The ANO vortex is obtained by imposing the background ansatz,

where are the polar coordinates for the extra dimensions,

and the integer 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) , the dimensionless functions and 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).

In order to search for a spinning vortex solution, we extend the background ansatz (2.5) as

In this case, the Hamiltonian density (2.7) becomes

Thus, from the condition that the vortex tension should be finite and the regularity at the origin, the dimensionless functions , , and must satisfy the following boundary conditions:

Under the ansatz (2.10), the equations of motion (2.4) become

where is a dimensionless radial coordinate and

are dimensionless parameters.

In fact, Eq. (2.13) does not have a solution that satisfies the boundary conditions in Eq. (2.12). Let us focus on a region . There, the second term of the equation for is neglected and . Thus the solution behaves as

where is a real constant. When 0$$]]>, we find that 0$$]]> and for . Therefore,

is positive for all regions because the second term on the right-hand side, which is positive, gets bigger and bigger as we approach the origin while the contribution of the first term, which is negative, decreases. This indicates that is always negative for any value of . For a similar reason, is always positive when . In both cases, we cannot satisfy the boundary conditions at the origin.⁴ The only possible case is . In this case, is a solution that satisfies the boundary conditions in Eq. (2.12). However, this solution is gauge-equivalent to the static solution in Sect. 2.1. Namely, a stationary spinning vortex solution does not exist in this model.

This fact is expected from the following reason.⁵ The vacuum of this model is

where 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 is decomposed as . The phase part is the would-be Nambu–Goldstone (NG) boson and is absorbed by the gauge boson, and the radial part gets a mass from the potential. Namely, no massless modes exist in the vacuum. This indicates that nonzero energy is necessary when we move the vortex in any direction. So we cannot rotate the vortex without an 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.

We extend the previous model by adding an extra charged scalar field. The Lagrangian is given by

where , , , , and are positive constants, and

The constant 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 U(1) global symmetry, which is denoted as U(1), under the transformation that changes the relative phase of and .

The vacuum structure of this model is summarized in Appendix A. In the following, we will focus on the case that

Then the vacuum (i.e., the global minimum of ) is

We will set the constant so that the vacuum energy is zero in the following. Namely,

The equations of motion are

This theory has the (static) ANO vortex solution,

For the purpose of finding an axially symmetric stationary spinning vortex solution, we make the following ansatz for the background:⁶

where , , and are dimensionless real functions, the integer is the vortex number, and the real constant 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 Eq. (3.7) are translated into the equations for the dimensionless functions as

where

are dimensionless coordinate and parameters.

Due to the U(1), this model has a massless mode corresponding to the fluctuation changing the relative phase between and 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.

From the equations in Eq. (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 , they behave as

where , , , and are real constants.

Next we consider a region . Then, using Eq. (3.11), Eq. (3.13) is reduced to

where

The solution of the second equation is expressed by the (modified) Bessel function as

up to the normalization factor, where

Namely, when 0$$]]>, it behaves as

where is a positive constant. Using this and the last two equations in Eq. (3.16), we find that

where and are real constants. Then, from the first equation in Eq. (3.16) with the above asymptotic forms, we obtain

where is a real constant.

Thus, when is small enough, the Hamiltonian density (3.10) is approximated as

for . Therefore, the vortex tension diverges when . This indicates that there is a maximum value of the (normalized) angular velocity :⁷

The equations in Eq. (3.13) with the boundary conditions (3.11) and (3.12) can be solved numerically. Figure 1 shows the profiles of the solution. The solid, the dot-dashed, the dashed, and the dotted lines represent , , and , respectively. The parameters are chosen as , , , , , and . For , and are exponentially small, and the background is almost that of the ANO vortex (3.8). For , and grow as increases, and the profiles of and are deformed due to the centrifugal force induced by the spin of the vortex. For 0.61$$]]>, the functions do not decay enough in the region of , and cannot satisfy the boundary condition (3.11).

These behaviors of the functions can be understood by noticing that the centrifugal force is proportional to the angular momentum of the vortex, rather than the angular velocity in the field space . The angular momentum is given by

where is the Noether current for the rotation in the – plane. Figure 2 shows the relation between and . The overlap integrals in Eq. (3.25) are exponentially small for ; increases linearly with in the region , and it diverges at some value around . In fact, with our parameter choice, Eq. (3.24) is

These behaviors indicate that we should parametrize the vortex configuration by rather than by .

The 6D scalar field is decomposed into the KK modes as

where . We choose the mode functions as solutions of the following mode equation:

where

are dimensionless ( is the KK mass).

Since Eq. (4.6) has rotational symmetry in the extra dimensions, the eigenvalues in Eq. (4.6) have degeneracy and the mode functions can be expressed as

where is an integer and labels the degenerate modes. Then, the mode equation becomes

where

A more explicit derivation of Eqs. (4.8) and (4.9) is given in Appendix B. This has the form of a 1D Schrödinger equation with the potential .

We can obtain the KK mass spectrum by solving Eq. (4.9). However, it depends on many parameters and it is hard to express it analytically. Thus, we illustrate its property by analyzing the Schrödinger equation with the potential approximated by a simple function.

