Revolving D-branes and Spontaneous Gauge Symmetry Breaking

We propose a new mechanism of spontaneous gauge symmetry breaking in the world-volume theory of revolving D-branes around a fixed point of orbifolds. In this paper, we consider a simple model of the T6/Z3 orbifold on which we put D3-branes, D7-branes and their anti-branes. The configuration breaks supersymmetry, but the R-R tadpole cancellation conditions are satisfied. A set of three D3-branes at an orbifold fixed point can separate from the point, but when they move perpendicular to the anti-D7-branes put on the fixed point, they are forced to be pulled back due to an attractive interaction between the D3 and anti-D7 branes. In order to stabilize the separation of the D3-branes at nonzero distance, we consider revolution of the D3-branes around the fixed point. Then the gauge symmetry on D3-branes is spontaneously broken, and the rank of the gauge group is reduced. The distance can be set at our will by appropriately choosing the angular momentum of the revolving D3-branes, which should be determined by the initial condition of the cosmological evolution of D-brane configurations. The distance corresponds to the vacuum expectation values of brane moduli fields in the world-volume theory and, if it is written as M/Ms^2 in terms of the string scale Ms, the scale of gauge symmetry breaking is given by M. Angular momentum conservation of revolving D3-branes assures the stability of the scale M against Ms.


Introduction
The dynamics of electroweak symmetry breaking has become even more mysterious after the discovery of the Higgs boson. It is widely believed that the standard model is merely an effective theory of the electroweak symmetry breaking and some unknown physics or dynamics should exist behind it. Many possibilities, mainly motivated by naturalness, have been examined, for example, dynamical symmetry breaking by strong coupling gauge interactions [1,2], radiative symmetry breaking with [3,4] or without supersymmetric extensions [5,6,7,8,9,10], Hosotani mechanism in the gauge-Higgs unification with extra dimensions [11] and so on.
It is also an important issue to pursue the origin of the masses of quarks and leptons, namely, how the electroweak symmetry breaking is mediated to them. In the standard model it is simply described by Yukawa couplings, and much efforts have been made to derive realistic Yukawa couplings. Model buildings with D-branes in String Theory are most attractive, because Yukawa couplings can be understood by configurations of D-branes in the six-dimensional compact space of ten-dimensional superstring theories. Some examples are intersecting Dbrane models [12,13,14,15,16,17] or models of D-branes at singularities [18,19,20,21]. In these models, massless states of string with appropriate quantum numbers are identified with the Higgs doublet fields, but it will be generically difficult to obtain small vacuum expectation values of the weak scale in the string theory setup.
In this paper we propose a new mechanism of spontaneous gauge symmetry breaking by assuming that our 3 dimensional space consists of revolving D-branes around a fixed point in 9+1 dimensional space-time. As a simple model, we study D-branes at fixed points of a T 6 /Z 3 orbifold in type IIB superstring theory [18]. We assume that all the moduli, except for the D-brane moduli, are stabilized. The six-dimensional torus is assumed to be factorizable, T 6 = T 2 × T 2 × T 2 , and the action of Z 3 is described by the twist vector v = (1/3, 1/3, −2/3). There are three fixed points in each T 2 and 27 fixed points in total. We distribute D3-branes and D7-branes and their anti-branes so as to cancel the twisted R-R tadpoles at each fixed point as well as the untwisted R-R tadpoles in the compact space. As a result all supersymmetries in the type IIB superstring theory are broken through the "brane supersymmetry breaking" mechanism [22,23,24,25,26]. In particular, we consider a system of four D3-branes and three anti-D7-branes discussed in [27]. Three of four D3-branes can move away from the fixed point in a Z 3 invariant way, and the separation of these D3-branes causes spontaneous gauge symmetry breaking of U(2)×U(1)×U(1) → U(1)×U(1) on the world-volume theory on the D3-branes. The rank of the gauge group is reduced due to the identification of three D3-branes by the Z 3 action. The reduction of rank is actually necessary in the electroweak symmetry breaking.
When we put anti-D7-branes on the fixed point as well as the D3-branes, there appears an attractive force between the D3-branes and the anti-D7-branes, and when D3-branes move away from the fixed point, they are forced to be pulled back. Thus D3-branes tend to be localized at the fixed point, and the gauge symmetry breaking does not occur unless there exists an additional balancing force that repels D3-branes from the fixed point. In this paper, we consider revolution of the D3-branes whose centrifugal force balances against the attractive force.
In section 2 we give a brief review of the models with D3-branes and D7-branes at the orbifold fixed points of a T 6 /Z 3 orbifold. We explain how the vacuum expectation values of the Dbrane moduli fields describe the displacements of the D3-branes. In section 3 the world-volume theory of revolving D3-branes is introduced. The effect of revolution is an introduction of the centrifugal potential in the rest frame of the revolving D3-branes. In section 4 the spectrum in the world-volume theory is discussed. We find that the spontaneous gauge symmetry breaking takes place and the scale of the expectation value is stable against the string scale perturbations. In section 5, we conclude and discuss some important issues not studied in details in the present paper.

