TIT/HEP–467

UT-939

hep-th/0107204

July, 2001

Simple SUSY Breaking Mechanism by Coexisting Walls

Nobuhito Maru
^{*}^{*}*e-mail address:
,
Norisuke Sakai
^{†}^{†}†e-mail address: ,
Yutaka Sakamura
^{‡}^{‡}‡e-mail address:
and Ryo Sugisaka
^{§}^{§}§e-mail address:

Department of Physics, University of Tokyo 113-0033, JAPAN

and

Department of Physics, Tokyo Institute of Technology

Tokyo 152-8551, JAPAN

Abstract

[5mm] A SUSY breaking mechanism with no messenger fields is proposed. We assume that our world is on a domain wall and SUSY is broken only by the coexistence of another wall with some distance from our wall. We find an model in four dimensions which admits an exact solution of a stable non-BPS configuration of two walls and studied its properties explicitly. We work out how various soft SUSY breaking terms can arise in our framework. Phenomenological implications are briefly discussed. We also find that effective SUSY breaking scale becomes exponentially small as the distance between two walls grows.

## 1 Introduction

Supersymmetry (SUSY) is one of the most promising ideas to solve the hierarchy problem in unified theories [1]. It has been noted for some years that one of the most important issues for SUSY unified theories is to understand the SUSY breaking in our observable world. Many models of SUSY breaking uses some kind of mediation of the SUSY breaking from the hidden sector to our observable sector. Supergravity provides a tree level SUSY breaking effects in our observable sector suppressed by the Planck mass [2]. Gauge mediation models uses messenger fields to communicate the SUSY breaking at the loop level in our observable sector [3].

Recently there has been an active interest in the “Brane World” scenario where our four-dimensional spacetime is realized on the wall in higher dimensional spacetime [4, 5]. In order to discuss the stability of such a wall, it is often useful to consider SUSY theories as the fundamental theory. Moreover, SUSY theories in higher dimensions are a natural possibility in string theories. These SUSY theories in higher dimensions have or more supercharges, which should be broken partially if we want to have a phenomenologically viable SUSY unified model in four dimensions. Such a partial breaking of SUSY is nicely obtained by the topological defects [6]. Domain walls or other topological defects preserving part of the original SUSY in the fundamental theory are called the BPS states in SUSY theories. Walls have co-dimension one and typically preserve half of the original SUSY, which are called BPS states [7, 8, 9]. Junctions of walls have co-dimension two and typically preserve a quarter of the original SUSY [10, 11].

Because of the new possibility offered by the brane world scenario, there has been a renewed interest in studies of SUSY breaking. It has been pointed out that the non-BPS topological defects can be a source of SUSY breaking [8] and an explicit realization was considered in the context of families localized in different BPS walls [12]. Models have also been proposed with bulk fields mediating the SUSY breaking from the hidden wall to our wall on which standard model fields are localized [13, 14, 15, 16]. The localization of the various matter wave functions in the extra dimensions was proposed to offer a natural realization of the gaugino-mediation of the SUSY breaking [17]. Recently we have proposed a simple mechanism of SUSY breaking due to the coexistence of different kinds of BPS domain walls and proposed an efficient method to evaluate the SUSY breaking parameters such as the boson-fermion mass-splitting by means of overlap of wave functions involving the Nambu-Goldstone (NG) fermion [18], thanks to the low-energy theorem [19, 20]. We have exemplified these points by taking a toy model in four dimensions, which allows an exact solution of coexisting walls with a three-dimensional effective theory. Although the model is only meta-stable, we were able to show approximate evaluation of the overlap allows us to determine the mass-splitting reliably.

The purpose of this paper is to illustrate our idea of SUSY breaking due to the coexistence of BPS walls by taking a simple soluble model with a stable non-BPS configuration of two walls and to extend our analysis to more realistic case of four-dimensional effective theories. We also examine the consequences of our mechanism in detail.

We propose a SUSY breaking mechanism which requires no messenger fields, nor complicated SUSY breaking sector on any of the walls. We assume that our world is on a wall and SUSY is broken only by the coexistence of another wall with some distance from our wall. We find an supersymmetric model in four dimensions which admits an exact solution of a stable non-BPS configuration of two walls and study its properties explicitly. We work out how various soft SUSY breaking terms can arise in our framework. Phenomenological implications are briefly discussed. We also find that effective SUSY breaking scale observed on our wall becomes exponentially small as the distance between two walls grows. The NG fermion is localized on the distant wall and its overlap with the wave functions of physical fields on our wall gives the boson-fermion mass-splitting of physical fields on our wall thanks to a low-energy theorem. We propose that this overlap provides a practical method to evaluate the mass-splitting in models with SUSY breaking due to the coexisting walls.

