Peristaltic modes of a single vortex in the Abelian Higgs model

Using the Abelian Higgs model, we study the radial excitations of single vortex and their propagation modes along the vortex line. We call such beyond-stringy modes peristaltic modes of single vortex. With the profile of the static vortex, we derive the vortex-induced potential, i.e., single-particle potential for the Higgs and the photon field fluctuations around the static vortex, and investigate the coherently propagating fluctuations which correspond to the vibration of the vortex. We derive, analyze, and numerically solve the field equations of the Higgs and the photon field fluctuations around the static vortex with various Ginzburg-Landau parameter $\kappa$ and topological charge $n$. Around the Bogomol’nyi-Prasad-Sommerfield value or critical coupling ${\kappa }^2=1/2$, there appears a significant correlation between the Higgs and the photon field fluctuations mediated by the static vortex. As a result, for ${\kappa }^2=1/2$, we find the characteristic new-type discrete pole of the peristaltic mode corresponding to the quasibound state of coherently fluctuating fields and the static vortex. We investigate its excitation energy, correlation energy of coherent fluctuations, spatial distributions, and the resulting magnetic flux behavior in detail. Our investigation covers not only usual type-II vortices with $n=1$ but also type-I and type-II vortices with $n\in Z$ for the application to various general systems where the vortexlike objects behave as the essential degrees of freedom.

Figure 1

The illustration for the peristaltic mode of the vortex. The axial-symmetric oscillation perpendicular to the vortex line propagates with the excitation of the Higgs and photon fields. The wavy solid line in the left side represents the surface where the amplitude of the Higgs field equals some constant value. The right-hand side represents the amplitude of the Higgs and the magnetic fields in the cross section perpendicular to the vortex line. Through this process, the conserved total magnetic flux repeatedly contracts and relaxes.

Figure 2

The profile for the Higgs field ${\varphi }_{\mathrm{cl}}(r)$, the magnetic field $H_z(r)$, and static field energy $\mathcal{E}(r)$ for ${\kappa }^2=0.1$ (type-I), ${\kappa }^2=0.5$ (BPS), ${\kappa }^2=1.0$ (type-II) in the case of $n=1$ vortex.

Figure 3

The plot for ${\kappa }^2$ dependence of the vortex mass divided by topological number $n$, $M_n/n$ for $n=1$, 2, 3 in the rescaled Abelian Higgs model in units of Eq. (2).

Figure 4

The plot for ${\kappa }^2$ dependence of the vortex mass divided by topological number $n$, $M_n/n$ for $n=1$, 2, 3 in the rescaled Abelian Higgs model in units of Eq. (2). The line with circles represents the mass of the $n=1$ vortex calculated with type-II limit (large ${\kappa }^2$) approximation in 1. The horizontal axis represents the log plot of ${\kappa }^2$ value in the much wider range than that in Fig. 3.

Figure 5

The plot of the vortex-induced potential $V_{\alpha \alpha }(r)$, $V_{\alpha \beta }(r)$, $V_{\beta \beta }(r)$ as the single-particle potential for fluctuations $\alpha (r)$ and $\beta (r)$ around single vortex, defined in Eq. (33). The Ginzburg-Landau parameter takes ${\kappa }^2=0.1$, 0.5, 1.0, and the topological number takes $n=1$.

Figure 6

${\kappa }^2$ dependence of $m_j^2$ for the $n=1$ case and the square of thresholds for $\alpha$ and $\beta$ fields, $4{\kappa }^2$ and 2, respectively. Around ${\kappa }^2=1/2$ (BPS value), as a collective mode of the Higgs and photon field fluctuations, there appear discrete poles in the lower-energy region below (square of) the continuum threshold $m_{\mathrm{th}}^2=\min (4{\kappa }^2,2)$.

Figure 7

The same plot as Fig. 6 except the replacement of $m_j^2$ with $m_j^{\prime 2}$, i.e., $m_j^2$ calculated in the case that the channel coupling between Higgs and photon field fluctuations is artificially dropped off. Without channel coupling, the bound state with ${\kappa }^2\simeq 1/2$ becomes the shallow bound state due to the lack of the interplay between the Higgs and photon field fluctuations.

Figure 8

The behavior of the low-lying mode of Higgs field fluctuation $\phi (r)\equiv r^{1/2}\alpha (r)$, the photon fluctuation $a_{\theta }(r)\equiv \sqrt{2}{r}^{1/2}\beta (r)$, and the magnetic field fluctuation $h_z(r)\equiv \frac{1}{r}\frac{d}{dr}(r{a}_{\theta })$ for $n=1$ vortex. For ${\kappa }^2=0.1$, 0.5 cases, we plot the lowest excitation mode with the radial mass $m_0$ smaller than the threshold $m_{\mathrm{th}}$. For the ${\kappa }^2=1.0$ case, where the discrete pole does not appear, we plot the typical low-lying excitation mode with the radial mass $m_j$, satisfying $2<m_j^2<4{\kappa }^2$.

Figure 9

The ${\kappa }^2$ dependence of $A^2\equiv {\int }_0^{\infty }dr\alpha (r{)}^2$, $B^2\equiv {\int }_0^{\infty }dr\beta (r{)}^2$ and $A·B\equiv {\int }_0^{\infty }dr\alpha (r)\beta (r)$ for the lowest excitation mode in the case of $n=1$.

Figure 10

Shown in the upper panel are the profiles for ${\varphi }_{\mathrm{cl}}(r)$, the magnetic field $H_z(r)$ and static field energy $\mathcal{E}(r)$ for $n=1$, 2, 3 vortices. The plots in the lower panel show the corresponding potential $V_{\alpha \alpha }(r)$, $V_{\alpha \beta }(r)$, $V_{\beta \beta }(r)$ for the fluctuations $\alpha$ and $\beta$. ${\kappa }^2$ is fixed to the BPS value, i.e., ${\kappa }^2=1/2$.

Figure 11

${\kappa }^2$ dependence of $m_j^2$ for the $n=2$ case and the square of thresholds for $\alpha$ and $\beta$ fields, $4{\kappa }^2$ and 2, respectively. The excitation energy is considerably lower than that in the $n=1$ case and the discrete pole appears up to considerably large ${\kappa }^2$.

Figure 12

The same plot as Fig. 11 except the replacement of the topological number $n$ from $n=2$ to $n=3$. The thresholds for $\alpha$ and $\beta$ are unchanged.