Localization of the Standard Model via Higgs mechanism and a finite electroweak monopole from non-compact five dimensions

We propose a minimal and self-contained model in non-compact flat five dimensions which localizes the Standard Model (SM) on a domain wall. Localization of gauge fields is achieved by the condensation of Higgs field via a Higgs dependent gauge kinetic term in five-dimensional Lagrangian. The domain wall connecting vacua with unbroken gauge symmetry drives the Higgs condensation which provides both electroweak symmetry breaking and gauge field localization at the same time. Our model predicts higher-dimensional interactions $|H|^{2n}(F_{\mu\nu})^2$ in the low-energy effective theory. This leads to two expectations: The one is a new tree-level contribution to $H \to \gamma\gamma$ ($H \to gg$) decay whose signature is testable in future LHC experiment. The other is a finite electroweak monopole which may be accessible to the MoEDAL experiment. Interactions of translational Nambu-Goldstone boson is shown to satisfy a low-energy theorem.


I. INTRODUCTION
The hypothesis that our four-dimensional world is embedded in higher-dimensional spacetime has been a hot topic in high energy physics for decades. Indeed, many mysteries of the Standard Model (SM) can be explained in this way. In particular, the discovery of D-branes in superstring theories [1] has intensified the research of the brane-world scenarios more than anything else. Then the seminal works [2][3][4][5] provided the basic templates for further studies.
The biggest advantage of models in extra dimensions is to utilize geometry of the extra dimensions. A conventional setup, common among the extra-dimensional models, is that extra dimensions are prepared as a compact manifold/orbifold. Namely, our four-dimensional spacetime is treated differently compared with extra dimensions.
In order to make things more natural, we can harness the topology of extra dimensions in addition to the geometry. The idea is quite simple and dates back to early 80's [6], namely that the seed of dynamical creation of branes in extra-dimensions is a spontaneous symmetry breaking giving rise to a topologically stable soliton/defect on which our fourdimensional world is localized. The topology ensures not only stability of the brane but also the presence of chiral matters localized on the brane [6,7]. In addition, graviton can be trapped [8][9][10][11][12][13]. Thus, the topological solitons provide a natural framework bridging gap between extra dimensions and four dimensions.
In contrast, localizing massless gauge bosons, especially non-Abelian gauge bosons, is quite difficult. There were many works so far . However, each of these has some advantages/disadvantages and there seems to be only little universal understanding. Then, a new mechanism utilizing a field dependent gauge kinetic term (field dependent permeability) came out in Ref. [38] where φ i are scalar fields. This is a semi-classical realization of the confining phase [2,[39][40][41][42][43][44] rather than Higgs phase outside the solitons. The authors have continuously studied brane-world models with topological solitons by using (I.1) [45][46][47][48][49][50][51].
Let us highlight several results: We investigated the geometric Higgs mechanism which is the conventional Higgs mechanism driven by the positions of multiple domain walls in an extra dimension in Ref. [49]. Then we proposed a model in which the brane world on five domain walls naturally gives SU (5) Grand Unified Theory in Ref. [50]. Furthermore, we have clarified how to derive a low-energy effective theory on the solitons in the models with a non-trivial gauge kinetic term (I.1) by extending the R ξ gauge in any spacetime dimensions D [51]. Another group also recently studied the SM in a similar model with β 2 taken as a given background in D = 5 [52,53]. They have also discussed phenomenology involving Nambu-Goldstone (NG) bosons for broken translation.
In this paper, we propose a minimal and self-contained model in non-compact flat five dimensions which localizes the SM on a domain wall. A striking difference from the previous works [45][46][47][48][49][50][51] is that we do not need extra scalar fields φ i which were introduced only for localizing gauge fields via Eq. (I.1). Instead, we put the SM Higgs in that role. As a consequence, localization of massless/massive gauge fields and the electroweak symmetry breaking have the same origin. In other words, the Higgs field is an active player in five dimensions with a new role as a localizing agent of gauge fields on the domain wall, in addition to the conventional roles giving masses to gauge bosons and fermions. Since our model does not need extra scalar fields φ i , it is not only very economical in terms of field content but also we are free from a possible concern that φ i would give an undesirable impact on the low-energy physics. We also study the translational NG boson Y (x µ ). Due to a low-energy theorem, it should have a derivative coupling with all other particles including Kaluza-Klein (KK) particles. We find a new vertexψ (KK) γ µ ∂ µ Y ψ (SM ) which provides a new diagram for the production of KK quarks ψ (SM ) + ψ (SM ) → ψ (KK) + ψ (KK) in the LHC experiment. This should be a dominant production process compared to the usual gluon fusion, and can easily violate experimental bounds. To avoid this, we will set a fundamental five-dimensional energy scale sufficiently large, providing all the KK modes supermassive.
