cm}\hfil\\\
d^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}d^{n}_{L(R)}(y)\end{array}\right.$
(41) $\displaystyle IV\left\\{\begin{array}[]{c c
c}q^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}q^{n}_{L(R)}(y)\\\
\vspace{-0.2cm}\hfil\\\
d^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}d^{n}_{L(R)}(y)\end{array}\right.$
(49)

Therefore, all the $f^{n}_{L/R}(y)$ profiles are systematically vanishing on
the whole $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold region, $y\in[-\pi
R^{+},0^{-}]\cup[0,\pi R]$. Such profiles conflict with the two (for $L/R$)
ortho-normalisation conditions over the full domain,

$\forall\,n,m\in\mathbb{N},\ \dfrac{1}{2\pi R}\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
f^{n*}_{L/R}(y)\,f^{m}_{L/R}(y)=\delta_{nm}\,,$ (50)

originating from the condition of a canonical form for the 4D effective
kinetic terms. Hence the solutions for the fields obtained through this first
method are not physically consistent.

###  3.2 Introducing the EBC

In fact, one necessary ingredient was missing in the naive approach of Section
3.1. In order to identify it, we have to study the conserved fermion
probability currents corresponding, via the Noether’s theorem, to the global
${\rm U(1)_{Q}}$ and ${\rm U(1)_{D}}$ symmetries of the action,

$S_{bulk}\,,$

involving the Lagrangian (5). The two independent global ${\rm U(1)_{Q,D}}$
transformations of the fields, letting $\mathcal{L}_{kin}$ invariant, act
respectively as,

$\displaystyle Q\mapsto\text{e}^{i\alpha}Q\,,\
\bar{Q}\mapsto\text{e}^{-i\alpha}\bar{Q}\,,\,{\rm and,}\
D\mapsto\text{e}^{i\alpha^{\prime}}D\,,\
\bar{D}\mapsto\text{e}^{-i\alpha^{\prime}}\bar{D}\,,$ (51)

where $\alpha,\alpha^{\prime}$ ($\in\mathbb{R}$) are continuous constants
entering for instance the infinitesimal field variations 171717Different clear
notations are used here for the infinitesimal field variations under specific
transformations, $\underline{\delta}F$, and the above generic field variations
in the variation calculus context of the least action principle, $\delta F$
[see typically Eq. (110)].:

$\underline{\delta}Q=i\alpha Q\,,\ \
\underline{\delta}\bar{Q}=-i\alpha\bar{Q}\,.$

Choosing instead to consider a unique symmetry ($\alpha=\alpha^{\prime}$ for
any field $F$) would correspond to a particular case only, among the general
Lagrangian symmetry possibilities. Besides, this particular case would not
provide the maximal information, since one symmetry would be associated to
only one conserved probability current. We thus well consider, in this
subsection, the transformations (51) [with both possibilities,
$\alpha\neq\alpha^{\prime}$ or $\alpha=\alpha^{\prime}$] and the two
independent ${\rm U(1)_{Q,D}}$ symmetries. Based on these two symmetries, and
the bulk EOM whose standard structure appears in Eq. (19), the Noether’s
theorem predicts the local conservation relation,

$\partial_{M}j_{F}^{M}=0\,,$ (52)

for the two probability currents,

$j^{M}_{Q}=-\alpha\bar{Q}\;\Gamma^{M}Q\,,\
j^{M}_{D}=-\alpha^{\prime}\bar{D}\;\Gamma^{M}D\,,$ (53)

as derived in details within the Appendix B of Ref. Angelescu:2019viv . This
relation holds over the whole $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold
domain, $y\in[-\pi R^{+},0^{-}]\cup[0,\pi R]$, since the sole bulk terms in
the action infinitesimal variation – under ${\rm U(1)_{Q,D}}$ transformation –
must vanish for any integration sub-region included inside the entire
integration domain of the action precisely defined for the model. The
mathematical consistency of the condition (52) imposes necessarily continuous
5-current components over all the model space-time and in particular a
continuous $j_{F}^{4}$ along $y\in[-\pi R^{+},0^{-}]\cup[0,\pi R]$
181818Notice that this condition is in agreement with Eq. (3) which guarantees
continuous fields along $y\in[-\pi R^{+},0^{-}]\cup[0,\pi R]$.. Furthermore, a
jump of the form, $j_{F}^{4}|_{0^{-}}\neq j_{F}^{4}|_{0}$, would not determine
any field at the fixed point and thus would not lead to vanishing variations
in Eq. (20) that would modify the BC (23) inducing non-physical solutions. A
similar argument applies at the other fixed point, $y=\pi R\equiv-\pi R$.
Hence, one has to consider the remaining model possibility,
$j_{F}^{4}|_{0^{-}}=j_{F}^{4}|_{0}$ and 191919A change must occur at both
fixed points to cure the problems of the solutions worked out in previous
subsection. $j_{F}^{4}|_{-\pi R^{+}}=j_{F}^{4}|_{\pi R}$, so that this current
component is continuous over all the range, $y\in(-\pi R,\pi R]$. In
particular, we can now write,

$j_{F}^{4}|_{0^{-}}=j_{F}^{4}|_{0}=j_{F}^{4}|_{0^{+}}\,.$ (54)

This obtained relation must be compared with the following one, coming
directly from the $\mathbb{Z}_{2}$ transformations of type (7)-(10) and
$\gamma^{5}$ properties,

$\displaystyle\left.j_{F}^{4}\right|_{0^{-}}$
$\displaystyle=-\alpha^{(\prime)}\left.\bar{F}\;\Gamma^{4}F\right|_{0^{-}}=-\alpha^{(\prime)}\left.\left(\pm\gamma^{5}F\right)^{\dagger}\gamma^{0}\left[-i\gamma^{5}\right]\left(\pm\gamma^{5}F\right)\right|_{0^{+}}$
$\displaystyle=\left.\alpha^{(\prime)}F^{\dagger}\gamma^{0}\gamma^{5}\left[-i\gamma^{5}\right]\left(\gamma^{5}F\right)\right|_{0^{+}}=\alpha^{(\prime)}\left.\bar{F}\;\Gamma^{4}F\right|_{0^{+}}=-\left.j_{F}^{4}\right|_{0^{+}}\,.$
(55)

The combination of Eq. (54) and Eq. (55) gives rise to a vanishing current
component at the fixed point:

$j_{F}^{4}|_{0^{-}}=j_{F}^{4}|_{0}=j_{F}^{4}|_{0^{+}}=0\,.$

Similar arguments regarding the second fixed point imply obviously that,

$j_{F}^{4}|_{\pi R^{-}}=j_{F}^{4}|_{\pi R}=j_{F}^{4}|_{-\pi R^{+}}=0\,,$

so that, using the generic chiral decomposition (112), we get the following
current conditions,

$\left.j_{F}^{4}\right|_{0,\pi
R}=i\alpha^{(\prime)}\left.\left(F^{\dagger}_{L}F_{R}-F^{\dagger}_{R}F_{L}\right)\right|_{0,\pi
R}=0\,,$ (56)

leading to the minimal Boundary Conditions (BC),

$\left\\{\begin{array}[]{r c l}F_{L}|_{0}&=&0\,,\\\ &\text{or}\\\
F_{R}|_{0}&=&0\,,\end{array}\right.and\,\ \left\\{\begin{array}[]{r c
l}F_{L}|_{\pi R}&=&0\,,\\\ &\text{or}\\\ F_{R}|_{\pi
R}&=&0\,.\end{array}\right.\ \ \ \ \ \ {\rm[EBC]}$ (57)

These BC induce systematically the vanishing of all the brane terms in the
varied action obtained in Eq. (21). Indeed, for example, the fixed value
$F_{L}|_{0}=0$ implies $F^{\dagger}_{L}|_{0}=0$ and in turn $\delta
F^{\dagger}_{L}|_{0}=0$ 202020Rigorously speaking, the action should not be
minimised with respect to the known fixed fields so that the terms with
vanishing field variations should not even appear. In fact, the brane terms of
Eq. (21) should originally be written as a generic sum over unfixed fields.
[considering more precisely their two respective components as is clear from
Appendix B.1]. Therefore the sole remaining BC are those of Eq. (57): there
are no more NBC generated from the brane terms of Eq. (21) and we name the BC
(57) as EBC since they are imposed by the $\mathbb{Z}_{2}$ transformations
(55) which contribute to define the studied model. From the point of view of
the methodology, notice interestingly that it was necessary to consider the
fermion probability currents to reveal the existence of the EBC. Now, solving
the new EBC (57) together with the unchanged bulk EOM (22) over the domain,
$y\in[0,\pi R]$, was precisely realised in Ref. Angelescu:2019viv  where the
same Eq. (3.3) and (3.16) [there] were solved over the interval, $y\in[0,L]$.
Once more, since the fields are continuous over $y\in[0,\pi R]$ [see Eq. (3)]
like there over $y\in[0,L]$, we can apply here the results derived in this
previous work: the 5D solutions found for Eq. (22)-(57) are given by Eq.
(24)-(25) and the following four possible sets of profiles over $y\in[0,\pi
R]$ together with the associated KK mass spectrum equations
($\forall\,n\in\mathbb{N}$),

$\displaystyle 1)$ $\displaystyle(--):\
f_{L}^{n}(y)=B^{n}_{L}\;\sin(m_{n}\,y)\,,\ (++):\
f_{R}^{n}(y)=B^{n}_{L}\;\cos(m_{n}\,y)\,;\ \sin(m_{n}\,\pi R)=0\,,$
$\displaystyle 2)$ $\displaystyle(++):\
f_{L}^{n}(y)=B^{n}_{R}\;\cos(m_{n}\,y)\,,\ (--):\
f_{R}^{n}(y)=-B^{n}_{R}\;\sin(m_{n}\,y)\,;\ \sin(m_{n}\,\pi R)=0\,,$

and,

$\displaystyle 3)$ $\displaystyle(-+):\
f_{L}^{n}(y)=B^{n}_{L}\;\sin(m_{n}\,y)\,,\ (+-):\
f_{R}^{n}(y)=B^{n}_{L}\;\cos(m_{n}\,y)\,;\ \cos(m_{n}\,\pi R)=0\,,$
$\displaystyle 4)$ $\displaystyle(+-):\
f_{L}^{n}(y)=B^{n}_{R}\;\cos(m_{n}\,y)\,,\ (-+):\
f_{R}^{n}(y)=-B^{n}_{R}\;\sin(m_{n}\,y)\,;\ \cos(m_{n}\,\pi R)=0\,.$

