^{3}

We propose a minimal and self-contained model in non-compact flat five dimensions that localizes the Standard Model (SM) on a domain wall. Localization of gauge fields is achieved by condensation of the Higgs field via a Higgs-dependent gauge kinetic term in the 5D 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

The hypothesis that our 4D 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 [

The biggest advantage of models in extra dimensions is the ability to utilize the

In order to make things more natural, we can harness the

In contrast, localizing massless gauge bosons, especially non-Abelian gauge bosons, is quite difficult. A great deal of work has been published so far [

In this paper, we propose a minimal and self-contained model in non-compact flat five dimensions that localizes the SM on a domain wall. A striking difference from the previous works [

The paper is organized as follows. In

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

A simple Abelian–Higgs-scalar model in ^{1}

The covariant derivative of the charged fermion field is defined by

There are two discrete vacua

However, spontaneous breaking of the

To understand the mechanism for the localized massless gauge field becoming 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 (

Inserting this ansatz into the Lagrangian and integrating over

The mass of the physical Higgs boson can be read from Eq. (

To figure out the spectrum of the gauge field, first of all, we use canonical normalization

Thus, the Kaluza–Klein (KK) spectrum is identical to eigenvalues of a 1D quantum mechanical problem with the Schrödinger potential

The black lines show the Schrödinger potentials

No other bound states exist and a continuum of scattering modes parametrized by the momentum

One can show that the fifth gauge field

Having Eq. (

Combining Eq. (

Finally, let us investigate the domain-wall fermions

Then we can treat the Yukawa term

The mass gap between the zero mode and the KK modes is again of order

The interaction between the lightest massive gauge boson

We can also easily derive an effective Yukawa coupling as follows:

Before closing this section, let us comment on the Higgs field. The Higgs condensation occurs at the 5D 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

In order to have a phenomenologically viable model, we need to explain the observed mass ^{2}^{3}

Fitting these masses to experimentally observed values, we still have one mass scale

For instance, if we choose the ratio of the high-energy scale and SM scale to be parametrized as

For the fermionic sector, we require Eqs. (

Thus, the 5D Yukawa couplings

In summary, to have the SM at low energy, all the dimensionful parameters in the 5D Lagrangian are set to be of the same order as

We need fine-tuning for two small parameters of mass dimension:

Here we study interactions of the translational Nambu–Goldstone (NB) mode, and their impact on low-energy phenomenology. The 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[^{4}^{5}

The precise value of the decay constant

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 an NG boson should vanish at zero momentum of the NG boson and will be suppressed by inverse powers of large mass scale such as

Let us compute the effective Lagrangian of the NG field

The wall position moduli in wave functions of fermions must also be promoted to the NG field

We finally obtain the effective Lagrangian for low-energy particles as:

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 ^{6}

Only when we take into account the heavy KK modes [^{7}

As explained above, our model provides a domain wall inside which all the SM particles are localized. All the KK modes are separated by the mass gap

Then the first term of Eq. (

Thus, there is a new tree-level amplitude for

To have a phenomenologically acceptable

The background configuration of the Higgs field

Then, determining the physical spectrum corresponds to solving the eigenvalue problem

If

Note that this is independent of

This is just the same as Eq. (

Now, let us turn to the problem of

As we see, the term

If the factor in front of the quartic term of Eq. (

Of course, the modification in Eq. (

The above consideration holds for another similar process of

Recently, another interesting signature was proposed from the localized heavy KK modes of gauge bosons and fermions [

Let us briefly describe how our mechanism works in the SM. The minimal 5D Lagrangian is

As before, there are two discrete vacua

The Higgs doublet

The details of the derivation will be given elsewhere [

For the fermions, we assume

This way, the SM particles are correctly localized on the domain wall in our framework.

Before closing, we evaluate the lower bound of the KK quark mass by using the KK quark production process in Eq. (

Feynman diagrams for the processes

The squark production

To obtain the bound for the production of heavy particles, we can expect that the cross section near threshold (

The simplified analysis for squark production gives

The SM has a point magnetic monopole, which is the so-called Cho–Maison (CM) monopole [

Neither CKY nor EMY discuss the underlying rationale for their modifications to the SM. In contrast, our 5D model has a clear motivation for the field-dependent gauge kinetic term, which is the domain-wall-induced Higgs mechanism. For example, one of EMY’s proposals is [

This can be derived from our model with

The background solution is still

Note that

CKY have claimed that discovery of an electroweak monopole is a real final test for the SM [

We proposed a minimal model in flat non-compact five dimensions that realizes the SM on a domain wall. In our approach, the key ingredients for achieving this result are the following: (i) the spacetime is 5D, (ii) there is an extra scalar 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

Contrary to the conventional wisdom in domain-wall model-building, where the formation of the domain wall happens separately from the Higgs condensation to break electroweak symmetry, in our model we succeeded in combining both mechanisms and keeping the Higgs field active even in five dimensions. In other words, our model is very economical in terms of field content. Naively, one may expect that this means that the domain-wall mass scale must coincide with the SM scale, but surprisingly, that does not have to be so. As we have argued in

In addition to the conceptual advantages listed above, we investigated a new interaction

If we introduce the other scalar fields

Although we have not explained it in detail, the absence of an additional light scalar boson from

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

Let us discuss the possible effects of radiative corrections in our low-energy effective theory. The particle content of effective theory below the mass scale

Models with warped spacetime [

Finally, our model offers an interesting problem for the study of the cosmological evolution of the universe. Let us restrict ourselves to the region of temperature around the scale

This work is supported in part by the Albert Einstein Centre for Gravitation and Astrophysics financed by the Czech Science Agency Grant No. 14-37086G (F.B.). This work is also supported in part by the Ministry of Education, Culture, Sports, Science (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No. S1511006) (N.S.), by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) Grant Numbers 26800119, 16H03984, and 17H06462 (M.E.), and by the program of the Czech Ministry of Education Youth and Sports INTEREXCELLENCE Grant Number LTT17018 (F.B.). F.B. was an international research fellow of the Japan Society for the Promotion of Science, and was supported by a Grant-in-Aid for JSPS Fellows, Grant Number 26004750.

Open Access funding: SCOAP

We can choose another BPS solution (

The linearized field equation, in this case, is decoupled with the Hamiltonian

We find exact mode functions in this case. We find two discrete bound states for

The massless mode gives an exact NG boson mode function in this case. For the fluctuation

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

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

After integrating over

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.

Here we calculate the differential cross section (

^{1} The bosonic part is a simple extension of the well studied model [

^{2} Here we have just one gauge coupling, because of our simplification of

^{3} Actually, they are somewhat less than unity experimentally, in conformity with the perturbativity of SM.

^{4} In our concrete model, we have fields such as

^{5} This definition is, in general, a nonlinear field redefinition of the effective field that arises in the mode analysis of fluctuation fields, such as in

^{6} Note that a nonderivative coupling

^{7} Note that