However, surprisingly, the Higgs dependent gauge kinetic term (I. In order to illustrate a novel role of the Higgs mechanism besides the conventional roles of giving masses to gauge fields and chiral fermions in a gauge invariant manner, let us consider a simple Abelian-Higgs-scalar model in D = 5 flat spacetime as a toy model. The following arguments are quite universal so that it is straightforward to apply them to non-Abelian gauge theories, such as the SM which we discuss in Sec. IV and also to models with D ≥ 5 [61].
A simple Abelian-Higgs-scalar model in D = 5 reads: 1 Here T is a real scalar field, and H is the Higgs field which interacts with A M not only via the covariant derivative D M H = ∂ M H + iq H A M H, but also through non-minimal gauge kinetic term with the field-dependent function β 2 defined by The covariant derivative of the charged fermion field is defined by 1 The bosonic part is a simple extension of the well-studied model [62][63][64] in which the Higgs field H is replaced by a real scalar field.
The five-dimensional Gamma matrix Γ M is related to four-dimensional one as Γ µ = γ µ and There are two discrete vacua T = ±v with H = 0. The vacua break the Z 2 symmetry but preserve U (1) gauge symmetry which is necessary to localize the massless U (1) gauge field on a domain wall [38,[45][46][47][48][49][50][51]. Therefore, the Higgs mechanism does not take place in the vacua.
However, spontaneous breaking of the Z 2 symmetry gives rise to a topologically stable domain wall, connecting these two discrete vacua. Depending on the values of the parameters, the following stable domain wall solutions are obtained We are not interested in the former solution (II.4) since the U (1) is unbroken everywhere and the gauge field is not dynamical due to β 2 = 0. On the other hand, as we will show below, the latter solution (II.5) localizes the U (1) gauge field by β 2 ∝ sech 2 Ωy. When the Higgs is neutral (q H = 0), the lightest mode of the localized gauge field is precisely massless [49][50][51] whereas, as we will see, it becomes massive when the Higgs is charged (q H = 0).
To understand the mechanism for the localized massless gauge field to become massive, let us compute the low-energy effective potential for the effective Higgs field in four dimensions in the parameter region From the linearized field equation around the background of the domain wall solution (II.5), we find that there is a mass gap of order Ω, and two discrete modes much lighter than the mass gap. The lowest mode is exactly massless Nambu-Goldstone (NG) boson corresponding to spontaneously broken translation symmetry along the y direction. Its interactions with all other effective fields are generally suppressed by inverse powers of large mass scale, whose characteristics will be discussed in Sec. III B and Sec. IV. Disregarding the NG boson, we retain only one light boson, whose wave function is well-approximated by the same functional form as the background solution H 0 (y) in (II.5). When λv = 0, this wave function gives the zero mode exactly, corresponding to the condensation mode at the critical point λv = Ω, where the H field begins to condense. After H condenses, this mode becomes slightly massive above the critical point (II.6) with the mass of order λv, whose wave function receives small corrections suppressed by powers of (including an admixture of fluctuations of T ). Combining the background solution and the fluctuation, we introduce the following effective field H(x) (a quasi-moduli) corresponding to the Higgs field in the low-energy effective field theory Inserting this Ansatz into the Lagrangian and integrating over y, we obtain effective action as where the effective gauge field in the covariant derivative D µ is more precisely defined below, see Eq. (II.17). The possible corrections suppressed by powers of 2 can be systematically computed as described in Appendix A. This is just a conventional Higgs Lagrangian which catches all the essential features. First, note that the sign of the quadratic term is determined  which is of order 2 as we expected. Thus, the y-dependent Higgs condensation H 0 (y) of Eq. (II.5) in D = 5 which is driven by the domain wall T 0 (y) connecting two unbroken vacua gives indeed the Higgs mechanism through Eq. (II.8). To complete the picture, we next calculate the mass of gauge bosons. We will assume vλ > Ω in the rest of paper, so that the solution (II.5) always applies.