.1 D-brane configurations
General ideas and formulations of D-branes at orbifold fixed points are discussed in [18]. In this section we particularly consider the models with D3-branes, D7-branes and their anti-branes in a T 6 /Z 3 orbifold. The six-dimensional torus is assumed to be factorized into three two-tori, T 6 = T 2 × T 2 × T 2 and the complex coordinates of the corresponding two-tori are denoted by with i = 1, 2, 3. Here X M with M = 0, 1, · · · , 9 are the coordinates of ten-dimensional spacetime of type IIB superstring theory. The coordinates are identified by the translations where R i are the radii of the two-tori and τ = (−1 + i √ 3)/2. We also identify points on the torus transformed by the Z 3 action where v = (1/3, 1/3, −2/3) is the twist vector. There are three fixed points in each two-torus under the above identifications, and in total 27 fixed points in the T 6 /Z 3 orbifold. Since the string world-sheet fields should follow the same identifications as the coordinates X M , there is an untwisted closed string propagating in the ten-dimensional space-time, and twisted closed strings localized at each fixed point.
We introduce D3-branes whose world-volume coordinates coincide with X µ with µ = 0, 1, 2, 3. In six-dimensional compact space the positions of these D3-branes are specified by the corresponding points which are distributed in a Z 3 symmetric way in the torus. Therefore, when D3-branes are put away from the fixed points, three D3-branes must move together which belong to the following regular representation of Z 3 . The cyclic permutation of three D3-branes in the torus is described by the matrix which can be diagonalized by a unitary transformation as where α = exp(2πi/3). This shows that the regular representation is not an irreducible representation of Z 3 . A D3-brane in the irreducible representation of Z 3 with one of the above three eigenvalues of γ is called a fractional D3-brane. It is localized at the fixed point and cannot move away from it. Only a set of three D3-branes can move. The above matrices also act as the Z 3 operations on the Chan-Paton indices of the open strings whose ends are confined in the corresponding D3-brane world-volume.
The twisted R-R tadpole cancellation condition of the D-branes at a fixed point is given by using the matrices of the Z 3 operation acting on the various D-branes: 3 Tr(γ 3 ) − 3 Tr(γ3) + Tr(γ 7 ) − Tr(γ7) = 0.
Here, γ 3 , γ3, γ 7 and γ7 are the matrices for D3, anti-D3, D7 and anti-D7 branes localized at the fixed point. A simple solutions to the condition of eq. (6) with four D3-branes and three anti-D7-branes is given by where 1 N denotes an N × N unit matrix [27]. Among four D3-branes at the fixed point, a set of three can move away from the fixed point by constructing the regular representation of eq. (5). The remaining D3 is in the irreducible representation 1 1 with an eigenvalue 1 of the matrix γ.
The above subsystem with eq. (7) can be embedded globally in the T 6 /Z 3 orbifold as shown in Figure 1. Since the codimension of D7-branes is 2 in ten-dimensional space-time, we can specify it by indicating directions along which its world-volume is not extended. D7 i denotes a D7-brane whose world-volume does not include the direction of i-th torus described by Z i . The Z 3 action matrices for other D-branes are given by All of the above D-branes are fractional D-branes. We also introduce Wilson line on the anti-D7-branes which is described by the operation matrix corresponding to the torus shifts of eq. (2). The twisted R-R tadpole cancellation conditions for the fixed points, which are fixed under Z 3 transformations up to the shift of eq. (2), are modified by replacing γ7 by γ W γ7 or γ 2 W γ7 in eq. (6) [28]. The untwisted R-R tadpoles are cancelled out, because the numbers of D3-branes and anti-D3-branes are the same, and the numbers of D7-branes and anti-D7-branes are the same. If moduli stabilization requires some other objects which are sources of untwisted R-R charges [29,30,31,32,33], this D-brane configurations are changed accordingly. In the following analysis, we simply assume that the model is constructed in a globally consistent way and focus on the subsystem of eq. (7) near the fixed point where four D3 and three D7 3 branes are localized. Since the supersymmetry is broken in the present model, the NS-NS tadpoles are not cancelled and the flat space-time with trivial dilaton and B-field background is no longer a solution to the equation of motion of string theory [34,35]. The effects of the NS-NS tadpoles on the background geometry are neglected for the moment and will be mentioned briefly in section 5.

DBI action
A bosonic part of the world-volume low-energy effective theory of a Dp-brane is given by the generalized DBI action where τ p = 1/(2π) p g s (α ′ ) (p+1)/2 is the tension of the Dp-brane while string coupling g s = e Φ is determined by the vacuum expectation value of the ten-dimensional dilaton field Φ = Φ + φ. The induced metric on the Dp-brane is defined as where X M (ξ) describe the embedding of the Dp-brane in ten-dimensional space-time. Here, we assume that the ten-dimensional space-time is flat Minkowski. Generalizations of the DBI action to n Dp-branes have been discussed in various literatures.
Let us now focus on the p = 3 case. The low-energy effective action of n D3-branes is described by four-dimensional N = 4 U(n) super-Yang-Mills theory whose bosonic part is obtained by expanding the non-Abelian generalization of the DBI action in eq. (10). The vector multiplet of four-dimensional N = 4 U(n) super-Yang-Mills theory consists of U(n) gauge bosons, four Weyl fermions and six real scalar fields. The six scalar fields are originated from the embedding fields X M (ξ) with M = 4, 5, · · · , 9 and can be interpreted as the brane moduli fields whose vacuum expectation values describe the positions of D-branes in six-dimensional compact space. We define the complex combinations Z i (i = 1, 2, 3) of these moduli fields as in eq. (1). In the absence of other sources of supersymmetry breaking such as the anti-D7-branes, there is no potential along the moduli directions. As we will see, some of these properties survive after the T 6 /Z 3 compactification, namely in the world-volume theory of D3-branes at the orbifold fixed points.

