###### Abstract

We propose a new mechanism for trapping bulk gauge field, giving rise to a massless photon on a flat Minkowski 3-brane in the Randall-Sundrum model in five space-time dimensions. The mechanism we propose employs the topological Higgs mechansim where a topological term and a 3-form gauge potential play an important role. This new mechanism might be considered as a gauge field’s analog of the localization of bulk fermions with the mass term of a ’kink’ profile.

EDO-EP-38

March, 2001

A New Mechanism for Trapping of Photon

Ichiro Oda
^{1}^{1}1
E-mail address:

Edogawa University, 474 Komaki, Nagareyama City, Chiba 270-0198, JAPAN

The gravity-localized models in a brane world offer an opportunity for a new solution to the hierarcy problem, an alternative compactification scenario and the cosmological constant problem and so on [1, 2]. The crucial ingredient in the models is the localization mechanism for all the familiar matter and gauge fields in addition to the graviton on a brane. In superstring theory, matter and gauge fields are naturally confined to D3 branes due to open strings ending on the branes while the gravity is free to propagate in a bulk space-time due to closed strings living in the bulk [3]. On the other hand, in local field theory, it is known that in contrast with the other fields, the gauge fields cannot be localized on a brane by a gravitational interaction [4, 5, 6, 7, 8]. (For a review see [9].) The non-localization of the bulk gauge fields on a brane is one of fatal drawbacks associated with the gravity-localized models since there certainly exists a massless ’’ in our world.

The aim of the present paper is to propose a new localization mechanism of the gauge fields on a 3-brane from the viewpoint of local field theory in the framework of the Randall-Sundrum model [2]. The possibility of constructing such a new mechanism has stemmed from an attempt of making a gauge field’s analog of the localization mechanism of the bulk fermions. So let us explain briefly our background idea behind the new mechanism in connection with the fermion localization.

The problem of localizing fermion zero-modes on a domain wall has been already solved by Jackiw and Rebbi [10] where the Yukawa interaction interpolating two different vacua at each side of a domain wall yields the mass term with a ’kink’ profile with the definitions of being a transverse dimension to the domain wall and being the step function given by and . Because of this type of mass term, the bulk fermions can be localized on the domain wall by a gravitational interaction. Then it is natural to ask ourselves whether we can apply this localization mechanism of fermions to gauge fields or not. However, it turns out that the introduction of a mass term in a simple manner does not lead to any localization of the bulk gauge fields [5].

To pursue an analogy to some extent, let us write down the actions for both spin-1/2 spinor and spin-1 vector fields. After taking a vacuum expectation value of a ’kink’ profile, the action for the spin-1/2 spinor field reads

(1) |

where the covariant derivative is defined as with being the spin connections. On the other hand, after spontaneous symmetry breakdown, the relevant action for the spin-1 vector field is given by

(2) |

where . As mentioned above, in the Randall-Sundrum model [1, 2] the zero-mode of the spinor field is localized on a brane, whereas the one of the gauge field is not so. Then one might wonder what differences there are between the two actions. One obvious difference lies in the form of mass term where the mass term takes a linear form depending on the step function in (1) while it is a quadratic form in (2). The other important difference is that the action (1) is invariant under gauge transformation while the action (2) is implicitly broken owing to spontaneous symmetry breakdown.

At this stage, it is of interest to ask ourselves if there is an action for the gauge field which is not only manifestly invariant under the gauge transformation but also has a linear mass term. To the best of our knowledge, there is only one action, which is nothing but the pure gauge theory with a topological term. Recall that in this theory the mass of the gauge field is generated by the so-called topological Higgs mechanism owing to the existence of a topological term instead of the conventional Higgs mechanism [11, 12, 13]. Therefore, in the present paper, we shall consider this theory and study the localization of the zero-mode of the bulk gauge field on a brane in the framework of the Randall-Sundrum model. Surprisingly enough, we will see that the theory provides us an ingenious mechanism for the localization of the gauge field on a flat 3-brane.

We shall start by fixing the model setup. The metric ansatz we adopt is of the Randall-Sundrum form [2]:

(3) | |||||

where denote five-dimensional space-time indices and , …four-dimensional brane ones. The metric on the brane denotes the four-dimensional flat Minkowski metric with signature . Moreover, where is a positive constant and the fifth dimension runs from to . We have the physical situation in mind where a single flat 3-brane sits at the origin of the fifth dimension, , and then ask if the bulk gauge field can be localized on the brane by a gravitational interaction. If we find a normalizable zero-mode of a bulk field with exponential falloff, we regard the zero-mode as a local field on our world corresponding to the bulk field. In this article, we will assume that the background metric is not modified by the presence of the bulk fields, that is, we will neglect the back-reaction on the metric from the bulk fields.

Now we are ready to consider the Maxwell’s action for massless gauge field plus the actions for a 3-form potential and a topological term whose total action is explicitly given by

(4) | |||||

where , and is a positive constant. This action has the following gauge symmtries as well as reducible symmetries [14]:

(5) |

Note that the gauge symmetry with respect to prohibits the mass term of a ’kink’ profile in the action.

Then, the equations of motion read

(6) |

As the gauge conditions of the symmetries (5), we shall take

(7) |

In particuar, note that the number of the latter gauge conditions precisely coincides with that of symmtries, which is . With these gauge conditions, we wish to look for a zero-mode solution with the forms of

(8) |

where we assume the equations of motion in four-dimensional flat space-time:

(9) |

with the definitions of and .

