Rigorous treatment of the $\mathcal{S}^1 / \mathbb{Z}_2$ orbifold model with brane-Higgs couplings

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 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.


Introduction
As it is well known since the 2000s, the paradigm of models with additional spatial dimensions 1 constitutes an attractive alternative to supersymmetry for addressing the Standard Model (SM) puzzle of the gauge hierarchy. Furthermore, the warped dimension framework [1] with SM fermions in the whole bulk [2] offers an elegant geometrical principle of fermion profile overlap generating the SM fermion mass hierarchy [3] (see concrete application models e.g. in Ref. [4][5][6][7][8][9]). In order to realise these two hierarchical features, the Brout-Englert-Higgs scalar field [10,11], which is at the origin of the SM particle masses through the electroweak symmetry breaking, must be stuck at the so-called TeV-brane 2 (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 3 . 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 [28,29] (see also Ref. [30,31] for the supergravity limit of string theories) 4 .
In this paper, we will study the original version [1] of the warped dimension scenario based on the S 1 /Z 2 orbifold [34,35] where the extra space is compactified on a circle respecting a spatial parity of the Lagrangian 5 . 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 S 1 /Z 2 orbifold background, by building rigorously the four-dimensional (4D) 6 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. [38] and references therein) and that, instead, one must introduce either Essential Boundary Conditions (EBC) on 5D fields 7 , originating from the 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. [38,39]), in the 4D or 5D approach, has been found as well [38] 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 higherdimensional 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 S 1 /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 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 [40,41], 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. [42] -where the 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 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 Z 2 scenario can only be treated through the 4D method. The fermion masses and couplings are also affected by this inclusive 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 [43,44], 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 [38] 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).

Geometry and symmetries: the proper action
We consider the 5D space-time model with the product geometry M 4 × S 1 /Z 2 described just below.
• M 4 represents the usual 4D Minkowski space-time whose coordinates are denoted by x µ where µ ∈ 0, 3 is the Lorentz index of the covariant formalism. The metric conventions are given in Appendix A.
• S 1 /Z 2 stands for the extra space orbifold obtained from modding out the circle S 1 by the discrete group 8 symmetry Z 2 . This circle S 1 is characterised by a radius R and its coordinate is y ∈ (−πR, πR], not double-counting the point y = πR since it is this point, by pure convention, that is chosen to be the junction point geometrically identified with the point y = −πR (which we note: −πR ≡ πR) 9 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 2πR and hence by limiting the 1D space to the fundamental domain (−πR, πR]. The (non-neutral) Z 2 transformation on space, y → −y 10 , has a representation on a generic 5D field, which must let the Lagrangian density invariant, by definition of the symmetry: We mention that this equation can define a class of equivalence of a given coordinate y 0 , defined as [y 0 ] = {y ∈ S 1 | y ∼ y 0 } with y ∼ ±y, as illustrated symbolically on 8 Factor element, e ±i 2π 2 = −1, and neutral element, 1. 9 Another possible mathematical convention would have been, for instance, − 3πR 2 ≡ πR 2 . 10 The convention above, of having taken the coordinate origin at the strict middle of the circle domain (or fundamental domain), renders the Z2 parity with respect to the origin more explicit and convenient to study. Two fixed points arise: (y = 0) → (−0 = 0) and (y = πR) → (−πR ≡ πR). At these fixed points, the Lagrangian condition of Eq. (2.2) is automatically satisfied: , so that T is naturally taken as the identity operator in Eq. (2.1) since no transformation needs to apply on the fields there. Another scenario will be analysed in Section 6. 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 = πR, 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 = πR), 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: . This choice is conventional and hence cannot affect numerical results, so let us choose conveniently 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: where > 0, S branes represents action terms located at the orbifold fixed points and 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. (2.3).
Besides, for L kin to be integrable over the entire region y ∈ [0, πR], this Lagrangian density, which will involve profile derivatives f (y), must be well-defined over this region.
with κ > 0, > 0, ω = + κ and similar equalities hold at the other region boundary y = −πR + . Furthermore, the worked out solutions f (y) will (well) be derivable over the two regions [−πR + , 0 − ] and [0, πR] (see Sections 3.2 and 5.3 respectively for the free and coupled fermion situations) so that L kin will be well-defined. For example, f (y) is derivable in the region [0, πR] 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. (2.3). Notice that from the point of view of the integration by pieces of the action in Eq. (2.4) precisely over the physical domain, the inclusion (or not) of the single points at y = 0 or y = πR ≡ −πR does not affect the integral results -given the continuous form of the even L kin over the two regions -so that only consistent action definition arguments were considered here.
Finally, the Lagrangians of the whole expression (2.4) will respect the Z 2 symmetry since the Lagrangian L kin will fulfill the condition (2.2) and the brane action will exclusively involve Lagrangians taken at fixed points like for example [see Eq.