Gauge symmetry breaking in the quiver gauge theory
The world-volume theory of the four D3-branes at the orbifold fixed point can be obtained by imposing the orbifolding conditions on N = 4 SYM theory and described by the fourdimensional N = 1 U(2)×U(1) 1 ×U(1) 2 quiver gauge theory as shown in Figure 2. It has the chiral multiplets in the bi-fundamental representations The subscript i with i = 1, 2, 3 denotes the index of three tori Z i and the superscript (a) with a = 1, 2, 3 denotes three different types of bi-fundamental representations of the U(2)×U(1) 1 ×U(1) 2 gauge symmetry. We use the same symbols Z (a) i to describe the scalar components of the corresponding chiral superfields. The covariant derivatives on the scalar fields are given by where the gauge fields of U(2), U(1) 1 and U(1) 2 are described as W A µ , B (1) µ and B (2) µ with A = 0, 1, 2, 3. The τ matrices are defined as τ A = (1 2 , σ i ) where σ i are the Pauli matrices. The gauge coupling constant is defined as g = √ 2πg s . Among the U(1) symmetries, the combination of U(1) 1 −U(1) 2 is the anomalous U(1) symmetry whose chiral anomaly is cancelled by the generalized Green-Schwarz mechanism [36].
These scalar fields acquire the F-term potential where W is given by and β is the index of the doublet representation of the U(2) group. There are also the D-term potential where The normalization of the Lie generators is fixed by the relation Tr(T a T b ) = δ ab /2. The flat directions of the potential V F + V D are parametrized by the six parameters (v i , θ i ) as The relative phases between Z with different a and b can be absorbed by some combinations of gauge transformations of U(1) component of U(2) and U(1) 1 and U(1) 2 , though the common phase cannot be gauged away, because all the fields are in bi-fundamental representations. If the set of three D-branes gets the vacuum expectation value of eq. (19), the gauge symmetry U(2)×U(1) 1 ×U(1) 2 is spontaneously broken down to U(1)×U(1). Expanding around the vacuum expectation value, the gauge bosons associated with the broken generators acquire the mass of i g 2 v 2 i . It has the following geometrical interpretation. These vacuum expectation values describe the configuration of three D3-branes away from the fixed point on T 6 /Z 3 . They are distributed in a Z 3 invariant way on each torus as shown in Figure 3. Since each Z (a) i is a scalar mode of an open string stretching from a D-brane to another, the mass is proportional to the distances, d or s, namely, d/2 √ 2πα ′ or s/2 √ 2πα ′ , respectively. Writing the position of three D-branes on i-th torus as (d i e iθ i , d i e i(θ i +2π/3) , d i e i(θ i +4π/3) ), the relative complex coordinates between them are given by, e.g., d i e iθ i (e i2π/3 − 1) = i √ 3d i e i(θ i +π/3) . If we take a different pair of D-branes, the constant part of the phase π/3 is changed. The distance s between the branes is related to the distance of the branes from the origin d = i d 2 i by the relation s = √ 3d as shown in Figure 3. The geometric interpretation of the coordinates and boson masses is made more explicit in the θ d s Figure 3: A Z 3 symmetric separation of three D3-branes away from the fixed point in T 2 . Blobs describe D3-branes. There are two distances between D3-branes: d and s = √ 3d.
following calculations, but we want to stress here that the total phase θ i represents the direction of the set of three D3-branes, as shown in Figure 3.
Within the N = 1 quiver gauge theory, there is no potential along the direction of increasing v i and the vacuum expectation value cannot be dynamically determined. In the next section, in order to study the dynamics of the gauge symmetry breaking in string theory, we will include the effect of the anti-D7-branes at the fixed point as shown in Figure 1, which breaks the supersymmetry. Furthermore we introduce revolution of D3-branes to balance the attractive force between D3 and anti-D7 branes. These two effects determine the vacuum expectation value v i .

Revolving D3-branes around anti-D7-branes
In this section we study the dynamics of the gauge symmetry breaking in the model of revolving D3-branes around the anti-D7-branes put at the orbifold fixed point. It is realized as a subsystem in the configuration of Figure 1.

Attractive potential between D3 and D7 branes
First we consider an effect of the anti-D7-branes. This breaks the supersymmetry of the D3brane system. Since the anti-D7-branes are not extended along the third torus Z 3 , it can be expected that the D3-brane flat direction in eq. (19) is lifted by an appearance of a masslike term along Z (a) 3 . The supersymmetry breaking potential cannot be understood by the closed string exchange, since the situation we have in mind is d < l s : the distance d between the D3-branes and the anti-D7-branes is much shorter than the string length l s ≡ √ α ′ (see ref. [37]). Instead, we can estimate it by calculating one-loop corrections of open strings stretching between D3 and anti-D7 branes, whose spectra do not preserve supersymmetry. For |Z (a) 3 | ≪ M s , the supersymmetry breaking potential can be expanded and the mass term is added to the F-term and D-term potentials V F + V D . An order estimation [27] gives the coefficient Since D3-branes are attracted to the anti-D7-branes due to the potential (20), they are bound to the anti-D7-branes at the fixed point on the third torus unless another repulsive interaction is added. Other anti-D-branes can be introduced far from the fixed point so that D3-branes are separated apart from the fixed point. But, it is generally difficult to stabilize the positions of D3-branes, and furthermore, even if they are stabilized, the typical length scale is given by the string scale, since there are no other dimensionful parameters in the theory.