To figure out the spectrum of the gauge field, first of all, we use canonical normalization The linearized equation of motion for A µ in the generalized R ξ gauge [51,61] is Thus, the Kaluza-Klein (KK) spectrum is identical to eigenvalues of 1D quantum mechanical problem with the Schrödinger potential V S = (∂ 2 y β)/β + 2q 2 H µ 2 . Fig. 1 (a) shows the corresponding Schrödinger potential. The eigenvalues m 2 n and eigenfunctions φ n (y) can be easily obtained [51]. There is a unique bound state (II.14) No other bound states exist and a continuum of scattering modes parametrized by the momentum k corresponds to the eigenvalues m 2 k = k 2 + Ω 2 + q 2 H µ 2 . Thus, the mass gap between the unique bound state φ 0 and the higher KK modes is of order Ω (under the assumption Ω µ) which is the inverse width of the domain wall. In terms of the original field A µ , the lightest massive gauge boson A (0) µ (x) is given by where the ellipses stand for the heavy continuum modes. The mass of the lightest massive gauge boson is One can show that the fifth gauge field A y has no physical degrees of freedom [51].
Having Eq. (II.15) at hand, we are now able to read the effective gauge coupling constant.
where ψ (n) L and ψ (n) R are left-handed (γ 5 ψ L = −ψ L ) and right-handed (γ 5 ψ R = ψ R ) spinors in four-dimensions (II. 22) and the mode functions f (n) with Q = ∂ y + ηT 0 , and Q † = −∂ y + ηT 0 . Assuming the five-dimensional Yukawa coupling to satisfy η > 0, we find a unique zero mode where N L,0 is a normalization constant. The number of excited bound KK states corresponds to n = ηv Ω ( is the floor function). For example, the first excited bound state exists when ηv Ω ≥ 1 and its wave function and mass are given by The mass gap between the zero mode and the KK modes is again of order Ω for the parameter region given in Eq. (II.20). The analysis forΨ can be done similarly by replacing η withη and by exchanging L and R.
The interaction between the lightest massive gauge boson A (0) µ and the fermionic zero mode ψ where the ellipses stand for the massive modes. Notice the gauge coupling is the same as in Eq. (II.18). We have to emphasize that the effective gauge coupling e is the same for any localized fields. The universality is ensured by the fact that the wave function of the lightest mode of A µ is always constant.
We can also easily derive an effective Yukawa coupling as follows, , where Γ(x) is the gamma function. Thus the Yukawa coupling in the four dimensions reads Before closing this section, let us comment on the Higgs field. The Higgs condensation occurs at the five-dimensional level leading to the localization of the massless/massive gauge bosons in our model. A new feature of our Higgs mechanism is that the order parameter H induced by domain wall is position-dependent. As a consequence, effective Higgs field is localized and only the massive physical Higgs boson h remains in the low-energy physics.
In contrast, if one uses other neutral scalar fields φ i to localize the gauge fields [45][46][47][48][49][50][51], one has to prepare another trick to localize the Higgs fields too. For example, in recent papers [52,53], the kinetic term of the Higgs field is not minimal but multiplied by a function β 2 (φ).
In such models, the Higgs field (massive Higgs boson and massless NG boson) is localized on the domain wall and the Higgs condensation occurs in the low-energy effective theory.
Namely, the Higgs field plays no active roles at the five-dimensional level.

A. Mass scales
In order to have a phenomenologically viable model, we need to explain observed mass Fitting these masses to experimentally observed values, we still have one mass scale Ω completely free. Therefore we can choose the energy scale Ω of the five-dimensional theory as large as we wish, leaving phenomenologically viable model at low-energies.
For instance, if we choose the ratio of the high energy scale and SM scale to be parametrized as we find the scale of parameters in the model as Thus, the five-dimensional Yukawa couplings η,η and χ are naturally set to be the same order. Note also that this justifies Eq. (II.20). To understand the hierarchy of lighter fermion masses, we can use the usual mechanism of splitting of position of localized fermions as explained briefly in Sec. IV.
