Spinning vortex braneworld

A spinning vortex is considered in the context of the braneworld. We numerically analyze the profiles of a stationary solution in a six-dimensional U(1) gauge theory, and clarify their dependence on the angular velocity in the field space ω. We find that there is an upper limit on ω, and the vortex configuration should be parameterized by the angular momentum rather than ω. We also discuss fermion zero-modes localized on the vortex, and comment on the violation of the four-dimensional Lorentz symmetry that they feel. E-mail address: sakamura@post.kek.jp


Introduction
The braneworld scenario is interesting both from the phenomenological and the stringtheoretical points of view, and has been extensively investigated in vast amount of papers.
However, many of them only discuss static brane configurations. This is mainly because such configurations are much easier to analyze, and moving branes generically violate the Lorentz symmetry on the branes. If we focus on a local region near the earth in the present universe, it may be a good approximation. However, when we discuss the cosmological evolution of the universe, we should take into account the brane motions in the past.
In the braneworld scenario, it is natural to imagine that branes were actively moving and colliding with each other in the early stage of the universe. Such motions become slower as the universe expands, and eventually the branes approach to the static configurations.
However, the brane motions in the past may affect the cosmological history since they generally leads to various symmetry breakings including the Lorentz violation. In addition, the brane collision process can induce the inflation [1,2]. Hence it is quite important to understand how such brane motions affect the four-dimensional (4D) effective theory.
There are various kinds of brane motions, such as the translation, the rotation, the collision and the merger of the branes, and so on. Especially, when a brane is a fieldtheoretical soliton rather than the D-branes in the string theory, it has a finite width. In such a case, the deformations and the spin of the brane are also possible. It is a nontrivial and intriguing subject to investigate how such brane motions affect the evolution of the 4D spacetime on the brane. This is the motivation of our work.
The simplest setup for the braneworld scenario is a five-dimensional (5D) theory. The moving branes with codimension-one in 5D are discussed in Refs. [3]- [7], and it has been shown that their motions affect the evolution of our 4D spacetime significantly. Here we will consider the next simplest case, i.e., the codimension-two case. In this case, a rotation of the branes in the extra-dimensional space becomes possible. 1 Specifically, we focus on a vortex soliton in six-dimensional (6D) theories. In this paper, as a first step for our purpose, we study a spinning vortex 2 in a 6D non-gravitational theory. The spin of the soliton may be understood as a trail of a grazing collision that the brane has experienced 1 A rotation of the D-branes in a compact space is discussed in Refs. [8]- [11]. 2 As another example of spinning codimension-two objects, a rotating hollow cylinder constructed by a domain wall is discussed in Ref. [12] in a 6D gravitational theory. In this case, the spin is necessary to stabilize the configuration against collapse due to the tension of the domain wall.
in the past. To simplify the discussion, we focus on a stationary field configuration. 3 We also discuss the fermion zero-modes localized on the vortex, and the violation of the 4D Lorentz symmetry that they feel.
The paper is organized as follows. In the next section, we briefly review the ANO vortex in the 6D Abelian-Higgs model, and discuss a possibility of its rotation. In Sec. 3, we extend the model to obtain a stationary solution for a spinning vortex. The profiles of the vortex background are numerically calculated, and their dependence on the angular velocity in the field space is clarified. In Sec. 4, we introduce the matter fermions and discuss the zero-modes localized on the vortex. We also comment on the violation of the 4D Lorentz symmetry in the 4D effective theory. Sec. 5 is devoted to the summary. In Appendix A, we summarize the vacuum structure of our model. In Appendix B, we collect the notations for the fermions.

Case of ANO vortex
First we consider the Abrikosov-Nielsen-Olesen (ANO) vortex [13,14]. The theory is the 6D Abelian-Higgs model whose Lagrangian is given by where M, N = 0, 1, · · · , 5, and The parameters λ and v are chosen to be positive. Since the scalar φ and the gauge field A M have mass dimension 2 in 6D, the dimensions of the parameters are given by 3) The equations of motion are

Static background
The ANO vortex is obtained by imposing the background ansatz, where (r, θ) are the polar coordinates for the extra dimensions, 6) and the integer n is the vortex number. The Hamiltonian density for this background is given by Thus, in order to have a finite vortex tension (i.e., 4D vacuum energy density) τ 3 = 2π ∞ 0 dr rH, the dimensionless functions f and α should satisfy Besides, the regularity of the fields at the origin requires With these boundary conditions, we obtain the (static) vortex solution by solving the equations of motion (2.4).