A toy model of revolution
In the following, we consider D3-branes revolving around an anti-D7-branes whose centrifugal force balances the attractive force from the anti-D7-branes. As a warm up exercise, let us consider a single complex scalar field φ which represents the moduli of the set of three D3branes on the third torus, namely, each component of Z which has the vacuum expectation value (see eq. (19)). The Lagrangian we consider is simply given by where the mass term comes from the attraction between the D3 and anti-D7 branes. The coefficient is µ ∼ M s /10. When we study time-dependent solutions, it is more convenient to consider the Hamiltonian density, Here, π φ =φ † . A solution to the equation of motion is given by with ω = µ. Since φ = (φ 1 + iφ 2 )/ √ 2 represents the position of three D3-branes, this solution can be interpreted as a revolving D3-brane with an angular velocity ω. In quantizing the system, we usually treat the zero mode wave function equally with the other plane waves with nonzero momenta, because we consider that φ itself represents a microscopic quantum field. The amplitude of the zero-mode v 0 fluctuates around v 0 = 0 and takes various values. But what if the solution is macroscopic (or classical)? In our case, φ represents the position of a D3-brane and the zero mode wave function φ = v 0 exp(iωt) has a geometrical meaning of the revolving D3-brane around the center; (φ 1 , φ 2 ) = ( √ 2v 0 cos ωt, √ 2v 0 sin ωt). Then the solution φ must be treated classically as the coordinate of the revolving D3-brane. We thus need to expand the scalar field around the solution (24) and quantize the fluctuations, just as we do in the case of the Higgs field around a nontrivial vacuum in a double well potential.
The next question is what determines the value of v 0 . It can not be fixed by the equation of motion derived from the Lagrangian (22). The situation is different from the case of the ordinary Higgs where it acquires a nontrivial vacuum expectation value at the minimum of the effective potential 1 . In the present case, the would-be moduli field will acquire a nonzero value of v 0 as a result of the angular momentum conservation of the revolving D3-brane. In this sense, the determination of v 0 needs some external input (or an initial condition) provided from outside of the four dimensional field theory. In order to see it, we first note that the Lagrangian (22) is invariant under the global U(1) phase rotation φ → e iθ φ. The corresponding Noether current is conserved; ∇ µ j µ = 0. In the geometrical interpretation in terms of the D3-brane configurations, the symmetry is nothing but the rotational symmetry of the D3-brane around the center. Inserting φ = v 0 exp(iωt) into the current conservation, we get the conservation of the angular momentum of the revolving D3-brane around the anti-D7-branes; The quantity l = v 2 0 ω is a constant of motion with a mass dimension 3, and related to the angular momentum of the revolving D3-brane. The angular momentum is given by is the volume of the D3-brane) and d is the distance from the center of the revolution. The mass M D3 can be read from the kinetic term in the DBI action of eq. (10), where the scalar fields X M are regarded as the coordinates of the brane embedded in the ten-dimensional space-time. The relations between d and v 0 , d = 2 √ 2πα ′ gv 0 , are obtained by identifying the mass of the ground state of stretched open string, d/2 √ 2πα ′ , and gauge boson mass through the Higgs mechanism, gv 0 . Hence, the value of l, which determines the value of v 0 , needs to be fixed by the initial condition of the cosmological evolution of D3-brane configurations. At the end of the discussions in section 5, we generalize the analysis to include the scale factor a(t) of the FRW universe on D3-branes. The essential points discussed here are not changed.
In order to see the appearance of the centrifugal potential, we focus on spacial homogeneous solutions and write the complex scalar field as φ = v(t)e iu(t) . It can be expanded as φ = (v 0 + δv) exp[i(ωt + π)] = v 0 e iωt + ϕ, where ϕ ∼ (δv + iv 0 π)e iωt . By inserting φ into the Hamiltonian density, we get On the other hand, the current conservation requires that the combination is a constant of motion which should be fixed by an initial condition. This introduces a constraint in the system, and reduces the dynamical degrees of freedom. The Hamiltonian is written in terms of δv only and becomes Due to the conservation of the angular momentum, the phase fluctuation π disappears from the Hamiltonian. It is physically reasonable since the conservation of angular momentum relates the angular velocity with the radius of the revolution. Of course, this reduction of dynamical degrees of freedom is applied only to the zero mode, and non-zero momentum modes of π do not disappear from the Hamiltonian because only the total angular momentum is conserved.
We will see this explicitly in section 4. Hence the role of the angular momentum conservation is not the reduction of degrees of freedom, but the determination of the value of v 0 . The minimum of the potential density V = l 2 /v 2 0 + µ 2 v 2 0 determines the value of v 0 as v 2 0 = l/µ. It is consistent with the relation l = v 2 0 ω and the frequency of the classical solution of the revolution ω = µ. In the toy model discussed above, we considered only the zero mode of field configuration which does not depend on the space coordinates. In obtaining the centrifugal potential for nonzero modes, we need to care about competition between the strength of the centrifugal force and the tension on the brane. There are two limiting cases depending on which force is stronger than the other. If the centrifugal force is stronger than the tension, each point on the brane behaves as if it is revolving independently of the other points. In this case, we can use an approximation that the conservation of the angular momentum at each point holds separately. Then each point is bound to the minimum of the potential in (29). The coefficient l 2 of the centrifugal potential can depend on positions x, and hence so is the minimum v 0 (x). Due to the balance between the attractive force F attractive = µv(x) and the repulsive force F repulsive = ω(x)v(x) at each point on the D3-brane, the brane revolves with the common angular frequency ω(x) = µ independent of the value of v 0 (x). On the other hand, in the opposite case where the tension is stronger, the only conserved quantity is the total angular momentum, and integral of (28) over the space, L = µT D3 v(x) 2 d 3 x. Then the brane can rotate around itself as it revolves. The degrees of freedom on the D3-brane can contribute to the angular momentum. In the following, we consider the first situation. Then each point on the brane is strongly constrained in the minimum of the potential. Furthermore, we consider the simplest case where v 0 (x) is independent of x 2 . General situations are left for future investigations.