Let us first consider the static vortex case (i.e., ). The typical form of the potential in this case is shown by the left figure in Fig. 3. In order to see the properties of the spectrum in this system, we approximate by the following simple function up to a constant:

where the constants and denote the depth and width of the potential (see the right figure in Fig. 3). Thus, Eq. (4.9) is approximated by

where (: constant). We will concentrate on the bound-state solutions whose eigenvalues satisfy . Then, the solution of Eq. (4.12) is

for , and

for \rho_{\rm min}+W$$]]>. Here, and are the Bessel functions of the first and second kinds, and is the modified Bessel function of the second kind. The mode function and its derivative should be continuous at . In order for these conditions to satisfy with nonvanishing and }$$]]>, it must be satisfied that

where , and

Figure 4 shows the plots of

The points of denote the eigenvalues. The plots show the following properties:

Since the eigenvalue of Eq. (4.9) is the square of the (normalized) KK mass, it must be non-negative. In particular, we consider a case that the lowest-mass eigenvalue is zero , which can be achieved by tuning the bulk squared mass appropriately.⁸ This corresponds to the choice that the constant is chosen as the lowest eigenvalue .

We can numerically read off the depth and the width of the potential (see Fig. 3). Qualitatively, is a decreasing function of the penetration length of the vortex, which is read off from Eq. (3.21), and an increasing function of the correlation length, which is read off from Eq. (3.22). The width is a decreasing function of the penetration length while having a nontrivial dependence on the correlation length.

The spin of the vortex induces nonvanishing and . Both of them have support near the core of the vortex, and affect the shape of the potential . However, since the signs of their contributions to the potential in Eq. (4.10) are opposite, their effects on , , and depend on the parameters , , , and . In particular, the negative contribution in Eq. (4.10) indicates that there can be a negative solution, which indicates that the background configuration is unstable. This reflects the fact that the vortex configuration becomes unstable when exceeds the critical value mentioned in the previous section.

For 0$$]]>, Eq. (4.12) has a continuous spectrum, which corresponds to unbounded states. Thus, Eq. (4.5) should be understood as

where denotes the number of localized modes, and is the lowest value of the continuous KK spectrum.

Making use of the orthonormal relations of the mode functions, we can rewrite the linearized equation of motion (4.4) as

where

If we move to the momentum basis by the Fourier transformation, this is rewritten as

By diagonalizing the matrix on the left-hand side, we obtain the dispersion relation for each KK mode. When the angular momentum of the vortex is small, each element of is small. Then, the contribution from the off-diagonal elements of to the eigenvalues of the matrix in Eq. (4.21) is negligible, and the dispersion relation for is read off as

Thus, the energy is expressed as⁹

Therefore, we can identify the effective KK masses as

Even if a massless localized mode exists in the static vortex case, it will obtain a nonvanishing mass by the spin of the vortex. Furthermore, we should note that this effective mass depends on the KK label . This means that the degeneracy in the KK spectrum, which Eq. (4.6) has, is resolved by the spin.

The equations of motion for the fermions are

In the two-component spinor notation, these are rewritten as

where the two-component spinors and are defined in Appendix C.

Since the background (3.9) breaks the 4D Lorentz symmetry SO(1,3) to SO(3), we need not discriminate the dotted and undotted indices. Thus, the linearized equations of motion for the fermions are expressed as

We have used the polar coordinates for the extra dimensions.

Each component of the fermions is decomposed into the KK modes as

where is the dimensionless coordinate.¹⁰ The sums in the above expansion contain integrals over the continuous spectrum, just like in Eq. (4.18). We choose the mode functions as solutions of the following mode equations:

where

are dimensionless ( is the KK mass).

Note that when are solutions with the eigenvalue , the functions or become solutions with the eigenvalue . Thus, we label the KK modes in such a way that

for 0$$]]> ( is the integer in Eq. (3.9)), and

for . These are consistent with the fact that only one chiral component has zero-modes, which is explicitly shown in Appendix D.1.¹¹ Making use of Eq. (5.9) and performing the partial integrals, we can show that

which leads to

Thus, we normalize the mode functions so that

To make the discussion more specific, we focus on the case of in the following. Similar to the scalar case in the previous section, the eigenvalues in Eq. (5.9) have degeneracy, and we can separate the variables as

Then the mode equations in Eq. (5.9) are expressed as

The relations in Eq. (5.11) are translated into

Since and are normalized by Eq. (5.15), and satisfy

Making use of the orthonormal relations, we can rewrite Eq. (5.7) as the linearized equations for the KK modes:

where

From Eqs. (5.18) and (5.19), we can see that

Note also that the KK modes with different are decoupled from each other in the linearized equations of motion (5.20). Thus, Eq. (5.20) is rewritten as

where

and the matrices are defined by

and the column vectors are defined by

for 0$$]]>. Note that are Hermitian. Since Eq. (5.23) is rewritten as

we obtain

In the momentum basis, this becomes

or

where

and

By diagonalizing the matrix on the right-hand side of Eq. (5.30), we can obtain the dispersion relations.

Let us consider a case in which is sufficiently smaller than in order to see how the dispersion relation for the lowest mode is distorted by the spin of the vortex. As we have seen in Sect. 3.4, is exponentially small in such a case, and we can expand Eq. (5.30) in terms of the elements of the matrix . We can immediately see that the contributions from the off-diagonal elements in Eq. (5.30) to the eigenvalues are . Hence, at , the dispersion relation for the lowest mode is read off as

When , this is reduced to¹²

which indicates that the lowest mode has a tiny but nonvanishing mass . As we mentioned in the previous section, the degeneracy in the KK spectrum is resolved due to the -dependence of the effective masses.

When is close to , we have to take into account the mixing with the higher KK modes by diagonalizing the matrix in Eq. (5.30).

In the static limit (), the 4D Lorentz symmetry is recovered and Eq. (5.30) provides ordinary relativistic dispersion relations:

Hence, the lowest modes become massless and chiral. In this limit, either or vanish depending on the sign of , as shown in Appendix D.