The opposite signs in front of the $(--)$ and $(-+)$ profiles, with respect to
the results in Ref. Angelescu:2019viv , just come from a different sign
convention for the $\Gamma^{4}$ matrix, between here [see Eq. (107)] and this
reference. In Eq. (LABEL:completeEBCsm0)-(LABEL:completeEBCcusto0), we use the
standard BC notations, i.e. $-$ or $+$ for instance at $y=0$ stands
respectively for the Dirichlet or Neumann BC: $f_{L/R}^{n}(0)=0$ or
$\partial_{4}f_{L/R}^{n}(y)|_{0}=0$. For example, the symbolic notation $(-+)$
denotes Dirichlet (Neumann) BC at $y=0$ ($y=\pi R$). These notations make
explicit the correspondence between the four EBC (57) and the four solutions
(LABEL:completeEBCsm0)-(LABEL:completeEBCcusto0). The equation
$\sin(m_{n}\,\pi R)=0$ possesses the following solutions for the KK mass
spectrum,

$\displaystyle m_{n}=\pm\frac{n}{R},\,n\in\mathbb{N}\,.$ (60)

Similarly, the equation $\cos(m_{n}\,\pi R)=0$ has the solutions:

$\displaystyle m_{n}=\pm\frac{2n+1}{2R},\,n\in\mathbb{N}\,.$ (61)

The part of the general $f_{L/R}^{n}(y)$ solutions in the complementary
domain, $y\in[-\pi R^{+},0^{-}]$, is now obtained via the four types of
$\mathbb{Z}_{2}$ transformations (26)-(49). Therefore, the inclusion of the
EBC based on the vanishing probability currents allows to obtain consistent
fermion profile and mass solutions.

| $\mathbb{Z}_{2}$ | Fields
---|---|---
Continuity | $Q_{L/R}$ | $D_{L/R}$
domains | $q^{n}_{L}(y)/e^{i\alpha_{Q}^{n}}$ | $q^{n}_{R}(y)/e^{i\alpha_{Q}^{n}}$ | $d^{n}_{L}(y)/e^{i\alpha_{D}^{n}}$ | $d^{n}_{R}(y)/e^{i\alpha_{D}^{n}}$
$[0,\pi R]$ | Any | $\sqrt{2}\;\cos(m_{n}\;y)$ | $-\sqrt{2}\;\sin(m_{n}\;y)$ | $\sqrt{2}\;\sin(m_{n}\;y)$ | $\sqrt{2}\;\cos(m_{n}\;y)$
$[-\pi R^{+},0^{-}]$ | I | $\sqrt{2}\;\cos(m_{n}\;y)$ | $-\sqrt{2}\;\sin(m_{n}\;y)$ | $\sqrt{2}\;\sin(m_{n}\;y)$ | $\sqrt{2}\;\cos(m_{n}\;y)$
II | $-\sqrt{2}\;\cos(m_{n}\;y)$ | $\sqrt{2}\;\sin(m_{n}\;y)$ | $-\sqrt{2}\;\sin(m_{n}\;y)$ | $-\sqrt{2}\;\cos(m_{n}\;y)$
III | $\sqrt{2}\;\cos(m_{n}\;y)$ | $-\sqrt{2}\;\sin(m_{n}\;y)$ | $-\sqrt{2}\;\sin(m_{n}\;y)$ | $-\sqrt{2}\;\cos(m_{n}\;y)$
IV | $-\sqrt{2}\;\cos(m_{n}\;y)$ | $\sqrt{2}\;\sin(m_{n}\;y)$ | $\sqrt{2}\;\sin(m_{n}\;y)$ | $\sqrt{2}\;\cos(m_{n}\;y)$
KK Masses |  $|m_{n}|=n/R$, $n\in\mathbb{N}$

Table 1: SM-like free fermionic $f^{n}_{L/R}(y)$ profiles – normalised to the
indicated complex phases – on the two orbifold domains $[-\pi R^{+},0^{-}]$
and $[0,\pi R]$, corresponding to the solution of line 1 (2) in Eq.
(LABEL:completeEBCsm0) for the field $D$ ($Q$). The associated mass spectrum
(60) is included as well for completeness. The profiles are given for the four
types of $\mathbb{Z}_{2}$ transformations (26)-(49). The phases
$\alpha^{n}_{Q/D}$ belong to $\mathbb{R}$. In the special case, $n=0$, the
$\sqrt{2}$ factors must all be replaced by the unity.

In Table 1, we present the explicit solutions over the whole orbifold domain
for the SM-like profile $d_{L/R}^{n}(y)$ ($q_{L/R}^{n}(y)$) taken from line 1
(2) of Eq. (LABEL:completeEBCsm0): see the discussion on SM chirality
configuration in Section 2.2.4. The mass spectrum for the 4D KK states is
defined by Eq. (25) and it is already determined by Eq. (60)-(61). Notice on
Table 1 that the same $m_{n}$ spectrum enters the profile solutions in both
regions, $y\in[0,\pi R]$, and, $y\in[-\pi R^{+},0^{-}]$. In this table, we
also give the general values of the $B_{L/R}^{n}$ complex constants, in Eq.
(LABEL:completeEBCsm0), obtained from the ortho-normalisation conditions (50)
212121Here, thanks to the profile parities, a change of variable, $y\to-y$,
could be applied to recover exclusively the integration domain $[0,\pi R]$..
We observe on Table 1 that the choice of type of $\mathbb{Z}_{2}$
transformation is just a convention since it can modify the profile signs but
it affects neither the mass spectrum nor the fermion chirality configuration –
as a certain chiral zero-mode profile vanishing on the region $[0,\pi R]$ is
also systematically vanishing over $y\in[-\pi R^{+},0^{-}]$. In contrast, the
chirality configuration and mass spectrum are fixed by the choice of EBC (57)
which can lead either to the two kinds of chiral solutions in Eq.
(LABEL:completeEBCsm0) or to the vector-like solutions
(LABEL:completeEBCcusto0).

In Figure 2, we draw the first two excitation profiles for each free solution
presented in Table 1 within the simple real case, $\alpha^{n}_{Q,D}=0$, and
for two different types of $\mathbb{Z}_{2}$ transformations from Eq.
(26)-(49). We see clearly on Figure 2 that for example with the type $II$ of
$\mathbb{Z}_{2}$ transformation, jumps appear for the profiles
$q^{0,1}_{L}(y)$ and $d^{0,1}_{R}(y)$ at the two fixed points at, $y=0$,
$y=\pi R\equiv-\pi R$, in the scenario without Yukawa couplings. The presence
of profile discontinuities here already justifies the treatment exposed in
Section 2.1. The precise prescription (3) regarding the action integration
domain, described in this section, renders the jumps of Figure 2 consistent
mathematically: the difference, e.g. $q^{1}_{L}(0^{-})\neq q^{1}_{L}(0)$, is
compatible with a well defined Lagrangian integrand over the action
integration domain, $y\in[-\pi R^{+},0^{-}]\cup[0,\pi R]$, where the profiles
are continuous.

Figure 2: Zero-mode and KK dimensionless wave functions $q^{n}_{L/R}(y)$,
$d^{n}_{L/R}(y)$, with $n=0,1,2$, along the $\mathcal{S}^{1}/\mathbb{Z}_{2}$
orbifold domain, $y\in[-\pi R^{+},0^{-}]\cup[0,\pi R]$, corresponding to the
free solutions of Table 1 in the simplified case, $\alpha^{n}_{Q,D}=0$,
$m_{n}>0$, and for the two different types of $\mathbb{Z}_{2}$
transformations, $I$, $II$ from Eq. (26)-(49). The two fixed points at, $y=0$,
$y=\pi R\equiv-\pi R$, and Dirichlet/Neumann BC, $(-)/(+)$, are indicated on
the graph.

###  3.3 Introducing the BBT

As suggested in Section 2.2.4, we can alternatively introduce the dimension 4
operators of Eq. (17) to study their effects with respect to the
inconsistencies raised in Section 3.1. Hence, to the action $S_{bulk}$ from
Eq. (4), we add now another part and consider:

$S_{bulk}+S_{B}\,.$

The variations of $S_{B}$ with respect to the generic field $\bar{F}$ [using
Eq. (111)],

$\delta_{\bar{F}}S_{B}=\displaystyle{\int d^{4}x\
\left(\sigma^{F}_{0}\left.\delta
F^{\dagger}_{L}F_{R}\right|_{0}+\sigma^{F}_{0}\left.\delta
F^{\dagger}_{R}F_{L}\right|_{0}+\sigma^{F}_{\pi R}\left.\delta
F^{\dagger}_{L}F_{R}\right|_{\pi R}+\sigma^{F}_{\pi R}\left.\delta
F^{\dagger}_{R}F_{L}\right|_{\pi R}\right)}\,,$

together with Eq. (21) allow to write down the variations of the free fermion
action:

$\displaystyle\delta_{\bar{F}}\left(S_{bulk}+S_{B}\right)=\displaystyle{\int
d^{4}x\,\left\\{\left(\int_{-\pi R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\delta{\bar{F}}\,i\Gamma^{M}\partial_{M}F\right.}+\left(\sigma^{F}_{0}1+1\right)\left.\delta
F^{\dagger}_{L}F_{R}\right|_{0}$ $\displaystyle\displaystyle{\ \ \ \ \ \
+\left(\sigma^{F}_{0}1-1\right)\left.\delta
F^{\dagger}_{R}F_{L}\right|_{0}}\displaystyle{\left.+\left(\sigma^{F}_{\pi
R}1-1\right)\left.\delta F^{\dagger}_{L}F_{R}\right|_{\pi
R}+\left(\sigma^{F}_{\pi R}1+1\right)\left.\delta
F^{\dagger}_{R}F_{L}\right|_{\pi R}\right\\}}.$ (62)

The individual vanishing of those volume and surface terms lead to the EOM
(22) together with the four following NBC, depending on the two
$\sigma^{F}_{0,\pi R}$ choices,

$\left\\{\begin{array}[]{r c l}F_{L}|_{0}&=&0\ (\sigma^{F}_{0}=-),\\\
&\text{or}\\\ F_{R}|_{0}&=&0\ (\sigma^{F}_{0}=+),\end{array}\right.and\,\
\left\\{\begin{array}[]{r c l}F_{L}|_{\pi R}&=&0\ (\sigma^{F}_{\pi R}=+),\\\
&\text{or}\\\ F_{R}|_{\pi R}&=&0\ (\sigma^{F}_{\pi R}=-).\end{array}\right.\ \
\ \ \ \ {\rm[NBC]}$ (63)

At this level, the EOM and NBC are effectively the same as the EOM (22) and
EBC (57) of the previous subsection, in the domain $y\in[0,\pi R]$, so that we
find again the solutions (LABEL:completeEBCsm0)-(LABEL:completeEBCcusto0)
together with the mass spectra (60)-(61). For instance, the SM-like choice
$\sigma^{Q}_{0(\pi
R)}=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}$
of Eq. (17) leads via Eq. (63) to the solution of line 2 in Eq.
(LABEL:completeEBCsm0). Then the parts of the general profile solutions in the
complementary region, $y\in[-\pi R^{+},0^{-}]$, are found out via the
different types of $\mathbb{Z}_{2}$ transformations (26)-(49) in the free
case, as in Section 3.2, so that the complete solutions are once more
identical and can also be illustrated by the Table 1 and Figure 2 both based
on the ortho-normalisation conditions (50). In conclusion, introducing the BBT
permits to rigorously work out profile and mass solutions. A second conclusion
in this approach is that the chirality set-up – one of the two chiral
solutions (LABEL:completeEBCsm0) or of the vector-like ones
(LABEL:completeEBCcusto0) – and associated mass spectrum are fixed by the
choice of NBC (63) and thus originally by the choices of $\sigma^{F}_{0,\pi
R}$ BBT signs in Eq. (17). In simpler words, the BBT (like the EBC previously)
control the chiral nature of the theory as well as each field chirality.