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 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 SU(2) L gauge doublet down-component quark and the singlet down-quark.
The 5D fields Q(x µ , y) and D(x µ , y) -of mass dimension 2 -have the following kinetic terms [entering Eq. (2.4)] 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 12 ): with M ∈ 0, 4 for the coordinates x M ∈ M 4 ×S 1 /Z 2 and Γ 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, in terms of the two two-component Weyl spinors Q L , Q R , L/R standing for the Left/Right chirality, and as usuallyQ = Q † γ 0 .
As stated at the end of previous section, the Lagrangian L kin must obey the condition (2.2). For this purpose, the Z 2 transformation (2.1) on the 5D fields Q(x µ , y) and D(x µ , y) can take four different forms which constitute Essential Conditions (EC) issued from the model definition: (2.10) under which the Lagrangian (2.5) is indeed invariant, as appears by using the properties of the γ 5 Dirac matrix and the odd parity of the fifth partial derivative ∂ 4 . Notice that the Z 2 parity (second order cyclic group) does not allow complex phase factors in the transformations: Using the γ 5 definition of Appendix A together with Eq. (2.6), we already deduce some information on the possible 5D chiral field parities with respect to y = 0, as indicated in Eq. (2.7)-(2.10).
Based on Eq. (2.6), we can rewrite the bulk Lagrangian of Eq. (2.5) 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. (2.7)-(2.10)], and the ∂ 4 odd parity: where the low double arrows indicate a replacement of 5D fields in the previous terms and the matrices σ µ ,σ µ are defined in Appendix A.

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 = πR, 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, with a potential V possessing a minimum which generates a non-vanishing Vacuum Expectation Value (VEV) for the field H expanded as 12) in analogy with the SM Higgs field.

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 = πR) to the introduced bulk fermions, 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 = πR, the 5D fields Q L/R (x µ , πR), D L/R (x µ , πR) are defined through the profile convention (2.3), 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 = e iα Y |Y 5 | and Y 5 = e iα Y |Y 5 | Yukawa coupling constants, entering Eq. (2.13), are independent and a well-defined 4D chirality holds for the fermion fields on the 3-brane strictly at y = πR [38,44]. 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 (2.14) where y 4 , y 4 are dimensionless coupling constants of 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. (5.22)-(5.23). 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 (2.13), with the compact notations X = vY 5 / √ 2 and X = vY 5 / √ 2. Based on Eq. (2.12), the complete action reads as, S Y = S X + S int , with the localised fermion-scalar interaction terms: (2.16) that allow to work out the 4D effective Yukawa coupling constants.