Revolving D3-brane in DBI action
When the embedding X µ represents revolution of a D3-brane, we can see that the DBI action (10) is reduced to the toy model discussed above. In the static gauge, a revolving D3-brane on the third torus is described by the following set of coordinates; whered ≡ √ τ 3 d. We set Z 1 = Z 2 = 0 for simplicity since we are interested in revolution of the D3-brane on the third torus. Then the induced metric on the world-volume of the D3-brane is given by whereD Setting B-field, dilaton and gauge fields zero, the DBI Lagrangian becomes Including the supersymmetry breaking mass term (20), this is the same as the toy model of the massive free scalar field of eq. (27) by identifying the parameter asd = √ 2v 0 . Then we obtain the relation d = 2 √ 2πα ′ gv 0 . A generalization to n D3-branes is straightforward.

Effects of D7-branes and revolution of D3-branes in T 6 /Z 3
The effects of the anti-D7-branes and the revolution of D3-branes in the world-volume field theory of the D3-branes can be summarized by adding a potential term l 2 /v 2 0 + µ 2 v 2 0 to the world-volume field theory of D3-branes. Here v 0 is proportional to the distance between the anti-D7 and D3 branes.
In the rotating reference frame where D3-branes are at rest, the branes feel the centrifugal potential l 2 /|φ| 2 . Here, φ is the moduli field representing the complex coordinate of the D3-branes. Originally it comes from the kinetic term but we can regard the term as one of the potential terms in the rotating reference frame. The discussion can be straightforwardly generalized to the case of revolving D3-branes on the T 6 /Z 3 orbifold. Since the three D3-branes are interacting each other, they exchange angular momenta so that only a sum of each angular momentum of D3-branes on the third torus is conserved. The conserved current is then a sum of each angular momentum of D3-branes Following the same argument in the toy model, the effects of the anti-D7-branes and the revolution of three D3-branes are to generate the attractive and the repulsive potentials respectively; The coefficient 3 in front of l is introduced for later convenience. In order to apply this to nonzero modes, as we discussed at the end of section 3.2, the centrifugal force must be stronger than the tension. In the next section, we discuss the low energy spectrum of the D3-anti-D7 brane system with the above potential. Since the potential V M (Z) depends only on the combination a |Z (a) 3 | 2 , mass term appears only in the breathing mode that changes the magnitude of the same combination. In the rotating reference frame on the third torus, without loss of generality, we can choose the classical solution of D3-branes Z Here, we assumed that there is no separations of D3-branes in the first and the second tori. This satisfies the D-flat condition. Note that we are now in the rotating reference frame of the D3-branes and the vacuum expectation value is set stationary without the phase. The breathing mode σ is defined as that to change the magnitude of a |Z (a) The normalization of the σ field is chosen so that the kinetic term becomes canonical. Inserting this into the potential V M , we can read the mass m σ of the breathing mode as m σ = 2µ. Hence it becomes as heavy as the string scale. The other modes are massless.
Finally in this section, let us play with the numerics of various quantities. D3-brane is a gigantically macroscopic object and its mass is obtained as M D3 = τ 3 V D3 . If we set M s = 10 18 GeV and V = (10 −33 eV) −3 (i.e. the present particle horizon), the mass becomes M D3 = 10 207 eV. Of course, it is much larger than the corresponding "mass" obtained from the critical density ρ crit of our current universe ρ crit V D3 ∼ 10 89 eV. It is nothing but the cosmological constant problem, since the energy scale of the tension of D3-brane is typically as large as the string scale and much larger than the energy scale of the dark energy at the present universe, i.e. meV. The total angular momentum of the D3-brane is thus given by In the second equality, we dropped numerical factors and used ω = µ = M s and d 2 = (gv 0 ) 2 /M 4 s . When we put, e.g. gv 0 = 100 GeV, the angular momentum is roughly L ∼ 10 148 . Hence, we can treat the revolution of the D3-brane classically. Though the angular velocity ω is as huge as the string scale, the velocity of the revolution is non-relativistic if the vacuum expectation value is much smaller than the string scale. For example, if gv 0 ∼ 100 GeV, the radius of the revolution is d ∼ gv 0 /M 2 s . Then the velocity of the revolving D3-brane is given by ωd ∼ gv 0 /M s ∼ 10 −16 , and the motion is quite non-relativistic. The acceleration is given by ω 2 d ∼ gv 0 , and it is again much smaller than the string scale. It justifies the validity of the DBI action and its expansions.

Gauge symmetry breaking by revolving D3-branes
In this section we investigate the low-energy spectrum of the world-volume theory with revolving D3-branes on the third torus, and calculate the masses of various fields. We also discuss stability of the vacuum expectation value.
If the centrifugal potential is stronger than the tension, we can treat each point on the brane is bound at the minimum of the potential (35) 3 . In such a situation, the Lagrangian we are going to study becomes 4 V F and V D are the F and D-term potentials. The potential V M is the supersymmetry breaking potential induced by the anti-D7-branes and the revolution of D3-branes discussed in the previous section. This term determines the position of D3-branes revolving around the fixed point.