Let us now discuss the probability currents. The addition of the $S_{B}$ part
in Eq. (17) to $S_{bulk}$ is not affecting the current equations (52)-(53)
since the new brane terms so induced in the infinitesimal action variation –
under the ${\rm U(1)_{Q,D}}$ transformations (51) – vanish due to their ${\rm
U(1)_{Q,D}}$ invariant form. In contrast with the previous subsection and with
the interval model in the free case with BBT Angelescu:2019viv , there exists
no demonstration here of Eq. (56). Nevertheless, we can check that
$j_{F}^{4}|_{0,\pi R}$ is well vanishing by using the obtained solutions
(LABEL:completeEBCsm0)-(LABEL:completeEBCcusto0): the product
$f_{L}^{n}(y)f_{R}^{m}(y)$ systematically vanishes at $y=0,\pi R$. Therefore,
the BBT play the rôle of making $j_{F}^{4}|_{0,\pi R}$ vanish
($\mathbb{Z}_{2}$ transformation consequence) like the EBC were guaranteeing
it in Section 3.2. Note that we could simultaneously apply the EBC and
introduce the BBT but those two processes would be physically redundant to
define the model.

##  4 Brane-localised scalar couplings in the orbifold: 4D approach

Once the free case is addressed, via the EBC (57) in Section 3.2 or the NBC
(63) induced by the BBT in Section 3.3, the free fermion mass spectrum and
profiles are known. Then how to take into account the effects of the action
part $S_{X}$ in the mass spectrum, the action (15) being induced by the Yukawa
interaction between a brane-localised scalar field and bulk fermions? The
considered action reads thus as,

$\displaystyle S_{bulk}+S_{X}\ (+S_{B})\ .$ (64)

A first method called the perturbation method, described in the present
section, is performed at the level of the 4D effective Lagrangian, that is by
calculating the mass mixings between the different levels of the KK towers.
Considering the SM-like profile solutions $d_{L/R}^{n}(y)$ ($q_{L/R}^{n}(y)$)
and associated free KK mass spectrum from line 1 (2) of Eq.
(LABEL:completeEBCsm0), all the initial 4D effective masses for the KK modes
of Eq. (24) in the interaction basis can be classified into two species: the
pure KK masses (60) and the mass contributions from the Yukawa interaction
given by the overlap between the wave functions and Higgs-brane,

$\left\\{\begin{array}[]{l}\displaystyle{\forall(i,j)\in\mathbb{N}^{2},\,\alpha_{ij}=X\,\dfrac{q_{L}^{i}(\pi
R)}{\sqrt{2\pi R}}\,\dfrac{d_{R}^{j}(\pi R)}{\sqrt{2\pi R}}}\,,\\\ \\\
\vspace{-0.8cm}\\\ \displaystyle{\forall(i,j)\in\mathbb{N}^{\star
2},\,\beta_{ij}=X^{\prime}\,\dfrac{d_{L}^{i}(\pi R)}{\sqrt{2\pi
R}}\,\dfrac{q_{R}^{j}(\pi R)}{\sqrt{2\pi R}}}\,.\end{array}\right.$ (65)

In particular, $\beta_{ij}=0$ as imply the respective SM solutions
(LABEL:completeEBCsm0) so that the coupling constant $X^{\prime}$ disappears
from the mass dependences. Note that for similar reasons [c.f. Eq. (112)], in
case of the presence of the BBT (17), those do not generate 4D mass terms. All
the 4D mass terms enter the 4D effective Lagrangian through the following mass
matrix,

$-\chi^{\dagger}_{L}\,\mathcal{M}\,\chi_{R}+\text{H.c.}$

within the field basis noted,

$\left\\{\begin{array}[]{r c
l}\chi^{t}_{L}(x^{\mu})&=&\left(Q_{L}^{0t},Q_{L}^{1t},D_{L}^{1t},Q_{L}^{2t},D_{L}^{2t},\cdots\right),\\\
\\\
\chi^{t}_{R}(x^{\mu})&=&\left(D_{R}^{0t},Q_{R}^{1t},D_{R}^{1t},Q_{R}^{2t},D_{R}^{2t},\cdots\right).\end{array}\right.$
(66)

The texture of this infinite mass matrix $\mathcal{M}$ involving the diagonal
$m_{n}$, off-diagonal $\alpha_{ij}$ and mixing the $Q$, $D$ fields together
can be precisely taken from the interval model context Angelescu:2019viv
(Section 3.2), with the replacement $L\leftrightarrow\pi R$, since the KK
masses and bulk profile solutions are then identical (up to extensions over
$[-\pi R^{+},0^{-}]$ as seen in Section 3.2 here) like the Yukawa interactions
localised at $y=\pi R$ [for any $\mathbb{Z}_{2}$ transformation (26)-(49)].
Now we can apply the results for the mass eigenvalues $M_{n}$ of the 4D
eigenstates $\psi^{n}_{L/R}(x^{\mu})$ obtained through the bi-diagonalisation
performed in this Ref. Angelescu:2019viv , based on the calculations of Ref.
Barcelo:2014kha , by re-normalising $X$ to $X/2$ since the two present
profiles (even or odd) entering $\alpha_{ij}$ are normalised via Eq. (50) over
a domain of double size $2L\leftrightarrow 2\pi R$ compared to the interval
case. Doing so, the obtained exact mass eigenvalues are determined by the
following equation, coming from the characteristic equation,

$\forall\,n\in\mathbb{N},\ \tan^{2}(\sqrt{|M_{n}|^{2}}\,\pi
R)=\left(\dfrac{X}{2}\right)^{2},$ (67)

in the case of a real $X$ parameter and the positive $m_{n}$ branch from Eq.
(60). Notice that the different conventional sign in front of the $(--)$
profiles found in Eq. (LABEL:completeEBCsm0) [here $q^{n}_{R}(y)$ and
$d^{n}_{L}(y)$], with respect to the interval study Angelescu:2019viv , does
not affect the final mass spectrum – as is clear from Eq. (65). Hence, the
physical absolute value of the mass spectrum reads as:

$|M_{n}|=\dfrac{1}{\pi
R}\left|\arctan\left(\dfrac{X}{2}\right)+(-1)^{n}\,\tilde{n}(n)\,\pi\right|,\,n\in\mathbb{N}\,,$
(68)

with the function $\tilde{n}(n)$ defined according to,

$\tilde{n}(n)=\left\\{\begin{array}[]{r c l}&\frac{n}{2}\ {\rm for}\ n\ {\rm
even}\,,\\\ \vspace{-0.2cm}\hfil\\\ &\frac{n+1}{2}\ {\rm for}\ n\ {\rm
odd}\,.\end{array}\right.$ (69)

##  5 Brane-localised scalar couplings in the orbifold: 5D approach

###  5.1 Applying the NBC

Let us now study the presence of Yukawa couplings at the fixed point, $y=\pi
R$, through the action,

$\displaystyle S_{bulk}+S_{X}+S_{B}^{0}\,,\ {\rm with,}\ S_{B}^{0}=\int
d^{4}x\
\left(\sigma^{Q}_{0}\,\left.\bar{Q}Q\right|_{0}+\sigma^{D}_{0}\,\left.\bar{D}D\right|_{0}\right)\,,$
(70)

within the 5D approach, that is by considering the mixings among KK excitation
states at the level of the 5D fields. The BBT introduced here at the fixed
point at $y=0$ are the ones of Eq. (17) leading to SM-like chirality
configurations: $\sigma^{Q}_{0}=+$, $\sigma^{D}_{0}=-$. Those guarantee a
correct treatment of the free brane, like the EBC, as analysed throughout
Section 3. Using Eq. (62) and Eq. (15), one gets directly the following action
variations with respect to the fields $\bar{Q}$ and $\bar{D}$,

$\displaystyle\delta_{\bar{Q}}(S_{bulk}+S_{X}+S_{B}^{0})$
$\displaystyle=\displaystyle{\int d^{4}x\,\left\\{\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\delta{\bar{Q}}\,i\Gamma^{M}\partial_{M}Q\right.}+$ (71)
$\displaystyle\displaystyle{\left.\left.\left[\delta
Q^{\dagger}_{L}\left(-XD_{R}-Q_{R}\right)+\delta
Q^{\dagger}_{R}\left(-X^{\prime}D_{L}+Q_{L}\right)\right]\right|_{\pi
R}+2\left(\left.\delta{Q^{\dagger}_{L}}Q_{R}\right)\right|_{0}\right\\}}\,,$
$\displaystyle\delta_{\bar{D}}(S_{bulk}+S_{X}+S_{B}^{0})$
$\displaystyle=\displaystyle{\int d^{4}x\,\left\\{\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\delta{\bar{D}}\,i\Gamma^{M}\partial_{M}D\right.}+$
$\displaystyle\displaystyle{\left.\left.\left[\delta
D^{\dagger}_{L}\left(-X^{\prime*}Q_{R}-D_{R}\right)+\delta
D^{\dagger}_{R}\left(-X^{*}Q_{L}+D_{L}\right)\right]\right|_{\pi
R}-2\left.\left(\delta{D^{\dagger}_{R}}D_{L}\right)\right|_{0}\right\\}}\,.$

The separate vanishings of these volume and surface terms, induced by the
least action principle, give rise respectively to the EOM (22) and the
following NBC,

$\left\\{\begin{array}[]{l}\left.\left(Q_{R}+X\;D_{R}\right)\right|_{\pi
R}=0\,,\ \ \ \left.\left(D_{L}-X^{*}\;Q_{L}\right)\right|_{\pi R}=0\,,\\\
\left.\left(Q_{L}-X^{\prime}\;D_{L}\right)\right|_{\pi R}=0\,,\ \ \
\left.\left(D_{R}+X^{\prime*}\;Q_{R}\right)\right|_{\pi R}=0\,,\\\
\left.Q_{R}\right|_{0}=0\,,\ \left.D_{L}\right|_{0}=0\,.\end{array}\right.$
(72)