In the next section, a model is introduced that allows a stable non-BPS two-wall configuration as a classical solution. We have also worked out mode expansion on the two-wall background, three-dimensional effective Lagrangian, and the single-wall approximation for the overlap of mode functions to obtain the mass-splitting. Matter fields are also introduced. Section 3 is devoted to study how various soft breaking terms arise in the three-dimensional effective theory. Soft breaking terms in four-dimensional effective theory are worked out in section 4. Phenomenological implications are discussed in section 5. Additional discussion is given in section 6. Appendix A is devoted to discussing the low-energy theorem in three dimensions and the mixing matrix relating the mass eigenstates and superpartner states. Low-energy theorems in four dimensions are derived in Appendix B. In Appendix C, we derive a relation among the order parameters of the SUSY breaking, the energy density of the configuration and the central charge of the SUSY algebra.

## 2 SUSY breaking by the coexistence of walls

### 2.1 Stable non-BPS configuration of two walls

We will describe a simple soluble model for a stable non-BPS
configuration that represents two-domain-wall system,
in order to illustrate our basic ideas.
Here we consider domain walls in four-dimensional spacetime
to avoid inessential complications.
We introduce a simple four-dimensional Wess-Zumino model
as follows.^{1}^{1}1
We follow the conventions in Ref.[21]

(2.1) |

where is a chiral superfield
,
.

A scale parameter has a mass-dimension one and a
coupling constant
is dimensionless, and both of them are real positive.
In the following, we choose as the extra dimension
and
compactify it on of radius .
Other coordinates are denoted as (), i.e.,
.
The bosonic part of the model is

(2.2) |

The target space of the scalar field has a topology of a cylinder as shown in Fig.1. This model has two vacua at , both lie on the real axis.

Let us first consider the case of the limit . In this case, there are two kinds of BPS domain walls in this model. One of them is

(2.3) |

which interpolates the vacuum at to that at as increases from to . The other wall is

(2.4) |

which interpolates the vacuum at to that at . Here and are integration constants and represent the location of the walls along the extra dimension. The four-dimensional supercharge can be decomposed into two two-component Majorana supercharges and which can be regarded as supercharges in three dimensions

(2.5) |

Each wall breaks a half of the bulk supersymmetry: is broken by , and by . Thus all of the bulk supersymmetry will be broken if these walls coexist.

We will consider such a two-wall system to study the SUSY breaking effects in the low-energy three-dimensional theory on the background. The field configuration of the two walls will wrap around the cylinder in the target space of as increases from to . Such a configuration should be a solution of the equation of motion,

(2.6) |

We can easily show that the minimum energy static configuration with unit winding number should be real. We find that a general real static solution of Eq.(2.6) that depends only on is

(2.7) |

where and are real parameters and the function denotes the amplitude function, which is defined as an inverse function of

(2.8) |

If , it becomes a periodic function with the period , where the function is the complete elliptic integral of the first kind. If , the solution is a monotonically increasing function with

(2.9) |

This is the solution that we want. Since the field is an angular variable , we can choose the compactified radius so that the classical field configuration contains two walls and becomes periodic modulo . We shall take to locate one of the walls at . Then we find that the other wall is located at the anti-podal point of the compactified circle. We have computed the energy of a superposition of the first wall located at in Eq.(2.3) and the second wall located at in Eq.(2.4). This energy can be regarded as a potential between two walls in the adiabatic approximation and has a peak at implying that two walls experience a repulsion. This is in contrast to a BPS configuration of two walls which should exert no force between them. Thus we can explain that the second wall is settled at the anti-podal point in our stable non-BPS configuration because of the repulsive force between two walls.

In the limit of , i.e., , approaches to the BPS configuration with near , which preserves , and to with near , which preserves . The profile of the classical solution is shown in Fig.2. We will refer to the wall at as “our wall” and the wall at as “the other wall”.

### 2.2 The fluctuation mode expansion

Let us consider the fluctuation fields around the background ,

(2.10) |

To expand them in modes, we define the mode functions as solutions of equations:

(2.11) |

(2.12) |

The four-dimensional fluctuation fields can be expanded as

(2.13) |

(2.14) |

As a consequence of the linearized equation of motion, the coefficient and are scalar fields in three-dimensional effective theory with masses and , and and are three-dimensional spinor fields with masses , respectively.