In summary, for having the SM at the low-energy, all the dimension full parameters in the five-dimensional Lagrangian are set to be of the same order as We need a fine-tuning for two small parameters of mass dimension: λv, µ ∼ 10 2 GeV.
Estimate of the lower bound for the parameter Ω ∼ 10 2+a GeV will be discussed in Sec. IV using constraints from the LHC data.

B. Translational zero mode
Here we study interactions of the translational Nambu-Goldstone (NB) mode, and their impact on low-energy phenomenology. Symmetry principle gives low-energy theorems, dictating that the NG bosons interact with corresponding symmetry currents as derivative interactions (no interaction at the vanishing momentum of NG bosons). Hence their interactions are generally suppressed by powers of large mass scale. In order to understand the interactions of the NG bosons, it is most convenient to consider the moduli approximation [75] where the moduli are promoted to fields in the low-energy effective Lagrangian. Let us consider a general theory with a number of fields 4 φ i (x, y) admitting a solution (soliton) of field equation, which we take as a background. When the theory is translationally invariant, the position Y of the soliton is a moduli. It is contained in the solution as φ i (x, y − Y ).
In the moduli approximation, we promote the moduli parameter Y to a field Y (x) slowly varying in the world volume of the soliton. We call this moduli field Y (x) as NG field 5 . By introducing the NG boson decay constant f Y to adjust the mass dimension of the NG field to the canonical value [Y (x)] = 1, we obtain The precise value of the decay constant f Y is determined by requiring the kinetic term of NG boson to be canonical as illustrated in the subsequent explicit calculation. By integrating over y, we can obtain the effective interaction of the NG field. One should note that the constant part Y of NG field Y (x) is nothing but the position of the wall, which can be absorbed into the integration variable y by a shift y → y − Y because of the translational invariance. Hence the constant Y disappears from the effective action after y-integration is done. This fact guarantees that Y (x) must appear in the low-energy effective theory always with derivatives, i.e. ∂ µ Y (x). Let us examine how this fact fixes the interactions of NG particle in the effective Lagrangian to produce the low-energy theorem. Derivative ∂ µ can only come from the derivative term in the original action L, giving terms linear in the NG particle Y (x) as where the energy-momentum tensor T M N of matter in five dimensions is given by This is the low-energy theorem of the NG particle for spontaneously broken translation. Thus we find that there are no nonderivative interactions that remain at the vanishing momentum of NG bosons, including KK particles. For instance, the possible decay amplitude of a KK fermion into an ordinary fermion and a NG boson should vanish at zero momentum of NG boson and will be suppressed by inverse powers of large mass scale such as Ω. In this way, we can compute the effective action of NG field in powers of derivative ∂ µ . Usually we retain up to second order in derivatives, but higher derivative corrections can be obtained systematically with some labor [76].
Let us compute the effective Lagrangian of NG field Y (x) more explicitly by using the moduli approximation in our model as The wall position moduli in wave functions of fermions must also be promoted to NG field A few features can be noted. First of all, the NG bosons have only derivative interactions, as required by the above general consideration. Secondly, the derivative interaction produces higher-dimensional operators coupled to NG bosons. The required mass parameter in the coefficient of the interaction term is given by the high energy scale as Ω/(2v 2 ) ∼ 1/Ω 2 .
Therefore the interaction is suppressed by a factor of (momentum)/Ω. Thirdly, the interaction linear in the NG particle in Eq.(III.7) happens to be absent in this model. This is a result of a selection rule in our model. 6 The Lagrangian (II.1) and the background solution Only when we take into account the heavy KK modes [53], we have interactions linear in ∂ µ Y . For example, including the lightest KK fermion given in Eq. (II.25) (b = ηv Ω > 1 in order to have a discrete state) we obtain a vertex where α is a dimensionless constant of order one defined by α ≡ y) is the beta function. 7 The above interaction gives the decay process ψ (III.14) 6 Note that a non-derivative coupling Yψ R from TΨΨ was recently studied in Ref. [53]. However, the symmetry principle of NG boson for translation does not allow coupling without the derivative ∂ µ . 7 Note that α → 0 as b → 1.
in the LHC experiment. This should be the dominant production mechanism because of large momentum fraction of quarks as given by their distribution function inside nucleons.
