#
Higgs Phase in a Gauge U(1) Non-Linear CP^{1}-Model. Two Species of BPS Vortices and Their Zero Modes

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## Abstract

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## 1. Introduction

## 2. A Gauge $\mathbb{U}\left(1\right)$ Massive Non-Linear ${\mathbb{CP}}^{1}$-Sigma Model with Two Self-Dual Vortex Species

- The south vortex species: If the ${\mathbb{S}}^{2}$-valued $\mathsf{\varphi}$ field describing the vortex solution points downwards at the vortex center, ${\mathsf{\varphi}}_{3}\left(0\right)=-1$, only the south chart enters the game. The first-order ODE system (11) determining the south vortex species reads:$$\frac{d{f}_{n}^{S}\left(r\right)}{dr}=\pm \frac{n}{r}{f}_{n}^{S}\left(r\right)[1-{\beta}_{n}^{S}\left(r\right)]\phantom{\rule{14.22636pt}{0ex}},\phantom{\rule{14.22636pt}{0ex}}\frac{n}{r}\frac{d{\beta}_{n}^{S}}{dr}=\frac{1}{2}\left[{\mathsf{\alpha}}^{2}-\frac{4{\mathsf{\rho}}^{2}{\left({f}_{n}^{S}\left(r\right)\right)}^{2}}{{\mathsf{\rho}}^{2}+{\left({f}_{n}^{S}\left(r\right)\right)}^{2}}]\right]$$
- The north vortex species: If the vortex configuration in $\mathsf{\varphi}$-space valued at the origin points upwards, ${\mathsf{\varphi}}_{3}\left(0\right)=1$, only the north chart plays a role. The self-duality ODE system solved by the north vortex species corresponds to:$$\frac{d{f}_{n}^{N}\left(r\right)}{dr}=\pm \frac{n}{r}{f}_{n}^{N}\left(r\right)[1-{\beta}_{n}^{N}\left(r\right)]\phantom{\rule{14.22636pt}{0ex}},\phantom{\rule{14.22636pt}{0ex}}\frac{n}{r}\frac{d{\beta}_{n}^{N}}{dr}=\frac{1}{2}\left[4{\mathsf{\rho}}^{2}-{\mathsf{\alpha}}^{2}-\frac{4{\mathsf{\rho}}^{2}{\left({f}_{n}^{N}\left(r\right)\right)}^{2}}{{\mathsf{\rho}}^{2}+{\left({f}_{n}^{N}\left(r\right)\right)}^{2}}]\right]$$