As usual, the 5D field solutions of the EOM (22) and NBC (72) have the form of
the following mixed KK decomposition [instead of Eq. (24)] Casagrande:2008hr ;
Azatov:2009na ,

$\left[\begin{array}[]{r c
l}Q_{L}\left(x^{\mu},y\right)&=&\dfrac{1}{\sqrt{2\pi
R}}\displaystyle{\sum_{n=0}^{+\infty}q^{n}_{L}(y)\,\psi^{n}_{L}\left(x^{\mu}\right)}\,,\\\
\vspace{-0.3cm}\hfil\\\ Q_{R}\left(x^{\mu},y\right)&=&\dfrac{1}{\sqrt{2\pi
R}}\displaystyle{\sum_{n=0}^{+\infty}q^{n}_{R}(y)\,\psi^{n}_{R}\left(x^{\mu}\right)}\,,\\\
\vspace{-0.3cm}\hfil\\\ D_{L}\left(x^{\mu},y\right)&=&\dfrac{1}{\sqrt{2\pi
R}}\displaystyle{\sum_{n=0}^{+\infty}d^{n}_{L}(y)\,\psi^{n}_{L}\left(x^{\mu}\right)}\,,\\\
\vspace{-0.3cm}\hfil\\\ D_{R}\left(x^{\mu},y\right)&=&\dfrac{1}{\sqrt{2\pi
R}}\displaystyle{\sum_{n=0}^{+\infty}d^{n}_{R}(y)\,\psi^{n}_{R}\left(x^{\mu}\right)}\,,\end{array}\right.$
(73)

with the 4D fields $\psi^{n}_{L/R}(x^{\mu})$, already mentioned in Section 4,
satisfying the Dirac-Weyl equations,

$\left\\{\begin{array}[]{r c
l}i\bar{\sigma}^{\mu}\partial_{\mu}\psi^{n}_{L}\left(x^{\mu}\right)-M_{n}\,\psi^{n}_{R}\left(x^{\mu}\right)&=&0\,,\\\
\vspace{-0.2cm}\hfil\\\
i\sigma^{\mu}\partial_{\mu}\psi^{n}_{R}\left(x^{\mu}\right)-M_{n}\,\psi^{n}_{L}\left(x^{\mu}\right)&=&0\,,\end{array}\right.$
(74)

the $M_{n}$ being the fermion mass eigenvalues including the contributions
from the Yukawa terms and these 4D fields the mass eigenstates including the
effects of mixings among the $Q$, $D$ fields as well as (infinite) KK levels.
The explicit profile solutions appearing in Eq. (73) over the domain,
$y\in[0,\pi R]$, were found out for the interval model studied in Ref.
Angelescu:2019viv  where the exactly identical EOM and same NBC, up to a sign
and a factor $2$ in front of each $X^{(\prime)}$ parameter, [respectively Eq.
(3.3) and (5.5) there] have been solved over $y\in[0,L]$. Because the fields
are continuous over $y\in[0,\pi R]$ [c.f. Eq. (3)] like there over
$y\in[0,L]$, one can apply here the conclusions obtained in this reference.
The opposite sign in front of the $X^{(\prime)}$ parameters originates from a
different Dirac matrix sign convention [see the $\Gamma^{4}$ sign in Eq.
(107)] and has thus no physical consequences. The relative factors $2$, at the
same places in the NBC (72), come from the existence of surface terms both at
$y=0$, $0^{-}$ and $y=\pi R$, $-\pi R^{+}$ as is clearly described in Eq.
(20)-(21). These factors turn out not to modify the relations between the
different profile solutions and to only induce a factor-$4$ change in the
final mass spectrum equation, both obtained in Ref. Angelescu:2019viv .
Besides, the two (for $L/R$) following ortho-normalisation conditions over the
full $\mathcal{S}^{1}$ domain [replacing Eq. (50)],

$\forall\,n,m\in\mathbb{N},\ \dfrac{1}{2\pi R}\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi
R}\right)dy\,\left[q^{n*}_{L/R}(y)\,q^{m}_{L/R}(y)+d^{n*}_{L/R}(y)\,d^{m}_{L/R}(y)\right]=\delta_{nm}\,,$
(75)

as induced by the decomposition (73), can be recast into the integration
relations of Ref. Angelescu:2019viv  over the region $[0,\pi R]$ but with a
global factor $2$, thanks to the change of variable $y^{\prime}=-y$, the fixed
odd/even parities of the profiles and Eq. (3), so that the demonstration about
profile solutions on the interval in Ref. Angelescu:2019viv  remains unchanged
here, from this point of view as well. Indeed, injecting the mixed KK
decomposition (73) into the first type of $\mathbb{Z}_{2}$ transformation (7),
we get the $\mathbb{Z}_{2}$ transformations directly on the profiles:

$Type\,I\left\\{\begin{array}[]{c c
c}\displaystyle{\sum_{n=0}^{+\infty}\left[q^{n}_{L(R)}(-y)\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}q^{n}_{L(R)}(y)\right]\,\psi^{n}_{L(R)}\left(x^{\mu}\right)}=0\,\Rightarrow\,q^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}q^{n}_{L(R)}(y)\\\
\vspace{-0.2cm}\hfil\\\
\displaystyle{\sum_{n=0}^{+\infty}\left[d^{n}_{L(R)}(-y)\mathbin{\vbox{\hbox{\oalign{\hfil$\scriptstyle+$\hfil\cr\kern-1.29167pt\cr$\scriptscriptstyle({-})$\cr}}}}d^{n}_{L(R)}(y)\right]\,\psi^{n}_{L(R)}\left(x^{\mu}\right)}=0\,\Rightarrow\,d^{n}_{L(R)}(-y)=\mathbin{\vbox{\hbox{\oalign{$\scriptstyle({+})$\cr\kern-1.29167pt\cr\hfil$\scriptscriptstyle-$\hfil\cr}}}}d^{n}_{L(R)}(y)\end{array}\right.$
(76)

In the same way, for the three other types of $\mathbb{Z}_{2}$ transformations
(8)-(10), one obtains the same profile parities as in Eq. (49). As a
conclusion, the same result as in Ref. Angelescu:2019viv  holds here for the
orbifold: the 4D effective Yukawa coupling constant for the lightest modes
($\psi^{0}_{L,R}$), induced by the found profiles, tends to zero within the
decoupling limit which is not compatible with the SM configuration expected.
The problematic characteristics of the solutions obtained in this naive
approach are confirmed by the final mass spectrum equation,
$\tan^{2}(M_{n}\,\pi R)=|X|^{2}$ (independent from the profile
normalisations), which conflicts analytically with the one obtained through
the 4D method in Eq. (67) for a real $X$ parameter. This failure motivates the
alternative 5D methods of the next two subsections.

###  5.2 Introducing the EBC

Following the same idea as for the free case in Section 3.2, we try now to
find consistent fermion mass solutions via considerations on their currents.
The currents permit a priori to fully define the geometrical field
configuration like here for the $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold
scenario. The complete relevant action including the brane-localised Yukawa
terms (15),

$\displaystyle S_{bulk}+S_{X}+S_{B}^{0}\,,$ (77)

like in Eq. (70), is invariant under the unique ${\rm U(1)_{F}}$ symmetry
defined via Eq. (51) only for,

$\alpha=\alpha^{\prime}\,,$ (78)

since the fermions $Q$ and $D$ are mixed on the brane at $y=\pi R$. Based on
this symmetry involving both $Q$ and $D$ as well as on the bulk EOM [whose
standard structure appears in the action variation (19)], the Noether’s
theorem predicts (c.f. Appendix B of Ref. Angelescu:2019viv ) the new local
probability conservation relation,

$\partial_{M}j^{M}=0\,,\phantom{0}\text{with,}\phantom{0}j^{M}=\displaystyle{\sum_{F=Q,D}j_{F}^{M}}\,,$
(79)

involving the individual currents given by Eq. (53)-(78) over the full
orbifold domain, $y\in[-\pi R^{+},0^{-}]$ $\cup[0,\pi R]$. Notice that the new
$S_{X}$ brane terms entering the infinitesimal action variation – under the
${\rm U(1)_{F}}$ transformations – vanish because of their invariant form and
have thus no direct effect on the conservation relation (79). The mathematical
consistency of the relation (79) implies necessarily the continuity of
5-current components over the whole space-time and in particular a continuous
$j^{4}$ along $y\in[-\pi R^{+},0^{-}]\cup[0,\pi R]$. Besides, a discontinuity
of the form, $j^{4}|_{-\pi R^{+}}\neq j^{4}|_{-\pi R}\equiv j^{4}|_{\pi R}$,
would not fix any field at this fixed point and in turn would not induce
vanishing variations in Eq. (71) possibly modifying the BC (72) which induce
the drawbacks already pointed out in Section 5.1. As a consequence, we must
consider the remaining model possibility:

$j^{4}|_{-\pi R^{+}}=j^{4}|_{-\pi R}\equiv j^{4}|_{\pi R}=j^{4}|_{\pi
R^{-}}\,,$ (80)

where Eq. (3) is also invoked. On the other side, the current $j^{4}$ is odd
under any type of $\mathbb{Z}_{2}$ transformation (7)-(10) as can be shown in
a similar way as in Eq. (55):

$\displaystyle\left.j^{4}\right|_{-\pi R^{+}}=-\left.j^{4}\right|_{\pi
R^{-}}\,.$ (81)

Combining Eq. (80) with Eq. (81) leads to,

$j^{4}|_{\pi R^{-}}=j^{4}|_{\pi R}=j^{4}|_{-\pi R^{+}}=0\,,$

so that, using Eq. (56) and (78), we get the relation (inducing EBC),

$\left.j^{4}\right|_{\pi
R}=i\alpha\left.\left(Q^{\dagger}_{L}Q_{R}-Q^{\dagger}_{R}Q_{L}+D^{\dagger}_{L}D_{R}-D^{\dagger}_{R}D_{L}\right)\right|_{\pi
R}=0\,,$ (82)

and its variation (for a non-trivial transformation with $\alpha\neq 0$),

$\displaystyle\left(\delta Q^{\dagger}_{L}Q_{R}+Q^{\dagger}_{L}\delta
Q_{R}-\delta Q^{\dagger}_{R}Q_{L}-Q^{\dagger}_{R}\delta Q_{L}\right.\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (83)
$\displaystyle\left.\left.+\,\delta D^{\dagger}_{L}D_{R}+D^{\dagger}_{L}\delta
D_{R}-\delta D^{\dagger}_{R}D_{L}-D^{\dagger}_{R}\delta
D_{L}\right)\right|_{\pi R}=0\,.$

At this level, we can consider the search for field solutions of vanishing Eq.
(71) and Eq. (82)-(83) first on the domain, $y\in[0,\pi R]$, which is
equivalent to the search performed for the interval model in Ref.
Angelescu:2019viv  with the replacement, $L\leftrightarrow\pi R$. Given that
the ortho-normalisation condition (75) written on the domain $[0,\pi R]$ is
the same within the orbifold and interval frameworks, up to an overall factor
$2$, we can apply the conclusion of Ref. Angelescu:2019viv  and claim that
there exists no SM-like consistent solution for the fields (over $y\in[0,\pi
R]$) for similar reasons as in Section 5.1. As a conclusion, the introduction
of EBC does not constitute the correct approach towards the treatment of
point-like Yukawa interactions at a fixed point of the
$\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold. Regarding the bulk fermion
probability currents, both the cases of a $j^{4}$ jump and a $j^{4}$
continuity at the Yukawa coupling location, $y=\pi R$, lead to inconsistent
field solutions so that, at this stage of the study, there exists no
theoretical proof of the $j^{4}$ continuity – and via Eq. (81) of its
vanishing – at this fixed point, in contrast with the interval model (case of
presence of boundary-localised Yukawa interactions) Angelescu:2019viv .