The production process (III.14) tells us the lower bound of the KK quark masses. We will estimate it in Sec. IV where the Standard Model is embedded in our framework.
Then the first term of Eq. (II.1) yields Thus, there is a new tree-level amplitude for h → γγ. In the SM, the Higgs boson decays into two photons mediated by top or W bosons at one-loop level. The operator of interest µν ) 2 , whose coefficient is bounded by the LHC measurement as c ∼ 10 −3 [65,66]. However, our simplest model has c = 1 2 , so is strongly excluded experimentally.

D. Generalized models
To have a phenomenologically acceptable h → γγ decay amplitude, we can modify the field dependent gauge kinetic term, for example, as The background configuration of the Higgs field H = H 0 (y) remains the same as in Eq. (II.5) since the β 2 F 2 M N term does not contribute to the background solution. The reason for selecting this specific modification will be explained below soon. Before that, however, let us mention that the modification comes with a price. The linearized equation of motion in the generalized R ξ gauge for the gauge field with a generic β reads [51,61] Then, determining the physical spectrum corresponds to solving the eigenvalue problem If β 2 is quadratic in H as was the case in Eq. (II.3), the third term on the left-hand side is constant. Therefore, the problem is of the same complexity as if q H = 0. On the other hand, when β 2 is not purely quadratic, the eigenvalue problem is essentially different from that of −∂ 2 y + (∂ 2 y β) β . Fig. 1 (b) shows the corresponding Schrödinger potential. In case of (III.17) the Schrödinger equation in terms of the dimensionless coordinate z = Ωy is given by Note that this is independent ofv because of H 0 =v sech z. Although we cannot solve this exactly, we can still solve this problem perturbatively for Ω µ by treating the third term on the left-hand side as a small correction. The lowest eigenfunction and eigenvalue are approximately given by This is just the same as Eq. (II.14), and, therefore, the mass of the lightest massive gauge boson is of order µ, which justifies our assumption Ω µ. Since the situation is almost the same as in the simplest model, we have v 2 h /2 = dy H 2 0 = 2v 2 /Ω, and the effective gauge coupling is e ∼ µ/v h ∼ 1. Thus the modified model defined by Eq. (III.17) provides the SM at low energies in the same manner as the simplest model does.
Now, let us turn to the problem of h → γγ. So we set q H = 0 and Eq. (III.21) becomes exact wave function of the massless photon. As before, we put H given in Eq. (III.15) into the gauge kinetic term −β 2 F 2 M N with β 2 given in Eq. (III.17). Then, we find As we see, the term h(F (0) µν ) 2 does not exist. Therefore, the modified model is compatible with the bound given by the current experimental measurement of h → γγ.
If the factor in front of the quartic term of Eq. (III.17) deviates slightly from 3 4 , the term h(F The above consideration holds for another similar process of h → gg (two gluons). An experimental signature should be the decay of physical Higgs particle to hadronic jets.
Moreover, it will affect the production rate of physical Higgs particles from hadron collisions.
Recently, it was proposed that another interesting signature from the localized heavy KK modes of gauge bosons and fermions [52,53], although the presence and/or the number of localized KK modes is more dependent on details of models. Our model has the same signatures too but they are subdominant in our model since they are 1-loop effects of the supermassive KK modes.

IV. THE STANDARD MODEL
Let us briefly describe how our mechanism works in the SM. The minimal five-dimensional Lagrangian is where and are absorbed by the W and Z bosons. Indeed, the spectrum of W ± µ = 2βW µ , and Z µ = 2βZ µ are determined by the 1D Schrödinger problems The details of the derivation will be given elsewhere [61]. On the other hand, the photon A µ = 2βA µ and gluon G µ = 2βG µ are determined by −∂ 2 y + (∂ 2 y β) β . Therefore, the lightest modes φ 0 ∝ β of photon and gluon are exactly massless. The results so far are independent of β 2 . To be concrete, let us choose the simplest function β 2 = |H| 2 /4µ 2 . Then the effective SU (2) W gauge couplings and the electric charge are given by where v h is given in Eq. (II.10). Masses of W and Z are easily read from Eq. (IV.4) as For the fermions, we assume η L > 0 and η R > 0. Then the left-handed fermion from Q is localized at the zero of T , while the right-handed fermion from U is localized at the zero of T − m. The Yukawa term χQHU is responsible for giving non-zero masses to the localized chiral fermions, which is necessarily exponentially small for m = 0 since the left-and righthanded fermions are split in space. By distinguishing parameters such as m for different generations as was done in many models with extra dimensions [73,74], the hierarchical Yukawa coupling can be naturally explained in our model.