Bilinear brane terms
Introducing all the covariant operators up to mass dimension 5 [like for the Yukawa couplings (2.13)] in this model, one should consider as well the dimension 4 operators given just below, that we call the BBT like in Ref. [38]. Furthermore, the presence of the BBT has several justifications: (i) they allow to avoid physical consistency problems both in the free case (see The following BBT lead to the SM chirality configuration, without Yukawa couplings, these terms will induce only a non-vanishing profile q 0 L (y) [see line 2 of Eq. (3.18) and Table 1 in case of the zero-mode with mass m 0 = 0] in the 5D field Q L (x µ , y) so that only the Left-handed 4D field Q 0 L (x µ ) will exist. This zero-mode Q 0 L (x µ ), 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 SU(2) L quark doublet in the direct extension to gauge symmetries (and three flavours). Given that, similarly, the BBT (2.17) will exclusively lead to a Right-handed 4D field D 0 18)] potentially representing the SM down quark type (gauge singlet). When adding the Yukawa couplings (2.13), this SM chirality set-up remains though it is no more explicit due to the Q n (x µ )-D n (x µ ) mixing, via vector-like KK state mixings, which induces some vector-like mass eigenstates ψ 0 L/R (x µ ) 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, , would lead to a chiral set-up for the zero-modes but different from the potential SM chirality configuration, namely: Finally, as will be described in the Sections 3.2 and 3.3, the possible signs, σ Q 0(πR) = ± (same sign for 0 and πR), would instead lead to the profile solutions (3.19) with two non-vanishing profiles for the lightest modes (as m 0 = 0) and hence to vector-like states: The same statement holds for σ D 0(πR) = ± and thus D 0 L/R (x µ ). Such massive vector-like states 13 can be used to build custodially protected warped models [50] and are then called custodians (see for instance Ref. [9]). Of course there exist 8 remaining cases combining the above Lagrangian sign configurations: 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 13 Extensive phenomenology at colliders has been developed about such vector-like particles [45][46][47][48][49].
by the action, (2.18) 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. [38].

Free bulk fermions on the orbifold
In this section, we calculate the fermionic mass spectrum for the free case where Y 5 = Y 5 = 0 in the action piece S Y given by Eq. (2.13).

Applying the NBC
We start by considering the bulk action part, of Eq. (2.4), from the considered action, S 5D , of Eq. (2.18). We apply the least action principle to it which leads to two relations of the kind, δF S bulk = 0, one for each of the unknown 5D fields F = Q, D, and two corresponding ones, δ F S bulk = 0, involving the complex conjugate fields 14 , since the elementary field variations δQ α , δQ α , δD α and δD α (see Appendix B.1) are generic and hence independent from each other. Using compact notations, like for example, we can write in particular 15 , 14 The 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. 15 We 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. [51]). This is realised by the local physics statement which induces vanishing fields at infinities due to the wave function normalisation conditions (see also Ref. [52]).
Based on the Lagrangian L kin of Eq. (2.5), these two bulk terms take the same form (the first one being calculated explicitly in Eq. (B.6) to clarify the spinor component treatment) and the two remaining brane terms can be calculated as well: where we have further invoked the Z 2 transformations (2.7)-(2.10) for the generic 5D field, Eq. (B.7) for its variation and γ 5 properties: Then thanks to Eq. (2.3) 16 and Eq. (B.4)-(B.5), respectively, the expression (3.2) simplifies to, 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), and the Natural Boundary Conditions (NBC), 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, where the implications come from the linear independence of mass eigenstates F n L/R (x µ ). Similarly, for the three other types of Z 2 transformations (2.8)-(2.10), we have the following profile parities: Therefore, all the f n L/R (y) profiles are systematically vanishing on the whole S 1 /Z 2 orbifold region, y ∈ [−πR + , 0 − ] ∪ [0, πR]. Such profiles conflict with the two (for L/R) ortho-normalisation conditions over the full domain, 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.