With these ansatzs, the equations of motion (6) reduce to the following differential equations:

(10) |

(11) |

(12) |

(13) |

Since Eqs. (11), (13) are the first-order differential equations with respect to -differentiation, we regard them as the gauge conditions in four-dimensional space-time. (Later, we will comment on these equations.) Then, the equations which we have to solve are Eqs. (10), (12). From the two equations, we can derive a differential equation to :

(14) |

Given our physical setup that a single brane sits at where there is a -functional source, a general solution to (14) is given by

(15) |

where is an integration constant. In this respect, it is worthwhile to notice that we have effectively selected the mass term with a ’kink’ profile as a solution to the equations of motion. When we impose the boundary conditions such that , a special solution takes the form

(16) |

We are now willing to check that this zero-mode solution of the gauge field as well as the 3-form potential leads to a normalizable mode and the localization on a brane. For this, let us plug Eqs. (7), (8) into the starting action (4), and then a simple calculation yields

(17) | |||||

where the topological term has disappeared from the above action by integration over . Whether the zero-mode (16) is normalizable or not can be checked by evaluating each -integral in (17). In particular, the first -integral in front of the kinetic term of the gauge field reads

(18) |

where is utilized. The finiteness of this integral means that the zero-mode (16) is certainly a normalizable mode at least for the gauge field. The second -integral is similarly evaluated as

(19) |

where and are used. The third -integral becomes

(20) |

Note that this integral takes the above finite value if , whereas it diverges if . The final integral is given by

(21) |

This integral also takes the finite value if , but it becomes divergent if .

First of all, let us consider the case that all the -integrals have the finite values, for which the inequality must be satisfied. Then in order to transform the kinetic terms to a canonical form, let us redefine the fields as

(22) |

As a result, the action (17) reduces to

(23) |

Moreover, the normalized zero mode of the gauge field in a flat space-time is of the form

(24) |

Here we encounter a problem. Recall that there is now the condition where both and are positive constants. Together with it, the massless condition of ’’ gives us the condition . Then the normalized zero mode of the gauge field, in (24), spreads more widely in a bulk, in other words, the brane gauge field is not sharply localized near a brane. This situation is very similar to the case of the locally localized model [15, 16, 17] where ’small extra dimensions’ scenario was proposed in order to avoid this problem.

Is there any possibility to circumvent this ’small extra dimensions’ scenario? Interestingly enough, in the case at hand, there an alternative possibility of trapping the bulk gauge field on a brane without invoking the ’small extra dimensions’ scenario. To do so, let us notice that the root of the problem exists in the inequality which has stemmed from the condition that the zero-mode of a 3-form potential should be normalizable. We can now relax this condition if we consider the situation where the 3-form potential does not have a normalizable zero-mode, thereby implying that the 3-form potential is not localized near a brane but resides in a whole bulk. In this situation, we have a condition and the brane action is effectively given by

(25) |

Then the massless condition of the gauge field requires the relation . These conditions are simply satisfied by taking . Hence, in this sense, we have succeeded in getting a massless ’’ which is sharply localized on a flat 3-brane. Note that under the condition , the 3-form potential resides in a bulk away from a brane since the zero-mode in a flat space has the behavior of .

Now some remarks are in order. The first remark is related to the gauge conditions (11) and (13) in four dimensions. Owing to the solution (16), these gauge conditions must take the different forms at each side of a 3-brane. Incidentally, we can rewrite them as

(26) |

Thus, on the brane, they reduce to the usual gauge conditions .

The second remark refers to the consistency of the equations of motion in four dimensions. In deriving the differential equations (10)-(13), we have assumed the four-dimensional equations (9). For self-consistency, we have to check that these equations are indeed valid in four dimensions. It is clear that the equations of motion, hold as far as as seen in (25). On the other hand, the equations of motion, need some attention since now reside in the region away from a brane. For this, we shall consider the action (17), from which we have the equations of motion for :

(27) |

Since live in the region away from a brane where and we have the condition , Eq. (27) means the desired equations, .

As a final remark, it is worthwhile to point out that the starting action (4) often appears in the context of superstring theory. For instance, it is well known that we need the term in the string-theory effective action for the Green-Schwarz anomaly cancellation [18]. This complicated term leads to a topological term upon compactification to five dimensions. Also, in type IIA superstring theory, there is the gauge field together with a 3-form potential from the Ramond-Ramond sector with a topological term [3]. Hence, the new localization mechanism for ’’ which we have proposed in this paper may have some applications within superstring theory.

In conclusion, we have proposed a new localization mechanism for the gauge fields in the Randall-Sundrum gravity-localized model. This new mechanism is very similar to that of fermions in the sense that in the both mechanisms the zero-modes share the same form and the presence of the mass term of a ’kink’ profile plays an essential role for trapping the zero-modes of the bulk fields on a flat Minkowski brane. So we have called the present localization mechanism a gauge field’s analog of the localization of fermions in the abstract. The model at hand naturally appears in superstring theory, so the present localization mechanism might also shed new light on superstring theory in addition to local field theory. In near future, we wish to understand various aspects of this new mechanism in connection with superstring theory.

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