This way, the SM particles are correctly localized on the domain wall in our framework.
Before closing, we evaluate the lower bound of KK quark mass by using the KK quark production process in Eq. (III.14) via Nambu-Goldstone boson exchange. If we take the initial quarks of different flavor for simplicity, we have only single Feynman diagram depicted in Fig. 2(a). In the process (III.14) followed by ψ L,R , the final state contains two SM fermion jets and a missing energy of the NG boson Y , whose signature is similar to squark pair production, where a squark decays into the partner SM quark and a gluino or neutralino in the simplified supersymmetric models [54][55][56]. In most of kinematical regions, a dominant processes for squark pair production is given by Feynman diagram depicted in Fig. 2(b). Since both processes involve the same valence quark distribution functions, we can compare these cross-sections directly to obtain an order of magnitude estimate of the lower bound for KK quark mass using the analysis for squark mass bound. As shown in Appendix C, the differential cross section dσ dt of (III.14) producing a pair of the first KK fermion with mass M 1 is given by summing contributons from initial state of different chiralities (LL, RR, LR, RL) as E is the center of mass energy of incoming particles, and θ is the scattering angle. We ignore masses of the SM quarks and all the parameters are taken to be common for the different quarks just for simplicity. We can assume v ≈ Ω 3/2 and M 1 ≈ Ω for simplicity.
The squarks production ud →ũd cross section [57][58][59] is The SU (3) C gauge coupling and gluino mass are denoted as g s and mg, and a common mass mq is assumed for squarks of different flavors and chiralities.
To obtain the bound for the production of heavy particles, we can expect that the crosssection near threshold (β = 0) is a good guide for the order of magnitude estimate. Both differential cross-sections become constants without angular dependence at the threshold, and their ratio is given as The simplified analysis for squark production gives mq > 1.5 TeV, assuming mq = mg [60]. The identical bound for the KK fermion mass M 1 ∼ Ω > 1.5 TeV is obtained for Since Ω = 10 2+a GeV, we have the lower bound for a as a 1. If the coupling α of KK fermion is larger than g s , we obtain larger lower bound for its mass. To determine how much larger requires a more detailed analysis of data.

V. FINITE ELECTROWEAK MONOPOLES
The SM has a point magnetic monopole which is the so-called Cho-Maison (CM) monopole [67]. It is different from either a Dirac monopole or a Nambu electroweak monopole [68]. Unfortunately, its mass diverges due to a singularity at the center of the monopole. Cho, Kim and Yoon (CKY) [69] have proposed a modification of the SM in four dimensions which includes the field dependent gauge kinetic term as In This can be derived from our model with The background solution is still H 0 =v sech Ωy. Fig. 1 (c) shows the corresponding Schrödinger potential. Then the wave function of the massive U (1) Y gauge field reads . As before, we identify the four-dimensional Higgs field H(x) as H =v H(x) v h sech Ωy with v h = 2 Ωv . We find the EMY's model from the five dimensions via the domain wall and the Higgs mechanism as where we ignored contributions from the massive KK modes.
Note that the β 2 modifies not only the gauge kinetic term of U (1) Y but also that of the SU (2) W . An electroweak monopole in such theory also has a finite mass [77].
CKY have claimed that discovery of an electroweak monopole is a real final test for the SM [69] . For us, it is not only the topological test of the SM but also would give constraints for restricting the β 2 factor of the five-dimensional theory.

VI. CONCLUSIONS AND DISCUSSION
We proposed a minimal model in flat non-compact five dimensions which realizes the SM on a domain wall. In our approach, the key ingredients for achieving this result are the following: (i) the spacetime is five-dimensional, (ii) there is an extra scalar field T which is responsible for the domain wall, (iii) there is a field-dependent gauge kinetic term as a function of the absolute square of the Higgs field.