Gauge boson masses
First let us calculate the masses of the gauge bosons. Since the gauge symmetry breaking is from U(2)×U(1) 1 ×U(1) 2 to U(1)×U(1), four of the six gauge bosons, W A µ of U(2), B (1) µ of U(1) 1 and B (2) µ of U(1) 2 , become massive by the Higgs mechanism. The other two gauge fields, associated with two unbroken generators, remain massless. With the normalizations of the charges in the covariant derivatives of eq. (12), the following two combinations are massless. The other four combinations become massive. They are classified into two different types. The first type is associated with the broken generators of U(2) and given by They acquire mass of gv 0 and are charged under the above unbroken U(1) symmetries 5 . The other type of combinations is orthogonal to the massless gauge fields (40) and written as They acquire masses of √ 3gv 0 and are neutral under the remaining two U(1) symmetries. The gauge boson Z

Scalar boson masses
Next let us calculate the mass spectrum of the scalar fields. As we discussed at the end of the previous section, since the potential V M is a function of a single combination of fields, only the breathing mode of eq. (37) acquires mass through the potential V M and the other modes are not affected by the supersymmetry breaking potential V M . We now consider the fluctuations of the scalar fields along the third torus Z + ϕ a , where the vacuum expectation value is defined in eq. (36) and ϕ a are decomposed into three types of scalar fields σ, ρ and π; The mass terms of these scalars can be obtained by expanding the potential V F + V D + V M around the classical solution of eq. (36). The F-term potential V F does not generate any mass 5 It is neutral under the combination of (A term, because the classical solution is non-vanishing only along the third direction Z (a) 3 . The potential V M generates the mass term for the breathing mode in σ only, and the other modes of σ, π and ρ are not affected by V M .
The mass matrix of the σ modes are obtained as follows. By redefining the fluctuations of σ a by the unitary transformation, only the first componentσ 1 , which corresponds to the breathing mode, acquires mass proportional to µ. Since this mode originally corresponds to the flat direction of the D-term potential, V D does not generate potential for the breathing mode. On the other hand, the other two combinations get their masses from the D-term potential only, and not affected by V M . Their masses are proportional to gv 0 , and independent of µ ∼ M s /10. Indeed, the mass matrix of "Higgs bosons", σ a , is given by where ǫ ≡ (gv 0 /2µ) 2 /3. The mass of the breathing modeσ 1 is 2µ and becomes very heavy. The other two modesσ 2 andσ 3 have equal masses of √ 3gv 0 .
The masses of the ρ modes are given as follows. Among four real scalar fields in ρ a , two combinations are would-be Nambu-Goldstone bosons of the gauge symmetry breaking of SU(2) into U(1). Since the broken generators acting on Z are σ a /2 and (−σ a ) * /2 with a = 1, 2 respectively, the combinations of are eaten by the longitudinal modes of W ± µ gauge bosons. Here, we wrote the real and imaginary part of ρ a as ρ a = (ρ a R + iρ a I )/ √ 2. The other two combinations of are "charged Higgs bosons" with mass gv 0 , For the π a modes, two of them are the would-be Nambu-Goldstone bosons of the gauge symmetry breaking of four U(1) symmetries (corresponding to two diagonal generators of U(2) and U(1) 1 and U(1) 2 ) into U(1)×U(1), and become the longitudinal modes of Z (1) µ and Z (2) µ . The last one is massless. It is the Nambu-Goldstone boson associated with the breaking of the global U(1) symmetry, by which the phases of the scalar fields Z 3 . This mode corresponds to changing the positions of revolving D3-branes without changing their relative positions and the distance from the origin. This global symmetry is the anomalous U(1) R symmetry of the full theory with the superpotential of eq. (14), and accordingly the corresponding scalar field becomes massive.

Stability of the symmetry breaking scale
The following arguments indicate stability of the mass spectrum of the scalar fields against radiative corrections. The spontaneous symmetry breaking occurs due to the potential V M in eq. (39). If gauge interactions were absent, we could consider the upper and the lower components of the doublet fields Z (a) 3 with a = 1, 2 as independent complex scalar fields. The action is then invariant under the U(5) rotation of 5 complex scalar fields, and the U(5) invariant potential V M has a minimum where U(5) is spontaneously broken down to U(4). Then the symmetry breaking of the global U(5) symmetry produces one massive "Higgs" boson and nine massless Nambu-Goldstone bosons. In presence of the gauge interactions, four of these nine Nambu-Goldstone bosons are absorbed into the gauge bosons through the Higgs mechanism, and the remaining five scalars become massive as pseudo-Nambu-Goldstone bosons (pNGBs), because the corresponding U(5) symmetries are explicitly broken by the gauge interactions. The radiative correction to the mass of the pNGB is protected by the approximate symmetry. The mass spectrum of the pNGB is generally written as the Dashen formula [39] where f is the decay constant and Q is the broken generator corresponding to the fluctuation of the pNGB around the minimum, and stable under radiative corrections so long as the coupling is weak [40,41,42,43]. In our scenario the vacuum expectation value v 0 is determined in terms of the initial angular momentum of the D3-branes which is conserved. Hence, it is fixed by the input of the initial condition, not by the field theory itself. This guarantees the stability of the value of the distance between the revolving D3-branes and the fixed point where anti-D7-branes are localized. In this geometrical way the symmetry breaking scale is invariant 6 . Furthermore, since the mass of theσ 1 field is much heavier than the other scalar fields, quantum fluctuations of the vacuum expectation value are highly suppressed in the low-energy effective theory. This is another indication of the stability of the scale of the spontaneous gauge symmetry breaking.