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 U(1) Q and U(1) D symmetries of the action, involving the Lagrangian (2.5). The two independent global U(1) Q,D transformations of the fields, letting L kin invariant, act respectively as, where α, α (∈ R) are continuous constants entering for instance the infinitesimal field variations 17 : Choosing instead to consider a unique symmetry (α = α 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 (3.11) [with both possibilities, α = α or α = α ] and the two independent U(1) Q,D symmetries. Based on these two symmetries, and the bulk EOM whose standard structure appears in Eq. (3.1), the Noether's theorem predicts the local conservation relation, for the two probability currents, as derived in details within the Appendix B of Ref. [38]. This relation holds over the whole , since the sole bulk terms in the action infinitesimal variation -under 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 (3.12) imposes necessarily continuous 5-current components over all the model space-time and in particular a continuous 18 . Furthermore, a jump of the form, j 4 F | 0 − = j 4 F | 0 , would not determine any field at the fixed point and thus would not lead to vanishing variations in Eq. (3.2) that would modify the BC (3.5) inducing non-physical solutions. A similar argument applies at the other fixed point, y = πR ≡ −πR. Hence, one has to consider the remaining model possibility, j 4 , so that this current component is continuous over all the range, y ∈ (−πR, πR]. In particular, we can now write, This obtained relation must be compared with the following one, coming directly from the Z 2 transformations of type (2.7)-(2.10) and γ 5 properties, The combination of Eq. (3.14) and Eq. (3.15) gives rise to a vanishing current component at the fixed point: Similar arguments regarding the second fixed point imply obviously that, so that, using the generic chiral decomposition (B.5), we get the following current conditions, leading to the minimal Boundary Conditions (BC), The part of the general f n L/R (y) solutions in the complementary domain, y ∈ [−πR + , 0 − ], is now obtained via the four types of Z 2 transformations (3.8)-(3.9). Therefore, the inclusion of the EBC based on the vanishing probability currents allows to obtain consistent fermion profile and mass solutions. In Table 1 Table 1 that the same m n spectrum enters the profile solutions in both regions, y ∈ [0, πR], and, y ∈ [−πR + , 0 − ]. In this table, we also give the general values of the B n L/R complex constants, in Eq. (3.18), obtained from the ortho-normalisation conditions (3.10) 21 . We observe on Table 1 that the choice of type of 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, πR] is also systematically vanishing over y ∈ [−πR + , 0 − ]. In contrast, the chirality configuration and mass spectrum are fixed by the choice of EBC (3.17) which can lead either to the two kinds of chiral solutions in Eq. (3.18) or to the vector-like solutions (3.19).
In Figure 2, we draw the first two excitation profiles for each free solution presented in Table 1 within the simple real case, α n Q,D = 0, and for two different types of Z 2 transformations from Eq. (3.8)-(3.9). We see clearly on Figure 2 that for example with the type II of 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 = πR ≡ −πR, in the scenario without Yukawa couplings. The presence of profile discontinuities here already justifies the treatment exposed in Section 2.1. The precise prescription (2.3) 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 − ) = q 1 L (0), is compatible with a well defined Lagrangian integrand over the action integration domain, where the profiles are continuous. 21 Here, thanks to the profile parities, a change of variable, y → −y, could be applied to recover exclusively the integration domain [0, πR].

Introducing the BBT
As suggested in Section 2.2.4, we can alternatively introduce the dimension 4 operators of Eq. (2.17) to study their effects with respect to the inconsistencies raised in Section 3.1. Hence, to the action S bulk from Eq. (2.4), we add now another part and consider: The variations of S B with respect to the generic fieldF [using Eq. (B.4)], together with Eq. (3.3) allow to write down the variations of the free fermion action: The individual vanishing of those volume and surface terms lead to the EOM (3.4) together with the four following NBC, depending on the two σ F 0,πR choices, [NBC] : the product f n L (y)f m R (y) systematically vanishes at y = 0, πR. Therefore, the BBT play the rôle of making j 4 F | 0,πR vanish (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.

