^{3}

Combining the semiclassical localization mechanism for gauge fields with

One of the most puzzling features of the Standard Model (SM) is the lack of explanation of the gauge hierarchy problem. To solve this problem, apart from other popular ideas such as supersymmetry (Refs. [

The possibility of dynamical realization of the brane-world idea via a domain wall was recognized quite early (Ref. [

It was pointed out in Ref. Refs. [

In this paper, we investigate the Higgs mechanism caused by the domain walls. In the previous works Refs. [

It is often the case that a non-Abelian global symmetry is realized in the coincident wall configuration. It has been found previously that the splitting of domain walls can break the global symmetry, and the moduli fields corresponding to the wall positions become massless Nambu–Goldstone (NG) bosons associated to the symmetry breaking (Refs. [

In our previous works, we have observed the geometric Higgs mechanism (Refs. [

Furthermore, we calculate the 4-dimensional low-energy effective Lagrangian in an arbitrary domain wall background in the so-called moduli approximation (Ref. [

Last, many similarities between domain walls and D-branes have been shown in the literature. For example, in Refs. [

The paper is organized as follows. In

Let us consider a (4+1)-dimensional

The first part

The potential (

The field-dependent gauge kinetic term

The mass dimensions of the fields and parameters.

Fields and parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Mass dimension |

The equations of motion for the above model are

The potential

The 2 vacua

In order to find the mass spectrum of each vacuum, let us first replace

Now, let us send

In short, we insist that there are no light scalar fields in any vacua. They are heavy since their masses are of the 5-dimensional mass scale

Let us look for static

We solve these with the boundary conditions

Note that these equations correspond to a non-Abelian extension of the well-known 2-scalar MSTB model (named after Montonen, Sarker, Trullinger, and Bishop), solutions of which have been studied in detail (Refs. [

In the MSTB model, 2 types of domain-wall-like solutions are known. The first type is

The second type, which is supported in the parameter region

The width of the wall is of order ^{1}

The bound is saturated when the energy equals the tension of the domain walls,

One can easily show that

Let us figure out the physical meaning of the parameters contained in the

The

Now, it is manifest that the eigenvalues

Let us next consider a small fluctuation of

To be concrete, let us consider

Amplitudes of the diagonal component of

Moreover, an infinitesimal global

Thus the 12 zero modes in

Let us verify the mass spectra and wave functions of each mode by considering small fluctuations around a background configuration. We again take the

Let us expand the fluctuation fields as

The wave functions of zero modes can be obtained explicitly as

The Schrödinger potentials

Defining the inner product for 2-component vectors of a function of

To understand where the effective fields are localized, let us define the profiles of kinetic terms for zero modes as

The profiles of the kinetic terms

As long as the

Finally, let us make comments on massive modes. In general, it is not easy to determine the massive modes because the linearized equations of motion (

First, we see that all the off-diagonal components vanish (see the dotted lines in the left panels of

In order to get a better insight, let us further simplify the global

Left: the Schrödinger potential

Thus we find that the off-diagonal components have a zero mode and light massive modes of order

Now, we come to main part of this work. Our aim here is to determine the physical spectrum around the background domain walls (

We continue to consider the

Let us derive linearized equations of motion for small fluctuations around the

There is a residual gauge transformation that depends only on the

On the other hand, the

To investigate the spectrum, we need to write down the linearized equations of motion for each component of Eqs. (

First, we find that the fluctuations

Let us next investigate the spectrum for the unbroken parts of the gauge fields given in Eq. (

We can decompose the gauge field into divergence-free and divergence components as

The

To find the spectrum, let us expand the divergence-free component as

This Schrödinger-type problem can be cast in the form

There are two benefits of this expression. First, the Hamiltonian is manifestly positive definite, so that we can be sure that no tachyonic modes exist in the spectrum. Second, the zero mode can be easily found by solving

The profiles of the kinetic terms for

In general, the Schrödinger equation with the Hamiltonian

The effective gauge coupling constants for the effective

Integrating this over

These are the dimensionless gauge coupling constants in

These relations are identical to the standard

Let us next investigate the off-diagonal parts in Eqs. (

Let us separate the zero mode and define fields

Note that the inner product should be taken by means of Eq. (

Equation (

Now, we are left with Eqs. (

Similarly, the same infinitesimal gauge transformation of the gauge field given in Eq. (

It is easy to see that gauge transformed

Note also that Eq. (

Let us decompose Eqs. (

The benefit of this decomposition is that the divergence-free part

Note that the Schrödinger equation can be written as

Note that the second term of the Hamiltonian

The zero mode (

Let us obtain the first massive mode. The Schrödinger potential

The Schrödinger potential

The mass

Note that peculiar phenomena in the geometric Higgs mechanism in our specific model appear in the opposite limit