###  5.3 Introducing the BBT

In order to get meaningful field solutions in the presence of brane-localised
Yukawa couplings at the fixed point, $y=\pi R$, let us finally try the
introduction of the SM-like BBT (17) as in the free case of Section 3.3 or as
in the interval model Angelescu:2019viv . We thus consider here the same
action as in Eq. (70)-(77) but adding now the BBT at $y=\pi R$:

$\displaystyle S_{bulk}+S_{X}+S_{B}\,.$

Using Eq. (62) and Eq. (71), we find the following action variations with
respect to $\bar{Q}$ and $\bar{D}$:

$\displaystyle\delta_{\bar{Q}}(S_{bulk}+S_{X}+S_{B})$
$\displaystyle=\displaystyle{\int d^{4}x\,\left\\{\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\delta{\bar{Q}}\,i\Gamma^{M}\partial_{M}Q\right.}$ $\displaystyle\ \ \
+\displaystyle{\left.\left.\left[-2\,\delta
Q^{\dagger}_{L}\left(Q_{R}+\dfrac{X}{2}D_{R}\right)-X^{\prime}\delta
Q^{\dagger}_{R}D_{L}\right]\right|_{\pi
R}+2\left.\left(\delta{Q^{\dagger}_{L}}Q_{R}\right)\right|_{0}\right\\}}\,,$
$\displaystyle\delta_{\bar{D}}(S_{bulk}+S_{X}+S_{B})$
$\displaystyle=\displaystyle{\int d^{4}x\,\left\\{\left(\int_{-\pi
R^{+}}^{0^{-}}+\int_{0}^{\pi R}\right)dy\
\delta{\bar{D}}\,i\Gamma^{M}\partial_{M}D\right.}$ $\displaystyle\ \ \
+\displaystyle{\left.\left.\left[-X^{\prime*}\delta
D^{\dagger}_{L}Q_{R}+2\,\delta
D^{\dagger}_{R}\left(D_{L}-\dfrac{X^{*}}{2}Q_{L}\right)\right]\right|_{\pi
R}-2\left.\left(\delta{D^{\dagger}_{R}}D_{L}\right)\right|_{0}\right\\}}\,.$

The individual vanishing of those volume and surface terms, due to the action
minimisation, leads to the EOM (22) and the following NBC,

$\left\\{\begin{array}[]{l}\left.\left\\{Q_{R}+(X/2)\;D_{R}\right\\}\right|_{\pi
R}=0\,,\ \ \left.\left\\{D_{L}-(X^{*}/2)\;Q_{L}\right\\}\right|_{\pi
R}=0\,,\\\ \left.X^{\prime}\;D_{L}\right|_{\pi R}=0\,,\ \
\left.X^{\prime*}\;Q_{R}\right|_{\pi R}=0\,,\ \
\left.Q_{R}\right|_{0}=\left.D_{L}\right|_{0}=0\,,\end{array}\right.\ $ (84)

which differ from the NBC (72) obtained without the BBT. As before, given the
continuity region defined by Eq. (3), we can start by considering the search
for profile solutions of 5D EOM (22) and 5D NBC (84) on the domain,
$y\in[0,\pi R]$, being equivalent to the search performed for the interval
scenario (with BBT) Angelescu:2019viv  after the replacement,
$L\leftrightarrow\pi R$. The 4D field solutions in the decomposition (73) obey
the known Eq. (74). First, the opposite sign in factor of each $X^{(\prime)}$
parameter in the NBC (84), with respect to Ref. Angelescu:2019viv  [see Eq.
(5.23) there], comes from the mentioned different Dirac matrix sign convention
[c.f. $\Gamma^{4}$ in Eq. (107)] and hence has no physical impact, neither on
the fermion masses nor on the 4D effective Yukawa couplings [see Eq. (90)].
Secondly, the factor $1/2$ difference at the same places in NBC (84), compared
to the interval NBC Angelescu:2019viv , comes from the existence of double
numbers of surface terms (at $y=0$, $0^{-}$ and $y=\pi R$, $-\pi R^{+}$) –
like in Section 5.1 – and leads to the factor $1/2$ in the final mass spectrum
relations (87)-(88) through a re-normalisation of the $X$ parameter as $X/2$.
Thirdly, the necessary ortho-normalisation condition (75) can be rewritten on
the domain $[0,\pi R]$ only, as [the subscript C stands for L or R],

$\displaystyle\delta_{nm}\,=\,\dfrac{1}{\pi R}\int_{0}^{\pi
R}dy\,\left[q^{n*}_{C}(y)q^{m}_{C}(y)+d^{n*}_{C}(y)d^{m}_{C}(y)\right]\,,$
(85)

thanks to the change of variable, $y^{\prime}=-y$, the fixed profile parities
(76)-(49) and the continuity relations (3):

$\displaystyle\int_{-\pi
R^{+}}^{0-}dy\left[q^{n*}_{C}(y)q^{m}_{C}(y)+d^{n*}_{C}(y)d^{m}_{C}(y)\right]=\int_{0^{+}}^{\pi
R^{-}}dy^{\prime}\left[q^{n*}_{C}(-y^{\prime})q^{m}_{C}(-y^{\prime})+d^{n*}_{C}(-y^{\prime})d^{m}_{C}(-y^{\prime})\right]$
$\displaystyle\ \ \ \ \ \ =\int_{0^{+}}^{\pi
R^{-}}dy^{\prime}\left[q^{n*}_{C}(y^{\prime})q^{m}_{C}(y^{\prime})+d^{n*}_{C}(y^{\prime})d^{m}_{C}(y^{\prime})\right]=\int_{0}^{\pi
R}dy\left[q^{n*}_{C}(y)q^{m}_{C}(y)+d^{n*}_{C}(y)d^{m}_{C}(y)\right]\,,$

recovering thus exactly and conveniently the interval condition, if $L=\pi R$.
Nevertheless, including the factor $1/\sqrt{2\pi R}$, the dimensional wave
functions [mass dimension $1/2$] are identical within the orbifold and
interval frameworks only up to an additional normalisation factor $1/\sqrt{2}$
here, due to the double compact space size: see the respectively used
decomposition normalisations (73) above and (4.1) in Ref. Angelescu:2019viv .
Therefore, we can finally apply the results of Ref. Angelescu:2019viv  here
for the SM-like consistent solutions of the fields over $y\in[0,\pi R]$: we
find, for the dimensionless profiles ($\forall\,n\in\mathbb{N}$),

$\left\\{\begin{array}[]{l}(+\times):\
q_{L}^{n}(y)=A^{n}_{q}\,\cos(M_{n}\,y)\,,\ \ \ (-\times):\
q_{R}^{n}(y)=-A^{n}_{q}\,\sin(M_{n}\,y)\,,\\\ (-\times):\
d_{L}^{n}(y)=A^{n}_{d}\,\sin(M_{n}\,y)\,,\ \ \ (+\times):\
d_{R}^{n}(y)=A^{n}_{d}\,\cos(M_{n}\,y)\,,\end{array}\right.\ $ (86)

for the two classes of real mass spectrum solutions ($X=|X|\,e^{i\alpha_{Y}}$
with $\alpha_{Y},\alpha_{0}^{n}\in\mathbb{R}$),

$\displaystyle\ \tan(M_{n}\;\pi R)=\,\left|\dfrac{X}{2}\right|,\
A_{q}^{n}=e^{i(\alpha_{0}^{n}+\alpha_{Y})},\ A_{d}^{n}=e^{i\alpha_{0}^{n}},$
(87) $\displaystyle\ \tan(M_{n}\;\pi R)=-\,\left|\dfrac{X}{2}\right|,\
A_{q}^{n}=e^{i(\alpha_{0}^{n}+\alpha_{Y}\pm\pi)},\
A_{d}^{n}=e^{i\alpha_{0}^{n}},$ (88)

and for the absolute values of the fermion masses [based on Eq. (69)],

$|M_{n}|=\dfrac{1}{\pi
R}\left|\,\arctan\left|\dfrac{X}{2}\,\right|+(-1)^{n}\,\tilde{n}(n)\,\pi\,\right|\,.$
(89)

We call $(\times)$ the new Yukawa coupling (in $X$) dependent BC, given by Eq.
(86), (87)-(88), (89) at the brane located at $y=\pi R$, in order to
distinguish them from the Dirichlet BC usually noted $(-)$ and the Neumann BC
$(+)$. Note that, similarly to the free solutions (LABEL:completeEBCsm0), the
opposite signs in front of the $(-\times)$ profiles (86), with respect to the
results of Ref. Angelescu:2019viv , simply come from a different sign
convention for the $\Gamma^{4}$ matrix. At this stage, the part of the profile
solutions on the complementary region, $y\in[-\pi R^{+},0^{-}]$, is deduced
through the four types of $\mathbb{Z}_{2}$ transformations (76)-(49). Hence,
the $M_{n}$ spectrum entering the profile solutions in both regions, $[0,\pi
R]$ and $[-\pi R^{+},0^{-}]$, is the same. As a first conclusion, the
introduction of the BBT allows to obtain realistic fermion wave functions and
consistent mass eigenvalues. The absolute mass spectrum obtained within the 5D
approach in Eq. (89) is analytically matching the one derived via the 4D
method in Eq. (68) for a real Yukawa coupling constant: this feature
represents a non-trivial confirmation of the present exact results. In
particular, the absence of $X^{\prime}$ parameter in the fermion 4D mass
matrix $\mathcal{M}$, described below Eq. (65), is interestingly recovered
through the mass independence from $X^{\prime}$ as induced as well by the
condition

$X^{\prime}=0\,,$

issued from the 5D NBC (84). Regarding the probability current, the component
$j^{4}|_{\pi R}$ at the Yukawa brane is still given by Eq. (82) since the BBT
do not affect it, as explained at the end of Section 3.3. The relations found
in the first line of the NBC (84), injected once into each term of this
current component expression, give rise to,