Brane-localised scalar couplings in the orbifold: 4D approach
Once the free case is addressed, via the EBC (3.17) in Section 3.2 or the NBC (3.23) 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 (2.15) being induced by the Yukawa interaction between a brane-localised scalar field and bulk fermions? The considered action reads thus as, 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 n L/R (y) (q n L/R (y)) and associated free KK mass spectrum from line 1 (2) of Eq. (3.18), all the initial 4D effective masses for the KK modes of Eq. (3.6) in the interaction basis can be classified into two species: the pure KK masses (3.20) and the mass contributions from the Yukawa interaction given by the overlap between the wave functions and Higgs-brane, All the 4D mass terms enter the 4D effective Lagrangian through the following mass matrix, within the field basis noted, (4. 3) The texture of this infinite mass matrix M involving the diagonal m n , off-diagonal α ij and mixing the Q, D fields together can be precisely taken from the interval model context [38] (Section 3.2), with the replacement L ↔ πR, since the KK masses and bulk profile solutions are then identical (up to extensions over [−πR + , 0 − ] as seen in Section 3.2 here) like the Yukawa interactions localised at y = πR [for any Z 2 transformation (3.8)-(3.9)]. Now we can apply the results for the mass eigenvalues M n of the 4D eigenstates ψ n L/R (x µ ) obtained through the bi-diagonalisation performed in this Ref. [38], based on the calculations of Ref. [53], by re-normalising X to X/2 since the two present profiles (even or odd) entering α ij are normalised via Eq. (3.10) over a domain of double size 2L ↔ 2πR compared to the interval case. Doing so, the obtained exact mass eigenvalues are determined by the following equation, coming from the characteristic equation, in the case of a real X parameter and the positive m n branch from Eq. (3.20). Notice that the different conventional sign in front of the (−−) profiles found in Eq. (3.18) [here q n R (y) and d n L (y)], with respect to the interval study [38], does not affect the final mass spectrum -as is clear from Eq. (4.2). Hence, the physical absolute value of the mass spectrum reads as: with the functionñ(n) defined according to,  5 Brane-localised scalar couplings in the orbifold: 5D approach

Applying the NBC
Let us now study the presence of Yukawa couplings at the fixed point, y = πR, through the action, 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. (2.17) leading to SM-like chirality configurations: σ Q 0 = +, σ D 0 = −. Those guarantee a correct treatment of the free brane, like the EBC, as analysed throughout Section 3. Using Eq. (3.22) and Eq. (2.15), one gets directly the following action variations with respect to the fieldsQ andD, The separate vanishings of these volume and surface terms, induced by the least action principle, give rise respectively to the EOM (3.4) and the following NBC, As usual, the 5D field solutions of the EOM (3.4) and NBC (5.3) have the form of the following mixed KK decomposition [instead of Eq. (3.6)] [5,44], with the 4D fields ψ n L/R (x µ ), already mentioned in Section 4, satisfying the Dirac-Weyl equations, In the same way, for the three other types of Z 2 transformations (2.8)-(2.10), one obtains the same profile parities as in Eq. (3.9). As a conclusion, the same result as in Ref. [38] holds here for the orbifold: the 4D effective Yukawa coupling constant for the lightest modes (ψ 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 πR) = |X| 2 (independent from the profile normalisations), which conflicts analytically with the one obtained through the 4D method in Eq. (4.4) for a real X parameter. This failure motivates the alternative 5D methods of the next two subsections.

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 S 1 /Z 2 orbifold scenario. The complete relevant action including the brane-localised Yukawa terms (2.15),  Besides, a discontinuity of the form, j 4 | −πR + = j 4 | −πR ≡ j 4 | πR , would not fix any field at this fixed point and in turn would not induce vanishing variations in Eq. (5.2) possibly modifying the BC (5.3) which induce the drawbacks already pointed out in Section 5.1. As a consequence, we must consider the remaining model possibility: Combining Eq. (5.11) with Eq. (5.12) leads to, so that, using Eq. (3.16) and (5.9), we get the relation (inducing EBC), and its variation (for a non-trivial transformation with α = 0), At this level, we can consider the search for field solutions of vanishing Eq. (5.2) and Eq. (5.13)-(5.14) first on the domain, y ∈ [0, πR], which is equivalent to the search performed for the interval model in Ref. [38] with the replacement, L ↔ πR. Given that the ortho-normalisation condition (5.6) written on the domain [0, πR] is the same within the orbifold and interval frameworks, up to an overall factor 2, we can apply the conclusion of Ref. [38] and claim that there exists no SM-like consistent solution for the fields (over y ∈ [0, πR]) 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 S 1 /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 = πR, 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. (5.12) of its vanishing -at this fixed point, in contrast with the interval model (case of presence of boundary-localised Yukawa interactions) [38].