Therefore, for large separations

Finally, we have to solve the coupled equations (

There is redundancy such that any function of

This formally determines the mass spectrum for

Instead of trying to solve Eq. (

If the compactification radius

It is worthwhile pointing out that the number of coincident walls corresponds to the rank of the gauge group preserved by the domain wall configurations. When

In this section, we derive a low-energy effective Lagrangian in the so-called moduli approximation (Ref. [

The goal of this section is to describe effective 4-dimensional dynamics of

To illustrate our approach, let us first present the effective Lagrangian in a fixed background with a

To pick up the

Here, the parameters

Notice that off-diagonal components in the second part of the decomposition (

Next, let us consider the gauge fields. We have established that the wave function of massless gauge fields is flat, hence we can just replace

The effective Lagrangian is obtained by inserting all the above decompositions into the full Lagrangian and integrating it over the

The contributions without 4-dimensional derivatives sum to the first term

This corresponds to those in Eq. (

The factor standing in front of the kinetic terms of moduli fields

The effective Lagrangian (

Let us adopt the same ansatz for scalar fields as in Eqs. (

Since we are working in an arbitrary background, there is no a priori distinction between unbroken and broken generators. Hence, the formula (

The effective Lagrangian is obtained by plugging the ansatz into the 5-dimensional Lagrangian (

This is the main result of this section. The effective Lagrangian (

In order to compare this with the results in

Note that this precisely coincides with

Note that, as we saw in Eq. (

The third term of the effective Lagrangian (

The functions

Although we consider terms up to 2 derivatives only, it is believed that effective dynamics of moduli fields of domain walls can be also captured by the Nambu–Goto-type action or, more generally, as a function of the Nambu–Goto action (Refs. [

In this paper we have presented a (4+1)-dimensional model that gives a framework of dynamical realization of the brane-world model by domain walls, incorporating two core ideas: the semiclassical localization mechanism for gauge fields and the geometric Higgs mechanism using a multi-domain wall background. Since the domain walls interpolate multiple vacua that preserve different subgroups of

Although we have not dealt with the moduli stabilization to find the symmetry breaking of grand unification theories (GUT) in this paper, natural and important application of the geometric Higgs mechanism is, doubtless, to realize the symmetry breaking of GUT dynamically on the domain walls. We will investigate it separately in the subsequent work Ref. [

As to the other side of our result, we pointed out a deep similarity between our domain walls and the D-branes in superstring theories. This similarity goes beyond the often-cited connections between field-theoretical solitons and D-branes. The similarities are threefold. First, the number

In this paper, we have not taken SUSY as our guiding principle. However, it may be advantageous to consider 5-dimensional SUSY with 8 supercharges as a master theory. The immediate benefit of implementing SUSY is that a gauge kinetic term as a function of scalar field occurs naturally via the prepotential (Ref. [

Another feature common to most non-Abelian gauge theories is magnetic monopoles, which originate from the breaking of the semisimple gauge group to a subgroup with a

A schematic depiction of the magnetic monopole from the 5-dimensional point of view as a string stretched between separated domain walls.

Let us also remark that this type of configuration has a direct analog in D-strings (Ref. [

Last, in this work we have not discussed gravity for simplicity. However, an interesting direction for future study may be to consider Randall–Sundrum-like theory (Refs. [

F.B. thanks M. Nitta for useful discussions and comments. F.B. was an international research fellow of the Japan Society for the Promotion of Science, and was supported by Grant-in-Aid for JSPS Fellows, Grant No. 26004750. 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, by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) Grant Nos. 25400280 (to M.A.), 26800119 and 16H03984 (to M.E.), and 25400241 (to N.S.), and by the Albert Einstein Centre for Gravitation and Astrophysics financed by the Czech Science Agency Grant No. 14-37086G (to F.B.).

Open Access funding: SCOAP

Let us consider the Abelian-Higgs model in 5 dimensions:

We compactify the fifth direction to

We also have

Finally, we have

Putting everything together, we find the quadratic Lagrangian

One immediately sees that

In conclusion, we have

Now we can read the mass spectrum as

The calculation of the effective Lagrangians in

Let us first consider a generic integral appearing in the kinetic terms of scalar fields, namely,

This leads to

Now we can formally shift the integration variable as

Further, we will use the fact that

This leads to the final result,

Let us also mention an identity relevant to our calculation of the moduli metric. If we decompose an adjoint field

^{1} This is the main reason for choosing the quadratic function in Eq. (