$j^{4}|_{\pi R}\,=\,0\,.$

The BBT are thus found to induce NBC leading to a vanishing current component
along the extra dimension at the fixed points of the orbifold, with (present
section) or without (c.f. Section 3.3) a brane-localised Yukawa coupling, and
in turn to a continuous current component along the extra dimension at those
points given the odd parities, demonstrated in Eq. (81) or (55) respectively.

| $\mathbb{Z}_{2}$ | Fields
---|---|---
Continuity | $Q_{L/R}$ | $D_{L/R}$
domains | $q^{n}_{L}(y)/(\pm e^{i(\alpha_{0}^{n}+\alpha_{Y})})$ | $q^{n}_{R}(y)/(\pm e^{i(\alpha_{0}^{n}+\alpha_{Y})})$ | $d^{n}_{L}(y)/e^{i\alpha_{0}^{n}}$ | $d^{n}_{R}(y)/e^{i\alpha_{0}^{n}}$
$[0,\pi R]$ | Any | $\cos(M_{n}\;y)$ | $-\sin(M_{n}\;y)$ | $\sin(M_{n}\;y)$ | $\cos(M_{n}\;y)$
$[-\pi R^{+},0^{-}]$ | I | $\cos(M_{n}\;y)$ | $-\sin(M_{n}\;y)$ | $\sin(M_{n}\;y)$ | $\cos(M_{n}\;y)$
II | $-\cos(M_{n}\;y)$ | $\sin(M_{n}\;y)$ | $-\sin(M_{n}\;y)$ | $-\cos(M_{n}\;y)$
III | $\cos(M_{n}\;y)$ | $-\sin(M_{n}\;y)$ | $-\sin(M_{n}\;y)$ | $-\cos(M_{n}\;y)$
IV | $-\cos(M_{n}\;y)$ | $\sin(M_{n}\;y)$ | $\sin(M_{n}\;y)$ | $\cos(M_{n}\;y)$
KK Masses |  $|M_{n}|=\left|\arctan\left|X/2\,\right|+(-1)^{n}\,\tilde{n}(n)\,\pi\right|/\pi R$, $n\in\mathbb{N}$

Table 2: SM-like coupled fermion profiles on the two orbifold continuity
domains $[-\pi R^{+},0^{-}]$ and $[0,\pi R]$, corresponding to the solutions
(86), (87)-(88), together with the associated absolute mass spectrum (89) for
completeness. The profiles are given for the four types of $\mathbb{Z}_{2}$
transformations (76)-(49).

In Table 2 are exhibited the explicit profile functions over the entire
orbifold domain for the SM-like solutions (86), (87)-(88), (89). We can see on
this table that the choice of type of $\mathbb{Z}_{2}$ transformation is
purely a convention because it can modify the profile signs but without
effects on the mass spectrum.

In Figure 3, we illustrate a set of excitation profiles, obeying the
$\mathbb{Z}_{2}$ transformations of types $I$ and $II$ in Eq. (76)-(49), for
the found Yukawa-coupled solutions (87), which are explicitly presented in
Table 2, within the simplified real case, $\alpha_{Y}=\alpha^{n}_{0}=0$. We
observe on this figure that all the wave function values at the Yukawa-brane
(at the fixed point, $y=\pi R$) are modified due to the presence of this
coupling. For example, under the type $I$ of $\mathbb{Z}_{2}$ transformation,
the profile values $d^{n}_{L}(\pi R)=d^{n}_{L}(\pi R^{-})$ are shifted from
zero as well as from $d^{n}_{L}(-\pi R^{+})$, in contrast with the free case
shown in Fig. 2. This shift creates profile jumps whose amplitude is depending
on the Yukawa coupling constant through the $X$ parameter [BC $(\times)$ from
Eq. (86), (87)-(88), (89)]. Under the type $II$ of $\mathbb{Z}_{2}$
transformation, the same figure shows clearly that the profile jump
$d^{n}_{L}(\pi R)=d^{n}_{L}(\pi R^{-})\neq d^{n}_{L}(-\pi R^{+})$ disappears
but then other kinds of jump arise like: $q^{n}_{L}(\pi R)=q^{n}_{L}(\pi
R^{-})\neq q^{n}_{L}(-\pi R^{+})$ and $q^{n}_{L}(0^{-})\neq
q^{n}_{L}(0)=q^{n}_{L}(0^{+})$. The presence of new possible profile
discontinuities justifies once more mathematically the prescriptions about the
field continuities and action integration domains introduced in Section 2.1.

Figure 3: Zero-mode and excitation wave functions $q^{n}_{L/R}(y)$,
$d^{n}_{L/R}(y)$, with $n=0,1,2$, along the $\mathcal{S}^{1}/\mathbb{Z}_{2}$
orbifold domain, $y\in[-\pi R^{+},0^{-}]\cup[0,\pi R]$, corresponding to the
Yukawa-coupled solutions (87), presented in Table 2, for the simplified case,
$\alpha_{Y}=\alpha^{n}_{0}=0$, and the two different types of $\mathbb{Z}_{2}$
transformations, $I$, $II$ from Eq. (76)-(49). The two fixed points at, $y=0$,
$y=\pi R\equiv-\pi R$, the BC, $(-)/(+)/(\times)$, the BBT and Yukawa coupling
brane-locations are indicated on the graph.

Finally, let us calculate, still without any kind of Higgs field
regularisation, the physical 4D effective Yukawa coupling constants between
the mass eigenstates $\psi_{L}^{n}(x^{\mu})$ and $\psi_{R}^{m}(x^{\mu})$ as
generated by the insertion of decompositions (73) into Eq. (16), based on the
obtained profile expressions (86), (87)-(88), (89):

$\displaystyle y_{nm}$ $\displaystyle\,\hat{=}-\dfrac{Y_{5}}{2\sqrt{2}\pi
R}\,q_{L}^{n*}(\pi R)\;d_{R}^{m}(\pi R)$
$\displaystyle=\mp\dfrac{|Y_{5}|}{2\sqrt{2}\pi
R}\,e^{i(\alpha_{0}^{m}-\alpha_{0}^{n})}\cos(M_{n}\;\pi R)\cos(M_{m}\;\pi R)$
$\displaystyle=\mp(-1)^{\tilde{n}(n)+\tilde{n}(m)}\,e^{i(\alpha_{0}^{m}-\alpha_{0}^{n})}\,\dfrac{|Y_{5}|}{2\sqrt{2}\pi
R(1+|X/2|^{2})}\,,$ (90)

where we have used a trigonometric identity 222222For $n\in\mathbb{Z}$, one
has, $\cos(\theta+n\pi)=(-1)^{n}\cos(\theta)$, and for $T\in\mathbb{R}$,
$\cos[\arctan(T)]=\dfrac{1}{\sqrt{1+T^{2}}}$. to get the last equality. In the
decoupling limit of extremely heavy KK modes, $R\rightarrow 0$, we can then
write the modulus of the lightest mode coupling constant, using Eq. (14), as,

$|y_{00}|\mathop{\to}\limits_{\begin{subarray}{c}R\rightarrow
0\end{subarray}}\,\frac{|Y_{5}|}{2\sqrt{2}\pi R}=\frac{|y_{4}|}{\sqrt{2}}\,,\
{\rm since,}\ X=\frac{v}{\sqrt{2}}\,Y_{5}=\frac{v}{\sqrt{2}}\,2\pi
R\,y_{4}\,,$ (91)

and the absolute mass eigenvalue of the lightest eigenstates as [from Eq.
(89)],

$|M_{0}|\mathop{\to}\limits_{\begin{subarray}{c}R\rightarrow
0\end{subarray}}\,\dfrac{|X|}{2\pi R}=\dfrac{v|Y_{5}|}{2\sqrt{2}\pi
R}\,\mathop{\to}\limits_{\begin{subarray}{c}R\rightarrow
0\end{subarray}}\,v\,|y_{00}|\,,$ (92)

so that the SM fermion set-up – for the assumed single family – is recovered
as expected from the decoupling condition. Besides, we can conclude that the
choice of type of $\mathbb{Z}_{2}$ transformation among Eq. (76)-(49) affects
neither the profile values taken at the point $y=\pi R$ – see Table 2 – nor
their global ortho-normalisation condition (75) – as described right below Eq.
(85) – so that the 4D effective Yukawa coupling constants (90) are insensitive
as well to this $\mathbb{Z}_{2}$ representation choice.

##  6 The inclusive $\mathbb{Z}_{2}$ parity

Let us study the alternative scenario whose definition is based on the
$\mathbb{Z}_{2}$ transformation of 5D fields extended to include the two fixed
points at $y=0$ and $y=\pi R$:

$\Phi(x^{\mu},-y)={\mathcal{T}}\Phi(x^{\mu},y)\,,\ \ \forall y\in(-\pi R,\pi
R]\,,$ (93)

in contrast with Eq. (1). This generic transformation still lets the
Lagrangian density invariant, exactly like in Eq. (2). At the two fixed
points, this Lagrangian invariance is once more automatically satisfied
without the need for any specific ${\mathcal{T}}$ transformation. Accordingly
to the simple Eq. (93), the operator ${\mathcal{T}}$ for the fixed points is
the same as the non-trivial one which must let the Lagrangian invariant in the
bulk. Let us consider in particular the realistic $\mathbb{Z}_{2}$
transformation leading to the SM chirality set-up: it is the bulk
transformation in Eq. (7), defined now over the same range as in Eq. (93),
which keeps well $\mathcal{L}_{kin}$ invariant in the bulk according to Eq.
(2):

$\left\\{\begin{array}[]{c c
c}Q\left(x^{\mu},-y\right)=-\gamma^{5}\,Q\left(x^{\mu},y\right)\\\
\vspace{-0.2cm}\hfil\\\
D\left(x^{\mu},-y\right)=\gamma^{5}\,D\left(x^{\mu},y\right)\end{array}\right.,\
\ \ \forall y\in(-\pi R,\pi R]\,.$ (94)

Focusing on the fixed points at $y=0$ and $y=\pi R\equiv-\pi R$, we obtain the
four non-trivial relations

$\displaystyle\left\\{\begin{array}[]{c c
c}Q\left(x^{\mu},0\right)=-\gamma^{5}\,Q\left(x^{\mu},0\right)\,\Rightarrow\,Q_{L}|_{0}=Q_{L}|_{0}\,,\,{\rm
and,}\ Q_{R}|_{0}=-Q_{R}|_{0}=0\\\ \vspace{-0.2cm}\hfil\\\
D\left(x^{\mu},0\right)=\gamma^{5}\,D\left(x^{\mu},0\right)\,\Rightarrow\,D_{L}|_{0}=-D_{L}|_{0}=0\,,\,{\rm
and,}\ D_{R}|_{0}=D_{R}|_{0}\end{array}\right.$ (98)
$\displaystyle\left\\{\begin{array}[]{c c c}Q\left(x^{\mu},\pi
R\right)=-\gamma^{5}\,Q\left(x^{\mu},\pi R\right)\,\Rightarrow\,Q_{R}|_{\pi
R}=0\\\ \vspace{-0.2cm}\hfil\\\ D\left(x^{\mu},\pi
R\right)=\gamma^{5}\,D\left(x^{\mu},\pi R\right)\,\Rightarrow\,D_{L}|_{\pi
R}=0\end{array}\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{\rm[EBC^{\prime}]}$ (102)

representing new EBC that we denote EBC’ to distinguish them from those in Eq.
(57).