Introducing the BBT
In order to get meaningful field solutions in the presence of brane-localised Yukawa couplings at the fixed point, y = πR, let us finally try the introduction of the SM-like BBT (2.17) as in the free case of Section 3.3 or as in the interval model [38]. We thus consider here the same action as in Eq. (5.1)-(5.8) but adding now the BBT at y = πR: Using Eq. (3.22) and Eq. (5.2), we find the following action variations with respect toQ andD: The individual vanishing of those volume and surface terms, due to the action minimisation, leads to the EOM (3.4) and the following NBC, recovering thus exactly and conveniently the interval condition, if L = πR. Nevertheless, including the factor 1/ √ 2πR, the dimensional wave functions [mass dimension 1/2] are identical within the orbifold and interval frameworks only up to an additional normalisation factor 1/ √ 2 here, due to the double compact space size: see the respectively used decomposition normalisations (5.4) above and (4.1) in Ref. [38]. Therefore, we can finally apply the results of Ref. [38] here for the SM-like consistent solutions of the fields over y ∈ [0, πR]: we find, for the dimensionless profiles (∀ n ∈ N),   In Table 2 are exhibited the explicit profile functions over the entire orbifold domain for the SM-like solutions (5.17), (5.18)-(5.19), (5.20). We can see on this table that the choice of type of 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 Z 2 transformations of types I and II in Eq. (5.7)-(3.9), for the found Yukawa-coupled solutions (5.18), which are explicitly presented in Table 2, within the simplified real case, α Y = α n 0 = 0. We observe on this figure that all the wave function values at the Yukawa-brane (at the fixed point, y = πR) are modified due to the presence of this coupling. For example, under the type I of Z 2 transformation, the profile values d n L (πR) = d n L (πR − ) are shifted from zero as well as from d n L (−πR + ), 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 (×) from Eq. (5.17), (5.18)- (5.19), (5.20)]. Under the type II of Z 2 transformation, the same figure shows clearly that the profile jump d n L (πR) = d n L (πR − ) = d n L (−πR + ) disappears but then other kinds of jump arise like: q n L (πR) = q n L (πR − ) = q n L (−πR + ) and q n L (0 − ) = 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.
where we have used a trigonometric identity 22 to get the last equality. In the decoupling limit of extremely heavy KK modes, R → 0, we can then write the modulus of the lightest mode coupling constant, using Eq. (2.14), as, 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 Z 2 transformation among Eq. (5.7)-(3.9) affects neither the profile values taken at the point y = πR -see Table 2 -nor their global ortho-normalisation condition (5.6) -as described right below Eq. (5.16) -so that the 4D effective Yukawa coupling constants (5.21) are insensitive as well to this Z 2 representation choice.