Other fields and fermions
So far we have considered only the scalar fields Z have the vacuum expectation values of eq. (36), they acquire their masses through the interactions described by the F-term potential (13). For their fermionic superpartners, they acquire the same masses through the Yukawa couplings, since the gauge symmetry breaking by the vacuum expectation value (36) does not break supersymmetries. All the fields, which are associated with the first and second tori, obtain masses of the order of gv 0 . Thus all the D-brane moduli fields corresponding to the first and the second tori are stabilized. They may also obtain their masses through one-loop effects of open strings between D3 and anti-D7 branes, which are not supersymmetric.
Yukawa couplings between the scalar fields Z (a) i and the fermion fields in the chiral multiplets come from the superpotential of eq. (14). Because of the ǫ-tensor in the superpotential, the Yukawa couplings must contain at least one field associated with each of the three tori. Hence, since the scalar fields Z with the nonzero vacuum expectation values couple to the fermions associated with the first and the second tori, the fermion in the first and the second tori become massive in pairs. Especially, the Yukawa coupling of the "Higgs boson",σ 1 , with the fermions are proportional to their masses. But it is not the case for light scalar fields,σ 2 andσ 3 . Note that the fermion fields associated with the third torus do not obtain their masses through the Yukawa couplings. A brief discussion towards more realistic model constructions of the standard model is given in section 5.