In the free case, Section 3.1 has shown that EBC(’) or BBT must be considered.
Starting with the EBC(’), in analogy with Section 3.2, the fixed
$\mathbb{Z}_{2}$ transformations (94) in the bulk lead to the EBC (57) while
the $\mathbb{Z}_{2}$ transformations (98) at the fixed points lead to the
EBC’. Those EBC’ select one general BC set among these four EBC sets for the
5D field $Q$, and same statement for $D$: the sets corresponding to the chiral
solution of line 1 (2) in Eq. (LABEL:completeEBCsm0) for the field $D$ ($Q$),
namely the SM-like chirality configuration. Finally, the complete profile
solutions over the whole orbifold domain are found out as before via the bulk
$\mathbb{Z}_{2}$ transformations (94).

Alternatively, the selected consistent BBT (17) can be included like in
Section 3.3 to obtain the same SM-like solutions. The corresponding EBC’ (98),
part of the EBC (57) and required by the model, are checked to be satisfied
afterwards, as consequences.

Once the free profiles are worked out as described right above – either
through the EBC(’) or the BBT – we can apply the 4D method of Section 4, based
on infinite matrix diagonalisation, in order to derive the mass spectrum in
the presence of brane-localised Yukawa couplings. Even the 4D effective Yukawa
coupling constants can be calculated in this way: the above EBC’ selection of
a specific chirality set-up and mass spectrum for the free fields would affect
as well these effective coupling constants, for instance via the KK mass
mixings.

In contrast, the analysis of point-like Yukawa interactions cannot be achieved
via the 5D approach within the present inclusive $\mathbb{Z}_{2}$ symmetry
model.

First, the EBC(’) motivated by Section 5.1 must be split into the EBC coming
directly from the vanishing probability currents – or say indirectly from the
fixed $\mathbb{Z}_{2}$ transformations (94) in the bulk – discussed in Section
5.2 and the EBC’ (98). These EBC’ combined with the surface terms at $y=\pi R$
in Eq. (71), including the Yukawa terms, give rise to the BC of type (72)
involving only single terms proportional to the Yukawa coupling constant and
equal to zero. Hence, the resulting mass spectrum looses its dependence on the
Yukawa coupling constant which conflicts with the decoupling limit argument
[see Eq. (92)].

Secondly, the BBT (17) could be added like in Section 5.3 to try obtaining SM-
like solutions. However the EBC’ (98), expected to be recovered afterwards,
are not compatible with the resulting BC (86) together with the spectrum
equations (87)-(88).

##  7 Result analysis

###  7.1 The higher-dimensional method

The present study confirms the general methodology depicted in Fig. 4 and
presented in Ref. Angelescu:2019viv . Within the present model, the
probability current condition on this schematic description is the vanishing
of fermion currents at the two fixed points (issued from $\mathbb{Z}_{2}$
symmetry criteria and inducing the EBC (57) in the free case). For the
interval model, the vanishing current condition is a direct implication of the
existence of boundaries for the matter fields. This current vanishing holds
both in the presence and absence of brane-localised Yukawa couplings.

In the framework of the orbifold version described in Section 6, the
additional field condition (98), coming from the $\mathbb{Z}_{2}$ symmetry at
the fixed points, accompanies the definition of the $\mathbb{Z}_{2}$ symmetry
of the bulk action and leads to the new EBC’.

Figure 4: Schematic inverse pyramidal picture describing the generic process
for determining the mixed KK mass spectrum and bulk field wave functions.
Acronym notations are the same as in the main text.

###  7.2 Discussion of the action content

In addition to the information contained in the action (18), the present
orbifold model is defined in a complementary way by other elements like: (i)
the $\mathcal{S}^{1}$ junction point at $y=\pi R\equiv-\pi R$, (ii) the
choices of $\mathbb{Z}_{2}$ transformations for the fields in the bulk [see
Eq. (7)-(10)] and possibly at the fixed points [c.f. Eq. (98)], (iii) the EBC
(57) imposed by the model definition when those are used instead of the BBT.
Regarding the point (iii), Table 3 summarises the obtained cases where the EBC
and the BBT can be used. This table is identical to the one obtained in the
interval model study Angelescu:2019viv .

|  No boundary characteristic |  Vanishing current condition [EBC] |  Bilinear brane terms [NBC]
---|---|---|---
4D Approach |  (Impossible) |  BC ($\pm$) |  BC ($\pm$)
5D Approach |  (Impossible) |  (Impossible) |  BC ($\times$)
Table 3: Types of boundary conditions for the bulk fermions at an orbifold
fixed point where is located their interactions with the Higgs boson, in
different brane treatments: presence of BBT, vanishing of probability current
or nothing specific. The 4D line holds as well for the 5D approach of the free
brane. As usually, the Dirichlet BC are noted $(-)$, the Neumann BC $(+)$ and
we denote $(\times)$ the new BC depending on the Yukawa coupling constant
[corresponding to Eq. (86) taken at $y=\pi R$]. The (N,E)BC acronym
definitions are the same as in the text.

###  7.3 About the orbifold/interval duality

The present $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold model and the $[0,L]$
interval scenario studied in Ref. Angelescu:2019viv  are physically different:
their geometrical set-ups and Lagrangian symmetries are not identical.
Nevertheless, the respective theoretical predictions for the observables like
the (brane-coupled) fermion mass spectra and 4D effective Yukawa coupling
constants are identical up to factors $2$, which may be called a duality.
Indeed, for a comparable dimension size $L=\pi R$, although the mass absolute
values (89) involve a new factor $1/2$ in front of $X$, with respect to the
interval analytical result, the measurable range of values for $|M_{n}|$ is of
the same order and the precise limits of this range rely on the approximate
perturbative limits of the 4D effective Yukawa coupling constants proportional
to $y_{4}$ [see Sections 2.2.3 and Eq. (90)-(91)-(92)]. Besides, the
dependence of the analytical mass formula on the Lagrangian parameters is
identical in the two models, up to this factor $1/2$ entering the coupling
constant definition, as can be seen from Eq. (89) and Section 2.2.3 –
including the free limiting case $X\to 0$. Similar comments hold for the 4D
effective Yukawa coupling constants (90) which have additional factors $1/2$
in front of $X$ and as an overall factor (latter one induced by normalisation
considerations), with respect to the interval case.

The orbifold version of Section 6 contains additional information at the fixed
point branes. It predicts thus a specific chirality configuration and mass
spectrum [among chiral or vector-like solutions respectively of type
(LABEL:completeEBCsm0)-(LABEL:completeEBCcusto0)] so that it is not dual to
the interval model.

Coming back to the case of duality, there exist similarities between the
orbifold and interval models, as it appeared throughout this work when solving
the EOM and BC to find out the fields. Let us now comment on the similarities
at the Lagrangian level. First, the BBT (17) have the same form as in the
interval framework Angelescu:2019viv  and the different factor $2$ is related
to the double size of the compactified space for the identification, $L=\pi
R$. The opposite front sign in the BBT (for a similar profile solution set-up)
is just due to a different Dirac matrix sign convention [see $\Gamma^{4}$ sign
in Eq. (107)].

In the global action (18), $S_{bulk}$ remains to be discussed, the other parts
being identical in the orbifold and interval models. Thanks to the orbifold
property (2), the change of variable, $y^{\prime}=-y$, allows the following
rewriting of Eq. (4),

$\displaystyle S_{bulk}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\left(\int_{-\pi R^{+}}^{0^{-}}dy\ \mathcal{L}_{kin}(y)+\int_{0}^{\pi
R}dy\ \mathcal{L}_{kin}(y)\right)$ $\displaystyle=$ $\displaystyle\int
d^{4}x\left(\int^{\pi R^{-}}_{0^{+}}dy^{\prime}\
\mathcal{L}_{kin}(y^{\prime})+\int_{0}^{\pi R}dy\
\mathcal{L}_{kin}(y)\right)=2\,\int d^{4}x\left(\int_{0}^{\pi R}dy\
\mathcal{L}_{kin}(y)\right),$

where the last step is based on Eq. (3). Therefore, using Eq. (18) and the
relevant identification, $L=\pi R$, we can express the orbifold action in
terms of the interval action pieces Angelescu:2019viv  (indicated by the $L$
exponent):

$\displaystyle S_{5D}$ $\displaystyle=$ $\displaystyle
2\,S^{L}_{bulk}+S^{(L)}_{H}+S^{(L)}_{X}+S^{(L)}_{int}+2\,S^{L}_{B}$
$\displaystyle=$ $\displaystyle
2\,\left[S^{L}_{bulk}+\frac{1}{2}\\{S^{(L)}_{X}+S^{(L)}_{int}\\}+S^{L}_{B}\right]+S^{(L)}_{H}$
$\displaystyle=$ $\displaystyle
2\,\left[S^{L}_{bulk}+S^{(L)}_{X/2}+S^{(L)}_{int}|_{X/2}+S^{L}_{B}\right]+S^{(L)}_{H}\,.$

This re-expression reveals an alternative method to derive the fermion masses
and couplings, which are independent from the pure scalar part, namely
$S^{(L)}_{H}$. The idea is that, within the orbifold model now described by
the action (7.3) importantly together with the description of the
$\mathbb{Z}_{2}$ symmetry over $\mathcal{S}^{1}$, we can first search for the
field parts along the limited domain $[0,\pi R]$. This search is in fact based
on the action $[S^{L}_{bulk}$ \+ $S^{(L)}_{X/2}$ \+ $S^{(L)}_{int}|_{X/2}$ \+
$S^{L}_{B}]$, since the overall factor $2$ in Eq. (7.3) affects neither the
EOM (global factor) nor the BC (same factor in front of the surface terms and
pure brane terms combined into BC) 232323This search could also be constrained
by vanishing currents at $y=0,\pi R$ instead of the $S^{L}_{B}$ presence, in
the free case, as shown in Sections 3.2 and 5.2., and is in turn strictly
equivalent to solving the interval model. Given this action, the solutions
obtained for the 4D masses (and 4D effective Yukawa coupling constants from
profile overlaps with the Higgs boson peak at $y=\pi R$) are those of Ref.
Angelescu:2019viv  but involving a normalised coupling parameter $X/2$. The
last stage of this technics is the extension of the obtained profiles over the
complete orbifold domain via the $\mathbb{Z}_{2}$ transformations, before
applying the ortho-normalisation condition. The 4D effective Yukawa coupling
constants are then changed by an additional factor $1/2$, as is clear from the
dimensional wave function normalisation forms (75)-(85), which confirms the
result (90). On the other side, we see as well that the fermion masses so
obtained (unchanged by the spatial domain extension) involve only a new
normalised parameter $X/2$, with respect to Ref. Angelescu:2019viv , which
confirms the found spectrum (89).