Background ansatz for spinning vortex
In order to search for a spinning vortex solution, we extend the background ansatz (2.5) In this case, the Hamiltonian density (2.7) becomes H = 1 2 Thus, from the condition that the vortex tension τ 3 should be finite and the regularity at the origin, the dimensionless functions f , α and β must satisfy the following boundary conditions.
Under the ansatz (2.10), the equations of motion (2.4) become where ρ ≡ v 1/2 r is a dimensionless radial coordinate, and are dimensionless parameters.
In fact, (2.13) does not have a solution that satisfies the boundary conditions in (2.12).
Let us focus on a region ρ ≫ 1. There, the second term of the equation for β is neglected and f (ρ) ≃ 1. Thus the solution behaves as where C is a real constant. When C > 0, we find that β ′′ (ρ) > 0 and β ′ (ρ) < 0 for ρ ≫ 1. Therefore, is positive for all region because the second term in the right-hand-side, which is positive, gets bigger and bigger as we approach to the origin while the contribution of the first term, which is negative, decreases. This indicates that β ′ (ρ) is always negative for any values of ρ. By a similar reason, β ′ (ρ) is always positive when C < 0. In both cases, we cannot satisfy the boundary conditions at the origin. 4 The only possible case is C = 0. In this case, β(ρ) = 1 is a solution that satisfies the boundary conditions in (2.12 This fact is expected from the following reason. 5 The vacuum of this model is where δ is a real constant. The fluctuation modes around this vacuum are as follows. The gauge boson gets a nonzero mass via the Higgs mechanism for the breaking of the U(1) gauge symmetry. The scalar field φ is decomposed as φ = (ϕ + v) e i(δ+χ) . The phase part χ is the would-be NG boson and is absorbed by the gauge boson, and the radial part ϕ gets a mass from the potential. Namely, no massless modes exists in the vacuum.
This indicates that nonzero energy is necessary when we move the vortex for any direction.
So we cannot rotate the vortex without energy cost. This is the reason why there is no stationary spinning vortex solution in this model.
According to the above perspective, we need a massless mode corresponding to the fluctuation along the phase direction in order to have a stationary spinning vortex. In the next section, we will extend the model in such a way.

Setup
We extend the previous model by adding an extra charged scalar field. The Lagrangian is given by where λ 1 , λ 2 , v 1 , v 2 and γ are positive constants, and The constant U 0 is irrelevant to the physics if we neglect the gravity.
The mass dimensions of the parameters are In addition to the U(1) gauge symmetry, the model has the U(1) global symmetry, which is denoted as U(1) gl , under the transformation that changes the relative phase of φ 1 and φ 2 .
The vacuum structure of this model is summarized in Appendix A. In the following, we will focus on a case that Then the vacuum (i.e., the global minimum of U) is We will set the constant U 0 so that the vacuum energy is zero in the following. Namely, The equations of motion are This theory has the (static) ANO vortex solution,

Background ansatz for spinning vortex
For the purpose of finding an axially-symmetric stationary spinning vortex solution, we make the following ansatz for the background. 6 where f 1,2 , α and β are dimensionless real functions, the integer n is the vortex number, and the real constant ω is the angular velocity.
Then the Hamiltonian density is In order to have a finite vortex tension, we should require the boundary conditions at infinity.
From the regularity at the vortex core, we obtain the boundary conditions at the origin.
With our ansatz, the equations of motion in (3.7) are translated into the equations for the dimensionless functions as are dimensionless coordinate and parameters.
Due to the U(1) gl , this model has a massless mode corresponding to the fluctuation changing the relative phase between φ 1 and φ 2 at every spacetime point. Thus it is expected for the above equations to have a solution that satisfies the boundary conditions (3.11) and (3.12), in contrast to the previous model.

Asymptotic behaviors of the solution
From the equations in (3.13) with the boundary conditions (3.11) and (3.12), we can read off the asymptotic behaviors of the dimensionless functions. In a region ρ ≪ 1, they behave as where C 0 f 1 , C 0 f 2 , C 0 α and C 0 β are real constants. Next we consider a region ρ ≫ 1. Then, using (3.11), (3.13) is reduced tô The solution of the second equation is expressed by the (modified) Bessel function as , (3.18) up to the normalization factor, where Namely, when a 2 > 0, it behaves as where C ∞ f 2 is a positive constant. Using this and the last two equations in (3.16), we find that where C ∞ α and C ∞ β are real constants. Then, from the first equation in (3.16) with the above asymptotic forms, we obtain where C ∞ f 1 is a real constant. Thus, when a 2 is small enough, the Hamiltonian density (3.10) is approximated as for ρ ≫ 1. Therefore, the vortex tension τ 3 ≡ 2π ∞ 0 dρ ρH diverges when a 2 ≃ 0. This indicates that there is a maximum value of the (normalized) angular velocityω. 7 (3.24)

Profiles of the solution
The equations in (3.13) with the boundary conditions (3.11) and (3.12) can be solved numerically. Fig. 1 shows the profiles of the solution. The solid, the dotdashed, the dashed  These behaviors of the functions can be understood by noticing that the centrifugal force is proportional to the angular momentum P θ of the vortex, rather than the angular velocity in the field space ω. The angular momentum P θ is given by where P θ (ρ) is the Noether current for the rotation in the x 4 -x 5 plane. Fig. 2 shows the relation between P θ and ω. These behaviors indicate that we should parameterize the vortex configuration by P θ rather than by ω.