Conclusions and discussions
In this paper, we have proposed a new mechanism of spontaneous gauge symmetry breaking in string theory with revolving D-branes around an orbifold fixed point.
In orbifold compactification of the string theory two types of D-branes are known: ordinary D-branes and fractional D-branes. The ordinary D-branes can move away from the fixed points by forming an invariant set under the orbifold projections while the fractional D-branes are localized at the fixed points. When the ordinary D-branes move away from the fixed point, the gauge symmetry is spontaneously broken with reduction of the rank of the gauge group. If the separation scale of the D-branes is shorter than the string length and their relative motion is non-relativistic, the dynamics can be well described by the low-energy world-volume theories of D-branes [44]. The separation length corresponds to the vacuum expectation value of the D-brane moduli fields. Unless supersymmetry is broken, the moduli fields have the flat potential along the vacuum expectation values. If we consider configurations of D-branes with broken supersymmetries, the flat directions will be lifted up by the one-loop corrections, whose mass scales are generically given by the string scale with a small correction of the one-loop suppression factors. Such effects are responsible for attractive forces between D-branes and anti-D-branes.
In the present paper, we consider a particular model of D3 and anti-D7 branes at a fixed point of a T 6 /Z 3 orbifold. In order to stabilize the separation of D3-branes from the fixed point, we introduced revolution of D3-branes around anti-D7-branes localized at the fixed point. The revolution can be described as the time-dependent phase of the classical solution of D-brane moduli fields. In the rest frame of the revolving D3-branes, the centrifugal potential appears. This force can balance the attractive force between the D3 and anti-D7 branes. The distance of the revolving D3-branes from the fixed point is determined by the value of the angular momentum, which should be given by the initial condition of the cosmological evolution of the D-brane configuration. An important point is that the scale of the distance, namely the vacuum expectation value of the moduli fields, is determined in terms of the angular momentum and can be taken much lower than the string scale. The situation is quite similar to the revolution radius of the earth around the sun, where it was determined by the initial condition of the solar system evolution. We can imagine that the early universe are filled with gas of D-branes and anti-D-branes. They collide and some of them are annihilated, but due to the conservation of the angular momentum, some branes revolve around others.
We calculated the mass spectrum of various moduli fields. The breathing mode, in which all the nonzero expectation values of the moduli fields change simultaneously, becomes super heavy as the string scale M s because the mass comes from the balance of the attractive potential between D3 and D7 and the repulsive centrifugal potential. On the other hand, the other scalar modes acquire their masses through the gauge symmetry breaking, and hence their masses are of order of the vacuum expectation value v 0 ≪ M s . They are stable under radiative corrections due to their nature of the pseudo-Nambu-Goldstone bosons. It is interesting to note that the stability of the scale v 0 is related to the stability of the geometrical distance of the revolving D3 branes from the fixed point. The distance is classically stable due to the conservation of angular momentum. Now several important remarks are in order. The first remark is possible loss of the angular momentum of the revolving D3-branes by emitting massless closed string states (gravitons or 4-form R-R field). If the effects are large, the revolution frequency ω decreases and the D3branes fall into the fixed point rapidly. As we discussed at the end of section 3, the velocity β = ωd ∼ gv 0 /M s and the acceleration α = ω 2 d ∼ gv 0 are very small and the emission may be expected to be tiny. However, it is not the case since both of the tension and the R-R charge are very large. Here let us estimate the emission rate of the gravitational radiation into the bulk from the revolving D3-brane by using the formula in [45]. The energy emission rate per unit time and unit volume of the D3-brane is estimated by M 5 s (ωd) 4 . Huge factor comes from the tension of the D3-brane. The rate is very large in comparison with the kinetic energy of the revolving D3-branes per unit volume τ 3 (ωd) 2 /2 ∼ M 4 s (ωd) 2 , even though ωd ∼ (ω/M s )(v/M s ) is very small. If it is really the case, the stability of the vacuum expectation value will be lost. But, the formula of the emission rate may not be applicable to the present case since the typical frequency of the radiation, which is of the same order of the angular velocity of the revolution ω ≃ M s /10, is close to the cut off scale of the low-energy effective theory, M s . Also, it is not certain whether radiation itself is possible since the length scale of the compact space is comparable to the typical wave length of the emitted radiation. Furthermore, the NS-NS tadpoles are not cancelled in this non-supersymmetric configurations of D-branes and we may need to take the effects of the backreaction to the space-time metric, dilaton and B-fields, which may drastically change the emission rate of radiation. So more detailed analysis will be necessary to give the correct estimates of the emission rate.
The second remark is a possibility of a modification of gravity in the IR region. One may expect that the revolution of D3-branes affects the gravity in the IR region since our model is similar to the ghost condensation [38]. In both of the ghost condensation and our setting, the time-dependent classical solutions of the scalar fields play an important role. In the ghost condensation, X 2 = |∂ µ φ| 2 has a non-vanishing vacuum expectation value at the minimum X 2 0 of the function P (X 2 ). Then the field can be expanded around the time-dependent solution φ = X 0 t, which breaks the Poincaré symmetry. Writing φ = X 0 t + ϕ, we have X 2 = (X 0 +φ) 2 − (∇ϕ) 2 . Then the kinetic term P (X 2 ) becomes P (X 2 ) ∼ P (X 2 0 ) + [2P ′′ (X 2 0 )X 2 0 ]φ 2 . The (∇ϕ) 2 term is absent, and the IR dispersion relation is drastically modified as ω 2 ∝ p 4 . In our case of the revolving D-branes, the time-dependent solution appears not because it gives a minimum of the kinetic term but it is due to the centrifugal potential in the rest frame of the D-branes. Because of this, if the centrifugal potential is stronger than the tension of the brane and each point of the brane is bound strongly at the minimum of the potential, the spectrum of the fluctuation seems to keep the Lorentz invariance and different from the case of the ghost condensation. But Lorentz invariance is generally broken due to the revolution, especially if the tension is comparable to the centrifugal potential. We come back to this issue in near future.
The third remark is a construction of more realistic models of particle physics. There have been many efforts to construct realistic models in the system of D-branes at orbifold fixed points or other singular points, in general (see for example [18,19,20,21]). Though the aim of this paper is not to propose a realistic model, some interesting structures of realistic models are included in the model we have discussed above. For example, following [18], we can identify the subgroup SU(2) in U(2) as the standard model SU(2) L group and a non-anomalous combination of U(1) symmetries, Q Y = −(Q/2 + Q 1 + Q 2 ), as the U(1) Y in the standard model. Here Q, Q 1 and Q 2 are charges of U(1) in U(2), U(1) 1 and U(1) 2 , respectively. In the identification, the value of the Weinberg angle becomes sin 2 θ W ≃ 0.27. Since the charges are normalized to take values of ±1, we can identify the scalar component of Z (1) 3 as a Higgs doublet field (Q Y = +1/2), the fermion components of Z (2) i=1,2 as the left-handed lepton doublets of the first and the second generations (Q Y = −1/2), and the fermion components of Z (3) i=2,1 as the righthanded neutrinos of the first and the second generations (Q Y = 0). In this simple setting, the Yukawa couplings are originated from the superpotential of eq. (14) and give large masses to neutrinos. It would be interesting to try to construct more realistic models with quarks and leptons in the framework of the spontaneous gauge symmetry breaking by the revolution of D-branes.
The fourth remark is a realization of an inflationary universe with the revolving D-branes. In the present paper, we treated the bulk space-time as the flat Minkowski, but of course it receives huge backreaction effects, since the configuration is not supersymmetric. In the following, instead of solving the bulk 10-dimensional Einstein equation, we study time-evolution of the induced metric on the D3-branes [46]. Since D3-branes are isotropic and homogeneous, the time-evolution of the metric will be described by the FRW universe 7 with the scale factor a(t). In the FRW universe, the solution to the equation of motion is changed from eq. (24) to This solution is valid when the oscillation frequency ω = µ is larger than the Hubble parameter H =ȧ/a. In the opposite case H > µ, the field is frozen so that the prefactor is almost constant. Such a scale factor dependence is understandable since the oscillation in a harmonic potential behaves as non-relativistic particles and the energy density µ 2 v 2 must decay as a −3 . Now Suppose that D3-branes, while they are rotating, start rolling down along the potential from a larger value of v to the stabilized point v 0 . The current j 0 is proportional to a −3 v 2 ω. Due to an additional factor |g| = a 3 in the current conservation ∇ µ j µ = a −3 ∂ µ (a 3 j µ ) = 0, the same quantity l = v 2 ω is conserved. The Hamiltonian density ρ for µ > H becomes The minimum of the potential is given by the same value v 0 = l/µ. The pressure density p is given by changing the sign of the second term as This is because the first term (centrifugal potential) is originated from the kinetic energy while the second one comes from the attractive interaction between D3 and anti-D7 branes. Denoting the ratio of v to v 0 = l/µ 0 by v/v 0 ≡ e θ/2 , ρ and p can be written as ρ = 2a −3 lµ cosh θ and p = −2a −3 lµ sinh θ. Then the equation of state becomes It is interesting that the equation of state of the revolving motion interpolates between w = −1 and w = 0. At the final stage of the evolution, D3-branes are stabilized at v = v 0 and the equation of state is w = 0. The energy density is then given by ρ = 2lµa −3 . On the other hand, in the initial stage of the cosmic evolution, the second term µ 2 v 2 in ρ and p dominates the first term and the equation of state is w = −1. In this case, the field is frozen and the prefactor a −3 is replaced by a constant. Then the energy density becomes ρ = µ 2 v 2 , which gives the Hubble constant of the inflationary universe caused by the revolving D3-branes. In this scenario, the radial direction of the revolving D-branes,σ 1 , plays a role of the inflaton field, similar to the brane-inflation model [37]. Similar angular motions of D-branes in the DBI inflation scenario has been discussed in [47,48,49].
The final remark is about possible modifications of the background geometry of the compact space due to introductions of the three-form fluxes and additional (anti)-D-branes for the moduli stabilization. The form of the angular momentum conservation might be modified as well as the attractive potential. But as far as the rotational symmetry is preserved and the angular momentum is conserved, the essential mechanism will not be changed much. The non-trivial modification of the background geometry could make it easier to obtain the small scale of the gauge symmetry breaking in comparison to the string scale, as is the case in the warped geometries [50].
Each of the above mentioned issues is very interesting, but needs more careful and detailed investigations. We want to come back to these issues in future.