Exact mode functions and mass-eigenvalues are known for several light modes of ,

(2.15) |

where functions , , are the Jacobi’s elliptic functions and are normalization factors. For , we can find all the eigenmodes

(2.16) |

The massless field is the Nambu-Goldstone (NG) boson for the breaking of the translational invariance in the extra dimension. The first massive field corresponds to the oscillation of the background wall around the anti-podal equilibrium point and hence becomes massless in the limit of . All the other bosonic fields remain massive in that limit.

For fermions, only zero modes are known explicitly,

(2.17) |

where is a normalization factor. These fermionic zero modes are the NG fermions for the breaking of -SUSY and -SUSY, respectively.

Thus there are four fields which are massless or become massless in the limit of : , , and . The profiles of their mode functions are shown in Fig.3 and Fig.4. Other fields are heavier and have masses of the order of .

In the following discussion, we will concentrate ourselves on the breaking of the -SUSY, which is approximately preserved by our wall at . So we call the field the NG fermion in the rest of the paper.

### 2.3 Three-dimensional effective Lagrangian

We can obtain a three-dimensional effective Lagrangian by substituting the mode-expanded fields Eq.(2.13) and Eq.(2.14) into the Lagrangian (2.1), and carrying out an integration over

(2.18) | |||||

where and an abbreviation denotes terms involving heavier fields and higher-dimensional terms. Here -matrices in three dimensions are defined by . The vacuum energy is given by the energy density of the background and thus

(2.19) |

and the effective Yukawa coupling is

(2.20) |

In the limit of , the parameters and vanish and thus we can redefine the bosonic massless fields as

(2.21) |

In this case, the fields and
form a supermultiplet
for -SUSY and their mode functions are both
localized on our wall.
The fields and are singlets
for -SUSY and are localized on the other
wall.^{2}^{2}2
The modes and form a
supermultiplet for
-SUSY.

When the distance between the walls is finite, -SUSY is broken and the mass-splittings between bosonic and fermionic modes are induced. The mass squared in Eq.(2.18) corresponds to the difference of the mass squared between and since the fermionic mode is massless. Besides the mass terms, we can read off the SUSY breaking effects from the Yukawa couplings like .

We have noticed in Ref.[18] that these two SUSY breaking parameters, and , are related by the low-energy theorem associated with the spontaneous breaking of SUSY. In our case, the low-energy theorem becomes

(2.22) |

where is an order parameter of the SUSY breaking, and it is given by the square root of the vacuum (classical background) energy density in Eq.(2.19). The low-energy theorem in three dimensions is briefly explained in Appendix A.1. Since the superpartner of the fermionic field is a mixture of mass-eigenstates, we had to take into account the mixing Eq.(2.21). The mixing in general situation is discussed and is applied to the present case in Appendix A.2 and A.3.

Fig.5 shows the mass-splitting as a function of the wall distance . As this figure shows, the mass-splitting decays exponentially as the wall distance increases. This is one of the characteristic features of our SUSY breaking mechanism. This fact can be easily understood by remembering the profile of each modes. Note that the mass-splitting is proportional to the effective Yukawa coupling constant , which is represented by an overlap integral of the mode functions. Here the mode functions of the fermionic field and its superpartner are both localized on our wall, and that of the NG fermion is localized on the other wall. Therefore the mass-splitting becomes exponentially small when the distance between the walls increases, because of exponentially dumping tails of the mode functions.

### 2.4 Single-wall approximation

Next we will propose a practical method of estimation for the mass-splittings. We often encounter the case where single-BPS-domain-wall solutions are known but exact two-wall configurations are not. This is because the latter are solutions of a second order differential equation, namely the equation of motion, while the former are solutions of first order differential equations, namely BPS equations. We can estimate the mass-splitting by using only informations on the single-wall background, even if two-wall configurations are not known. As mentioned in the previous subsection, the mass-splitting is related to the coupling constant and the order parameter . So we can estimate by calculating and .

When two walls are far apart, the energy of the background in Eq.(2.19) can be well-approximated by the sum of those of our wall and of the other wall.

(2.23) |

Considering the profiles of background and mode functions, we can see that the main contributions to the overlap integral of come from neighborhood of our wall and the other wall. These two regions give the same numerical contributions to the integral, including their signs. Thus we can obtain by calculating the overlap integral of approximate background and mode functions which well approximate their behaviors near our wall, and multiplying it by two.

In the neighborhood of our wall, the two-wall background can be well approximated by the single-wall background with . So,

(2.24) |

Next, we will proceed to the approximation of mode functions. From the mode equations in Eq.(2.12), we can express the zero-modes and as

(2.25) | |||||

(2.26) |

where and are normalization factors.

Since the function has its support mainly on our wall, it is simply approximated near our wall by

(2.27) |

Then we can determine by the normalization condition.