In our model, all spatial dimensions are treated on the same footing at the beginning.
The effective compactification of the fifth dimension happens as a result of the domain wall formation breaking the Z 2 symmetry spontaneously. The presence of domain wall automatically localizes chiral fermions [6,7]. The key feature of our model is that the Higgs dependent gauge kinetic term drives the localization of SM gauge bosons and the electroweak symmetry breakdown at the same time. The condensation of the SM Higgs field inside the wall for Ω < λv can be understood as follows. As we let the parameter Ω decrease across λv, we find a massless mode emerges at the critical point Ω = λv, which becomes tachyonic below the critical point and condenses until a new stable configuration is formed.
It is interesting to observe that this thought-process is analogous to a second-order phase transition if we regard the parameter Ω as temperature. In addition to the conceptual advantages listed above, we investigated a new interaction hγγ (and hgg) coming from Eq. (I.1). This should be bounded by the LHC measurement [65,66], therefore it gives a constraint to β 2 . However, a small deviation from exactly vanishing amplitude hγγ from tree-level coupling is allowed, which can be a testable signature in the future experiment at the LHC. This possibility of the tree-level coupling of hγγ is a new signature of our model of domain-wall-induced Higgs condensation and gauge field localization. This feature is in contrast to similar models of gauge field localization without the active participation of Higgs field in the localization mechanism [52,53]. For instance, these models generally give only loop-effects of KK particles, instead of the treelevel hγγ coupling. Therefore we can have a testable signature of hγγ even if there are no low-lying KK particles, unlike these models. Furthermore, our five-dimensional model explains higher dimensional interaction as Eq. (V.1) that allows the existence of a finite electroweak monopole, whereas previous studies have failed to provide the origin of such higher-dimensional operators [69,70]. The monopole mass was estimated [69,70] as 5.5 TeV, so that it can be pair-produced at the LHC and accessible to the MoEDAL experiment [71,72]. If an electroweak monopole will be found, it provides an indirect evidence for the extra dimensions and the domain wall. Our domain wall model can account for the hierarchical Yukawa coupling in the SM from position difference of localized wave functions of matters as was done in many models with extra dimensions [73,74].
If we introduce the other scalar fields φ i to localize the gauge field and the Higgs field via β(φ i ) as in Eq. (I.1), they would give an impact on the low-energy physics like φ i → hh, φ i → γγ, and φ i → gg. Therefore, we have to be very cautious for including the extra scalar fields φ i . Our model is free from this kind of concern, which is one of the important progress achieved in this work.
Although we did not explain it in detail, the absence of additional light scalar boson from A y is one of the important properties of our model [51][52][53]. Moreover, the fact that the localization of gauge fields via Eq. (I.1) automatically ensures the universality of gauge charges is also important.
In summary, the particle contents appearing in the low-energy effective theory on the domain wall are identical to those in the SM. All the KK modes can be sufficiently separated from the SM particles as long as we set Ω ∼ 10 2+a GeV be sufficiently large. Nevertheless, our model is distinguishable from the SM by the new tree-level decay h → γγ (h → gg) and a finite electroweak monopole. A possible concern in our model is the additional massless particle Y (x) which is inevitable because it is the NG mode for spontaneously broken translational symmetry. However, thanks to the low-energy theorems, all the interactions including Y (x) must appear with derivatives ∂ µ Y (x). Consequently, they are suppressed by the large mass scale Ω and have practically no impact on phenomena at energies much lower than the large mass scale Ω. The KK quark pair production via NG particle exchange gives a lower bound for Ω which is larger than 1.5 TeV. Larger Ω requires severer fine-tuning, but is safer phenomenologically, whereas smaller Ω requires less fine-tuning and can be disproved more easily by experimental data.
Let us discuss possible effects of radiative corrections in our low-energy effective theory.
The particle content of effective theory below the mass scale Ω is identical to SM except the value v h . We need to assume that the higher dimensional coupling of Higgs boson and gauge fields are fine-tuned to that value when the Higgs vacuum expectation value is finetuned to a value much smaller than Ω. With this assumption, we expect that the radiative corrections to quantities such as physical Higgs boson mass should be essentially the same as nonsupersymmetric SM. For instance we need to implement supersymmetry if we wish to make the fine tuning less severe in our model.