The inclusive Z 2 parity
Let us study the alternative scenario whose definition is based on the Z 2 transformation of 5D fields extended to include the two fixed points at y = 0 and y = πR: Φ(x µ , −y) = T Φ(x µ , y) , ∀y ∈ (−πR, πR] , (6.1) in contrast with Eq. (2.1). This generic transformation still lets the Lagrangian density invariant, exactly like in Eq. (2.2). At the two fixed points, this Lagrangian invariance is once more automatically satisfied without the need for any specific T transformation. Accordingly to the simple Eq. (6.1), the operator 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 Z 2 transformation leading to the SM chirality set-up: it is the bulk transformation in Eq. (2.7), defined now over the same range as in Eq. (6.1), which keeps well L kin invariant in the bulk according to Eq. (2.2): , ∀y ∈ (−πR, πR] . (6.2) 22 For n ∈ Z, one has, cos(θ + nπ) = (−1) n cos(θ), and for T ∈ R, cos[arctan(T )] = 1 Focusing on the fixed points at y = 0 and y = πR ≡ −πR, we obtain the four non-trivial relations representing new EBC that we denote EBC' to distinguish them from those in Eq. (3.17).
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 Z 2 transformations (6.2) in the bulk lead to the EBC (3.17) while the Z 2 transformations (6.3) 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. (3.18) 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 Z 2 transformations (6.2).
Alternatively, the selected consistent BBT (2.17) can be included like in Section 3.3 to obtain the same SM-like solutions. The corresponding EBC' (6.3), part of the EBC (3.17) 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 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 Z 2 transformations (6.2) in the bulk -discussed in Section 5.2 and the EBC' (6.3). These EBC' combined with the surface terms at y = πR in Eq. (5.2), including the Yukawa terms, give rise to the BC of type (5.3) 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.

Result analysis 7.1 The higher-dimensional method
The present study confirms the general methodology depicted in Fig. 4 and presented in Ref. [38]. 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 Z 2 symmetry criteria and inducing the EBC (3.17) 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 branelocalised Yukawa couplings.
In the framework of the orbifold version described in Section 6, the additional field condition (6.3), coming from the Z 2 symmetry at the fixed points, accompanies the definition of the Z 2 symmetry of the bulk action and leads to the new EBC'.   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 [38].

Bilinear brane terms [NBC] 4D Approach
(Impossible) BC (±) BC (±) 5D Approach (Impossible) (Impossible) BC (×)  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 (2.17) have the same form as in the interval framework [38] and the different factor 2 is related to the double size of the compactified space for the identification, L = πR. 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 Γ 4 sign in Eq. (A. 3)].

About the orbifold/interval duality
In the global action (2.18), S bulk remains to be discussed, the other parts being identical in the orbifold and interval models. Thanks to the orbifold property (2.2), the change of variable, y = −y, allows the following rewriting of Eq. (2.4), where the last step is based on Eq. (2.3). Therefore, using Eq. (2.18) and the relevant identification, L = πR, we can express the orbifold action in terms of the interval action pieces [38] (indicated by the L exponent): This re-expression reveals an alternative method to derive the fermion masses and couplings, which are independent from the pure scalar part, namely S H . The idea is that, within the orbifold model now described by the action (7.2) importantly together with the description of the Z 2 symmetry over S 1 , we can first search for the field parts along the limited domain [0, πR]. This search is in fact based on the action [S L bulk + S (L) , since the overall factor 2 in Eq. (7.2) affects neither the EOM (global factor) nor the BC (same factor in front of the surface terms and pure brane terms combined into BC) 23 , 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 = πR) are those of Ref. [38] 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 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 (5.6)-(5.16), which confirms the result (5.21). 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. [38], which confirms the found spectrum (5.20).
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 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) ∝ (e iMn y + e −iMn y ), coming in factor (via the KK decomposition) of the energy coefficients e ±iE t in the 4D Dirac fields, which gives rise to wave planes propagating in both y-directions of the interval with momenta ±p n = ±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 → 0 and then coming back along 0 → L, reproduces the propagation along S 1 , following consecutively the two fundamental domains −πR → 0 − and 0 + → πR of the orbifold (effectively equivalent orientations of the circle in the bulk so a unique propagation direction chosen along it): exactly the same L [Φ(x µ , y)] 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 Z 2 symmetry depicted in the drawing 1.

Conclusions
In the study of the S 1 /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 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 [54].

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. (2.6) can be written in terms of its four explicit components F α [α = 1, 2, 3, 4]: and similarly,F can be expressed in terms of its own four componentsF α :

B.2 A typical compact form calculation
Using the Lagrangian L kin of Eq. (2.5), let us work out explicitly the following quantity entering Eq.