Similarly, the mode can be approximated near our wall by

(2.28) |

Unlike the case of , however, we cannot determine by using this approximate expression because the mode is localized mainly on the other wall. Here it should be noted that from Eq.(2.12) and the property of the background: . Thus,

(2.29) | |||||

and we can obtain a relation:

(2.30) |

In the region of , the background is well approximated by

(2.31) |

with and , and thus

(2.32) | |||||

Thus the normalization factor can be estimated as

(2.33) |

Here we used the fact that and . As a result, the mode function of the NG fermion can be approximated near our wall by

(2.34) |

In the limit of , the -SUSY is recovered and thus the mode function of the bosonic field in Eq.(2.21), , is identical to . However, when the other wall exist at finite distance from our wall, this bosonic field is mixed with the field localized on the other wall. Because the masses of these two fields and are degenerate (both are massless), the maximal mixing occurs. (See Eq.(2.21).)

(2.35) |

where is the mode function of . Thus the mode function of the mass-eigenmode is approximated near our wall by

(2.36) |

Then by using Eqs.(2.24), (2.27), (2.34) and (2.36), we can obtain the effective Yukawa coupling constant ,

(2.37) |

As a result, the approximate mass-splitting value is estimated as

(2.38) |

by using Eq.(2.23) and the low-energy theorem Eq.(2.22). From this expression, we can explicitly see its exponential dependence of the distance between the walls. We call this method of estimation the single-wall approximation.

In our model, we know the exact mass-eigenvalue . So we can check the validity of the above approximation by comparing the approximate value and the exact one . Fig.6 shows the ratio of to as a function of the wall distance . As this figure shows, we can conclude that the single-wall approximation is very well.

### 2.5 Matter fields

Let us introduce a matter chiral superfield

(2.39) |

interacting with in the original Lagrangian (2.1) through an additional superpotential

(2.40) |

which will be treated as a small perturbation^{3}^{3}3
We can take the interaction like
as in Ref.[18] in order to localize the mode function
of the
light matter fields on our wall.
The choice of like Eq.(2.40) is completely
a matter of convenience.
.

Let us decompose the matter fermion into two real two-component spinors and as . Then these fluctuation fields can be expanded by the mode functions as follows.

(2.41) |

The mode equations are defined as

(2.42) |

Thus zero-modes on the two-wall background (2.7) can be solved exactly

(2.43) |

the mode is localized on our wall and the mode is on the other wall.

Besides these zero-modes, there are several light modes of localized on our wall when the coupling is taken to be larger than . Those non-zero-modes can be obtained analytically in the limit of . For example, the low-lying mass-eigenvalues are discrete at with , and the corresponding mode functions for the fields are

(2.44) |

where is the hypergeometric function and is normalization factors. The mode functions for the fields have forms similar to those of .

Although we do not know the exact mass-eigenvalues and mode functions in the case that the wall distance is finite, we can estimate the boson-fermion mass-splittings by using the single-wall approximation discussed in the previous subsection. For example, let us estimate the mass-splitting between and its superpartner . After including an interaction like Eq.(2.40), the effective Lagrangian has the following Yukawa coupling terms.

(2.45) |

(2.46) |

Just like the case of and , the degenerate states and are maximally mixed with each other and their masseigenvalues split into two different values and . By calculating the effective coupling in Eq.(2.46) in the single-wall approximation, we can obtain the following mass-splitting. (See Appendix A.3.4.)

(2.47) |

Thanks to the approximate supersymmetry, -SUSY, we can use the mode function in Eq.(2.44) as both and . Then we obtain the mass-splitting in the single wall approximation

(2.48) |

This result is independent of the level number . However, it is not a general feature of our SUSY breaking mechanism. It depends on the choice of the interaction . If we choose as an example, we will obtain a different result that becomes larger as increases, just like the result in Ref.[18].

## 3 Soft SUSY breaking terms in 3D effective theory

In this section, we discuss how various soft SUSY breaking terms in the three-dimensional effective theory are induced in our framework.

Firstly, we discuss a multi-linear scalar coupling, a generalization of the so-called A-term. Such a “generalized A-term” is generated from the following superpotential term in the bulk theory

(3.1) | |||||

(3.3) | |||||

where is the fundamental mass scale of the four-dimensional bulk theory, is a dimensionless holomorphic function of , and are chiral matter superfields,

(3.4) |

The equation of motion for is given by

(3.5) |

Note that the the superpotential term Eq.(3.1) is a generalization of Eq.(2.40). In Eq.(3.3), we used the following Kaluza-Klein (KK) mode expansions,

(3.6) |