Localized fermion zero-modes
In this section, we introduce matter fermions in the bulk, and consider the zero-modes localized on the vortex brane in the previous section. We introduce 6D Weyl fermions Ψ ± whose Lagrangian is given by where χ 6 denotes the 6D chirality, and The constants q ± are the U(1) charges of Ψ ± , respectively. Due to the charge conservation, they are related as The coupling constants y 1 and y 2 are chosen to be real and have the mass dimension The notations for the gamma matrices and the fermions are collected in Appendix B.

Mode equations
The equations of motion for the fermions are In the 2-component spinor notation, these are rewritten as where the 2-component spinors χ ± andζ ± are defined in Appendix B.
In the following analysis, we move to a gauge in which the background for A 0 is zero.
Namely, the background (3.9) becomes 8 Then the linearized equations of motion for the fermions are We have used the polar coordinates for the extra dimensions.
Each component of the fermions is decomposed into the KK modes as where ρ = v are dimensionless (m k is the KK mass).
In the following, we will focus on the zero-modes, i.e.,m 0 = 0. When both Yukawa coupling constants y 1 and y 2 are nonvanishing, it is hard to separate the coordinate-dependence of the mode functions. So we leave the analysis in such a case to our future works, and consider cases in which at least one of the Yukawa couplings vanish in this paper.

y 1 = 0 and y 2 = 0 case
First we consider the case that the fermions do not couple to φ 2 . In this case, we can separate the variables by assuming that (4.12) The single-valuedness of the mode functions require that the parameters m and m ′ are integers. We have degenerate zero-modes parameterized by m or m ′ .
The mode equations for b R± (ρ) and b L± (ρ) are [m] where A Thus we have |n| right-handed zero-modes.
In either case, |n| 4D chiral fermions are obtained in the effective theory, as in the static case [15]. 9 The solution proportional to e |ỹ1|ρ is non-normalizable, and is excluded.

y 1 = 0 and y 2 = 0 case
Next we consider the case that the fermions do not couple to φ 1 . Then, we can separate the variables by assuming that (4.18) The mode equations for b In a region of ρ ≫ 1, the mode functions are damped as b [m] In contrast to the previous case, the mode functions do not decay exponentially because f 2 (ρ) in (4.19) is negligible for ρ ≫ 1. Since the normalization conditions require that the integers m and m ′ must satisfy Namely, they are constrained as q − n < m < q + n − 1 = (q − − 1)n − 1, q + n + 1 = (q − − 1)n + 1 < m ′ < q − n. (4.23) We have used (4.3).
Next we consider a region ρ ≪ 1. From (4.19) with (3.12), the mode functions behave as b [m] (4.24) The regularity at the origin requires that either coefficient A or B must vanish. For example, for a positive m, the coefficients B The coefficients A No constraints on m and m ′ come out from the regularity at the origin.

y 1 = y 2 = 0 case
Finally, we consider the case that the fermions do not couple to either scalar. In this case, the four equations in (4.10) are decoupled and can be solved independently. We can separate the variables by assuming that where m ± and m ′ ± are integers. The solutions are L± are normalization constants. In a region of ρ ≫ 1, they behave as The normalization conditions require that In a region of ρ ≪ 1, (4.28) is approximated as Hence, the regularity at the origin requires that From (4.30) and (4.32), the integers m ± and m ′ ± are constrained as 0 ≤ m + < q + n − 1, q − n + 1 < m − ≤ 0, Thus, the number of the zero-modes depends on the charges q ± , in contrast to the previous cases. In the absence of the Yukawa interactions, it is determined by the charge and the flux threading the extra-dimensional space, as the index theorem insists [16,17]. However, because the extra-dimensional space is non-compact in our model, some of the zero-modes are non-normalizable and are dropped in the spectrum. So the number of the zero-modes is smaller than that of the compact case.
When we turn on the Yukawa coupling y 1 or y 2 , some of the zero-modes allowed in this subsection obtain masses via the Yukawa coupling and are decoupled in low energies.
The number of the remaining zero-modes only depend on the vortex number n and are independent of the charges [15], as we have seen in the previous subsections.