Models with warped spacetime [4,5] exhibit features similar to our model, except that the usual assumption of delta-function-like brane in models with warped spacetime is replaced by a smooth localized energy density (fat brane) in our model. Previously we have studied BPS domain-wall solutions embedded into four-and five-dimensional supergravity [11][12][13] As we heat up the universe starting from this situation, finite temperature effects come in to raise the effective potential for nonzero values of Higgs field. Eventually around a certain temperature of order λv, we will find a phase transition to the phase without Higgs condensation, namely SU (2)×U (1) gauge symmetry restoration. To estimate this transition temperature, we need to study the change of effective potential during this process. As we noted, the coefficient λ 2 2 is likely to change gradually from −4(λv) 2 /3 to −(λv) 2 . Therefore we need to take account of the change of λ 2 2 besides the finite temperature effects. This is an interesting new challenge to determine the transition temperature in this kind of models.
We leave this issue for a future study.
The quadratic part of the bosonic Lagrangian is given by means of Hamiltonians K T R , K I Once we obtain eigenfunctions of these Hamiltonians, we can obtain mode expansions of the 5D fields into KK towers of effective fields, such as where the n-th eigenstate generally has components in both 5D fields δT and δH R , since they have coupled Hamiltonian K T R . The label of eigenstates n contains also continuum states.
Since the δH I will be absorbed by the gauge boson by the Higgs mechanism, we will consider only the coupled linearized field equation for δT and δH R . Since the coupled equation is difficult to solve exactly, we solve it starting from the λv = 0 case as a perturbation series in powers of the small parameter 2 = (λv/Ω) 2 .
At λv = 0, the Hamiltonian K T R becomes diagonal and the T and H R linearized field equations decouple Eigenvalues of the Hamiltonian give mass squared m 2 of the corresponding effective fields.
In the parameter region (II.6), we find two discrete bound states for δT , and a continuum of states with the threshold at (m . This mode will become massive physical Higgs particle when we switch on the perturbation (λv) 2 > 0.
We can now systematically compute the perturbative corrections in powers of small parameter . The lowest order correction to the eigenvalue can be obtained by taking the expectation value of the perturbation Hamiltonian in terms of the lowest order wave function. Therefore we obtain the mass eigenvalue of the physical Higgs particle up to the leading order (m h ) 2 = dyu We find exact mode functions in this case. We find two discrete bound states for δT and a continuum of states with the threshold at (m This is precisely the tachyonic mode at the unstable background solution. We note that the value of (negative) mass squared is different from the corresponding value −4(λv) 2 /3 of the off-shell extension to H = 0 of the effective potential computed on the stable BPS solution in Eq.(II.9). This is due to the fact that a different background solution gives a different spectrum of fluctuations, even though they are qualitatively similar.
Once the exact mode function is obtained, on the background of the unstable solution, we only need to insert the following Ansatz into the 5D Lagrangian and integrate over y, in order to obtain the effective potential of the effective Higgs field H (x). The quadratic term agrees with the mass squared eigenvalue of the mode equation of fluctuations. It is interesting to observe that the coefficient of the quadratic term is different from that computed on the stable BPS solution as background, although the quartic term is identical.

Appendix C: Cross section for KK fermion pair production by NG boson exchange
Here we calculate the differential cross section (IV.7). First we consider the process L , whose Feynman diagram is shown in Fig. 2(a). The amplitude is given in terms of spinor wave functions u uL and u dL of incoming SM fermions, and u u (1) L and u d (1) L of outgoing KK quarks as iM = α 2 Ω v 2 i t (ū u (1) L (k 1 )i( / p 1 − / k 1 )u uL (p 1 ))(ū d (1) L (k 2 )i( / p 1 − / k 1 )u dL (p 2 )), (C.1) with t = (p 1 − k 1 ) 2 . We approximate SM quarks to be massless, and assume the same vertex couplings α for uu (1) Y and dd (1) Y for simplicity, although they can be different since fermion wave functions for u, u (1) and d, d (1) are in general different. The squared amplitude is which leads to the differential cross section with s = 4E 2 . Other combinations of initial quark chiralities RR, LR, RL are found to give identical differential cross sections. Hence we find (IV.7).