Beyond these action correspondences, there are other elegant similarities. For
example, as illustrated by Fig. 4, both the interval and orbifold scenarios
lead to the same vanishing probability current conditions at the two branes
(and hence to identical EBC); those current conditions come, respectively,
directly from the interval boundary criteria and indirectly from
$\mathbb{Z}_{2}$ symmetry considerations. Besides, Table 3 shows that the same
treatments of the two branes, at the fixed points or interval boundaries, must
be adopted in identical situations and that the same BC are generated.

Finally, let us propose an intuitive description for understanding the
orbifold versus interval model duality. The obtained wave functions for the
bulk fermions on the interval are of the kind
$\cos(M_{n}\,y)\propto(e^{iM_{n}\,y}+e^{-iM_{n}\,y})$, coming in factor (via
the KK decomposition) of the energy coefficients $e^{\pm iE\,t}$ in the 4D
Dirac fields, which gives rise to wave planes propagating in both
$y$-directions of the interval with momenta $\pm p_{n}=\pm M_{n}$ – as for
oscillations left-moving and right-moving along opposite directions in the
world-sheet parameter space of strings. The associated particle, going in the
direction $L\to 0$ and then coming back along $0\to L$, reproduces the
propagation along $\mathcal{S}^{1}$, following consecutively the two
fundamental domains $-\pi R\to 0^{-}$ and $0^{+}\to\pi R$ of the orbifold
(effectively equivalent orientations of the circle in the bulk so a unique
propagation direction chosen along it): exactly the same
$\mathcal{L}\left[\Phi(x^{\mu},y)\right]$ Lagrangian evolution is felt by this
particle during those dual travelings along the extra $y$-dimension, in the
two different models, as is clear from the Lagrangian $\mathbb{Z}_{2}$
symmetry depicted in the drawing 1.

##  8 Conclusions

In the study of the $\mathcal{S}^{1}/\mathbb{Z}_{2}$ orbifold, the proper
action definition through improper integrals has allowed to obtain consistent
bulk profile solutions with possible discontinuities at the fixed points. In
particular the point-like interaction of Yukawa creates a profile jump.

These solutions have been obtained without brane-Higgs regularisation, by
relying on the necessary EBC, coming from vanishing fermion probability
currents, or alternatively on the introduction of BBT in the action. The
associated calculations have been confirmed by the matching, between the 4D
and 5D approaches, of the analytical results for the fermion mass spectrum and
4D effective Yukawa coupling constants.

The orbifold version, with $\mathbb{Z}_{2}$ transformations of the fields
extended to the fixed points, was shown to be able to generate the chiral
nature of the theory and even to select the expected SM chirality
configuration for the 4D states.

The duality between the interval and orbifold scenarios has been deeply
described. It has also constituted the opportunity to point out an alternative
method for calculating the tower of excitation masses and 4D Yukawa couplings.

We are now working on the introduction of distributions in this context WIP .

Acknowledgments

R.L. is supported by the agreement signed between the China Scholarship
Council (CSC) and the Université Paris-Saclay.

## Appendix

##  Appendix A Notations & conventions

Throughout the paper, we use the conventions of Ref. Schwartz:2013pla . The 5D
Minkowski metric is,

$\eta_{MN}=\text{diag}(+1,-1,-1,-1,-1)\,,$

where $M,N=0,1,...,4$.

The 4D Dirac matrices are taken in the Weyl representation,

$\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{pmatrix}\phantom{000}\text{with}\phantom{000}\left\\{\begin{array}[]{r
c l}\sigma^{\mu}&=&\left(\mathbb{I},\sigma^{i}\right)\,,\\\
\bar{\sigma}^{\mu}&=&\left(\mathbb{I},-\sigma^{i}\right)\,,\end{array}\right.$
(105)

where $\mu=0,1,2,3$ and $\sigma^{i}$ ($i=1,2,3$) are the three Pauli matrices:

$\sigma^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,,\phantom{000}\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\,,\phantom{000}\sigma^{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,.$

One has also the 4D chirality operator,

$\gamma^{5}=i\prod_{\mu=0}^{3}\gamma^{\mu}=\begin{pmatrix}-\mathbb{I}&0\\\
0&\mathbb{I}\end{pmatrix}\,.$ (106)

In our conventions, the 5D Dirac matrices $\Gamma^{M}$ ($M=0,1,...,4$) obey
$\left\\{\Gamma^{A},\Gamma^{B}\right\\}=2\,\eta^{AB}$ ($A,B=0,1,...,4$) and
read as,

$\Gamma^{M}=\left(\gamma^{\mu},-i\gamma^{5}\right)\,.$ (107)

##  Appendix B From spinor components to compact notations

###  B.1 Spinor components and their variations

The generic spinor field $F$ ($F=Q,D$) introduced via Eq. (6) can be written
in terms of its four explicit components $F_{\alpha}$ [$\alpha=1,2,3,4$]:

$F=\begin{pmatrix}F_{1}\\\ F_{2}\\\ F_{3}\\\ F_{4}\end{pmatrix}\,,$ (108)

and similarly, $\bar{F}$ can be expressed in terms of its own four components
$\bar{F}_{\alpha}$:

$\bar{F}=\begin{pmatrix}\bar{F}_{1},&\bar{F}_{2},&\bar{F}_{3},&\bar{F}_{4}\end{pmatrix}\hat{=}\,F^{\dagger}\gamma^{0}=\begin{pmatrix}F^{*}_{3},&F^{*}_{4},&F^{*}_{1},&F^{*}_{2}\end{pmatrix}\,.$
(109)

These 8 components constitute the fundamental variables of the Lagrangian (5).
Hence, the variation of the associated Action, $S_{bulk}$ [see Eq. (4)],
involves the following 8 elementary variations, that we can group into new
4-component (transposed) vectorial objects defined as:

$\delta F\,\hat{=}\begin{pmatrix}\delta F_{1}\\\ \delta F_{2}\\\ \delta
F_{3}\\\ \delta F_{4}\end{pmatrix}\,,\ \ \
\delta\bar{F}\,\hat{=}\begin{pmatrix}\delta\bar{F}_{1},&\delta\bar{F}_{2},&\delta\bar{F}_{3},&\delta\bar{F}_{4}\end{pmatrix}\,,$
(110)

introducing the 8 components $\delta F_{\alpha}$ and $\delta\bar{F}_{\alpha}$.
We then define,

$\delta F=\begin{pmatrix}\delta F_{L}\\\ \delta F_{R}\end{pmatrix},\
\delta\bar{F}\,\hat{=}\begin{pmatrix}\delta{F}^{\dagger}_{R},&\delta{F}^{\dagger}_{L}\end{pmatrix},\
\rm{with\ for\ instance,}\
\delta{F}^{\dagger}_{R}\,\hat{=}\begin{pmatrix}\delta\bar{F}_{1},&\delta\bar{F}_{2}\end{pmatrix}\,,$
(111)

inspired by the following generic relations, based on Eq. (6),

$F=\begin{pmatrix}F_{L}\\\
F_{R}\end{pmatrix},\,\bar{F}\,\hat{=}\,F^{\dagger}\gamma^{0}=(F^{\dagger}_{R},\,F^{\dagger}_{L})\,.$
(112)

###  B.2 A typical compact form calculation

Using the Lagrangian $\mathcal{L}_{kin}$ of Eq. (5), let us work out
explicitly the following quantity entering Eq. (19) in a compact form (no
explicit spinor index of type $\alpha$),

$\displaystyle\delta{\bar{F}}\,\dfrac{\partial\mathcal{L}_{kin}}{\partial\bar{F}}$
$\displaystyle\,\hat{=}$
$\displaystyle\sum_{\alpha=1}^{4}\delta\bar{F}_{\alpha}\dfrac{\partial\mathcal{L}_{kin}}{\partial\bar{F}_{\alpha}}=\sum_{\alpha=1}^{4}\delta\bar{F}_{\alpha}\dfrac{\partial}{\partial\bar{F}_{\alpha}}\left(\frac{i}{2}\,\bar{F}\Gamma^{M}\partial_{M}F\right)$
(113) $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{4}\delta\bar{F}_{\alpha}\dfrac{\partial}{\partial\bar{F}_{\alpha}}\left(\frac{i}{2}\,\sum_{\beta=1}^{4}\bar{F}_{\beta}[\Gamma^{M}\partial_{M}F]_{\beta}\right)=\sum_{\alpha=1}^{4}\delta\bar{F}_{\alpha}\frac{i}{2}\,[\Gamma^{M}\partial_{M}F]_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{i}{2}\,\delta\bar{F}\Gamma^{M}\partial_{M}F\,,$

where the spinor components of Eq. (108) and (109) have appeared, as well as
the variations of Eq. (110).

###  B.3 $\mathbb{Z}_{2}$ transformations of field variations

Finally, we can derive the $\mathbb{Z}_{2}$ transformation for the compact
form $\delta\bar{F}$ of Eq. (110). Accordingly to the $\mathbb{Z}_{2}$
transformations (7)-(10), we have,

$\displaystyle\bar{F}|_{-y}=F^{\dagger}|_{-y}\gamma^{0}=\left(\pm\gamma^{5}F\right)^{\dagger}|_{y}\gamma^{0}=\pm
F^{\dagger}|_{y}\gamma^{5}\gamma^{0}=\mp
F^{\dagger}|_{y}\gamma^{0}\gamma^{5}=\mp\bar{F}|_{y}\gamma^{5}\,,$

due to the anti-commutator relation
$\left\\{\gamma^{5},\gamma^{\mu}\right\\}=0$. Then one must rewrite this
relation by making the spinor components of Eq. (109) appear explicitly:

$\displaystyle\bar{F}_{\alpha}|_{-y}=\mp\sum_{\beta=1}^{4}\bar{F}_{\beta}|_{y}\gamma^{5}_{\beta\alpha}\,,$

in order to deduce the relation on the variations of these components:

$\displaystyle\delta\bar{F}_{\alpha}|_{-y}=\mp\sum_{\beta=1}^{4}\delta\bar{F}_{\beta}|_{y}\gamma^{5}_{\beta\alpha}\,.$

Thanks to Eq. (110), this equation can be contracted back to the compact
notation as,

$\displaystyle\delta\bar{F}|_{-y}=\mp\delta\bar{F}|_{y}\gamma^{5}\,.$ (114)

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