4D Lorentz violation
Our background (3.9) breaks 4D Lorentz symmetry due to the nonvanishing A bg 0 and the explicit t-dependence of φ bg 2 . In the fermionic sector, this Lorentz-violating effect appears through the explicit t-dependence of the mode functions. When y 2 = 0, such t-dependences are proportional to the U(1) charges (see (4.12) or (4.27)). Thus, they are cancelled due to the gauge symmetry except for the coupling to φ 1 . The situation is similar even if we also introduce matter scalar fields that do not couple to φ 2 . Namely, the 4D Lorentz violation is not observed in the matter sector at tree level in this case.
On the other hand, when y 1 = 0 and y 2 = 0, the t-dependences of the mode functions cannot be cancelled. In this case, if we move to the gauge in which the t-dependences of the mode functions become proportional to the charges, and are cancelled in the 4D effective theory. However, due to the nonvanishing A bg 0 , the dispersion relations deviate from the relativistic ones, just like the situation in Ref. [10].
The fluctuations around φ bg 1 and φ bg 2 couple with those of A bg 0 , A bg r and A bg θ , and directly receive the Lorentz-violating effects of the background.

Summary
We considered a situation that the 3-brane we live is spinning in the extra-dimensional space. The ANO vortex in the Abelian-Higgs model does not have a degree of freedom to rotate the vortex configuration without energy cost. So the stationary spinning solution does not exist. We have extended the model by adding an extra charged scalar so that an extra U(1) global symmetry appears, and the stationary spinning vortex solution is allowed.
We find that the vortex profile has a nontrivial dependence on the angular velocity in the field space ω only in a limited region, and there is an upper limit on ω. Thus the vortex configuration should be parameterized by the angular momentum for the rotation in the extra-dimensional space, rather than ω. In contrast to the ANO vortex, the U(1) gauge symmetry is not restored at the core of the vortex due to the nonvanishing background of the second scalar φ 2 .
As in the static vortex case, fermion zero-modes are localized on the vortex by interactions with the vortex fields, and 4D chiral fermions can be easily realized. The number of such modes is determined by the vortex number n, and is independent of the U(1) charges in the presence of the Yukawa coupling to the vortex scalars.
Due to the spin of the vortex, the 4D Lorentz symmetry is violated in the effective theory. The dispersion relations of the fluctuation modes around the background deviate from the relativistic ones. On the other hand, when y 2 = 0, the localized fermion zeromodes do not feel the 4D Lorentz violation in the matter sector at tree level . Of course, even in such a case, the 4D Lorentz-violating effects will be induced by quantum corrections.
We will evaluate these effects in a separate paper.
There are many directions we should proceed in. We would like to generalize the situation by considering various kinds of vortices in various models, and extract universal properties of the spinning vortices. If we extend the model in a supersymmetric way, we can also discuss the SUSY-breaking effects in the 4D effective theory induced by the spin of a BPS vortex. The vortex in motion on the compact space is also an intriguing subject.
In this paper, we have only considered classical motion. However, the spinning vortex may radiate some particles by a quantum effect and lose the energy. In such a case, the vortex solution is no longer stationary, but the angular velocity will slow down, and the configuration will be reduced to be static. It is interesting to pursue this process and study how it affects the cosmological history. Besides, we have neglected the gravity in our analysis to simplify the discussion, but it is important to investigate effects of the spin on the 4D cosmological evolution in 6D gravitational theories. We will discuss these issues in subsequent papers.
A Vacuum structure of the model in Sec. 3 Here we summarize the vacuum structure of the model (3.1). 10 The minimization conditions of the potential U are By solving these, we find the following stationary points of U.
This solution is possible only when In order to investigate the stability of the vacua, we divide the complex scalar fields as and evaluate the Hessian matrix, (A.4) 10 See Ref. [18] for a similar setup.
For the stationary point 1, the Hessian is Thus this is a local maximum. In the stationary point 4, we choose without loss of generality. Then we obtain Thus, we have two massless modes, which correspond to the NG-modes for the breakings of the U(1) gauge and U(1) global symmetries. This stationary point is a local minimum iff det 4 λ 1 λ 2 − γ 2 is positive. Combining with the condition (A.2), this indicates that Λ 1 > 0 and Λ 2 > 0.
Namely, the point 2 or 3 and the point 4 cannot be local minima simultaneously. The potential value at this vacuum is 14) In the case that the points 2 and 3 are local minima, i.e., Λ 2 < 0 and Λ 1 < 0, we find that 15) which indicates that 16) In fact, the interaction parametrized by γ prevents both scalars φ 1 and φ 2 having nonvanishing VEVs.
When the conditions are satisfied, the point 2 becomes a global minimum of the potential.

B Notations
Here we collect the notations for the fermions. The 6D chirality matrix Γ 7 is defined as The 6D Weyl fermions Ψ ± are expressed by where the 4-component spinors ψ ± are further decomposed as