## 3. Zero Mode Fluctuations around Cylindrically-Symmetric BPS Vortices of the Two Species

#### 3.1. Analytical Description of the Zero Modes of Fluctuation of Cylindrically-Symmetric BPS Vortices

- Regularity of the function ${s}_{nk}\left(r\right)$ at the origin: The Frobenius method can be applied to the linear differential Equation (33) at the regular singular point $r=0$. We expand the function ${s}_{nk}\left(r\right)$ as a power series:$${s}_{nk}\left(r\right)={r}^{s}\sum _{j=0}^{\infty}{c}_{j}^{(n,k)}{r}^{j}={r}^{s}{h}_{nk}\left(r\right)\phantom{\rule{22.76228pt}{0ex}},\phantom{\rule{22.76228pt}{0ex}}{h}_{nk}\left(r\right)=\sum _{j=0}^{\infty}{c}_{j}^{(n,k)}{r}^{j}$$$$\parallel \xi (\overrightarrow{x},n,k){\parallel}^{2}=2\pi \int dr\phantom{\rule{0.166667em}{0ex}}{r}^{2s+1}\left[{h}_{nk}^{2}\left(r\right)+\frac{{\left({h}_{nk}^{\prime}\left(r\right)\right)}^{2}}{g\left[{f}_{n}^{2}\left(r\right)\right]\phantom{\rule{0.166667em}{0ex}}{f}_{n}^{2}\left(r\right)}\right]$$$$\begin{array}{ccc}& & \sum _{j=0}^{2n+1}\left[(1+j+k-n+s)(1-j+k+n-2)g\left(0\right){c}_{j}^{(n,k)}\right]{r}^{j}+\hfill \\ & & +\sum _{j=2}^{2n+1}\left[-g\left(0\right)2n{e}_{2}(-1+j+k-n+s){c}_{j-2}^{(n,k)}\right]{r}^{j}+\hfill \\ & & \sum _{j=2n}^{2n+1}\left[{g}^{\prime}\left(0\right){d}_{n}^{2}\left((-1+4n)(s+j-2n)+(1+k-n)(1+k+3n)\right){c}_{j-2n}^{(n,k)}\right]{r}^{j}+\mathcal{O}\left({r}^{2n+2}\right)=0\hfill \end{array}$$$$g\left[{f}_{n}^{2}\left(r\right)\right]=g\left(0\right)+{g}^{\prime}\left(0\right)\phantom{\rule{0.166667em}{0ex}}{r}^{2n}\phantom{\rule{0.166667em}{0ex}}{\left(\sum _{\ell =0}^{\infty}{d}_{n+2\ell}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{r}^{2\ell}\right)}^{2}$$Notice that in (36), the recurrences are cut at order $2n+1$; we will see shortly that there is no need for taking into account more terms to ensure regularity at the origin and, henceforth, ${L}^{2}$-integrability, accounting for only the dominant terms near the vortex center.From hypothesis ${c}_{0}^{(n,k)}\ne 0$, the indicial Equation (37) with $j=0$ fixes the value of the characteristic exponents: $s=n-k-1$ and $s=n+k+1$. Both possibilities are equivalent: simply redefine k, $k\to -k-2$. Thus, we shall stick to the first option in the sequel. This choice of s in Equation (37) for the index $j=1$ implies that, necessarily, ${c}_{1}^{(n,k)}=0$. Near the origin, the first summand in the integrand of (35) (recall that $s=n-k-1$) is therefore:$${r}^{2(n-k)-1}{h}_{nk}^{2}\left(r\right)\simeq \left({c}_{0}^{(n,k)}\right){r}^{2(n-k)-1}+o\left({r}^{2(n-k)+1}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$$Poles at the origin in the integrand are skipped if:$$2(n-k)-1\ge 0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k\le n-1\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},$$The two-term recurrence relations for the next group of indices $j=3,\cdots ,2n-1$, and the characteristic exponent $s=n-k-1$ becomes:$$j(2k+2-j)\phantom{\rule{0.166667em}{0ex}}{c}_{j}^{(n,k)}=2n\phantom{\rule{0.166667em}{0ex}}{e}_{2}(j-2)\phantom{\rule{0.166667em}{0ex}}{c}_{j-2}^{(n,k)}$$Starting from ${c}_{1}^{(n,k)}=0$, it is easily checked that (39) implies ${c}_{2i+1}^{(n,k)}=0$ for all of the odd indices $j=2i+1$ in the range $3<2i+1\le 2n-1$. The recurrence (39) for even indices, however, $j=2i$ reads:$$i(k-i+1){c}_{2i}^{(n,k)}={e}_{2}(i-1)n{c}_{2i-2}^{(n,k)}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$$Insertion of the values $i=1,2,3,\cdots ,k$ in (40) means that all of the coefficients ${c}_{2}^{(n,k)}={c}_{4}^{(n,k)}=\cdots ={c}_{2k}^{(n,k)}=0$ vanish. ${c}_{2}^{(n,k)}$ is zero because the factor $i(k-i+1)$ appearing in the left-hand side of (40) is non-null, while $i-1$ present in the right-hand side is zero, for $i=1$. If $2\le i\le k$, a similar situation happens: all of the right side members in (40) are zero because the coefficients are zero, but the left-hand sides must be also zero, restricting the values of the coefficients up to ${c}_{2k}^{(n,k)}$ to be zero. The first non-null coefficient after ${c}_{0}^{(n,k)}$ is ${c}_{2k+2}^{(n,k)}$ because $k-i+1=0$ in this case. The first two terms of the ${s}_{nk}\left(r\right)$-power series expansion near $r=0$ are thus:$${h}_{nk}\left(r\right)={c}_{0}^{(n,k)}+{c}_{2k+2}^{(n,k)}\phantom{\rule{0.166667em}{0ex}}{r}^{2k+2}+O\left({r}^{2k+3}\right)$$$${r}^{2(n-k)-1}\frac{{\left({h}_{nk}^{\prime}\left(r\right)\right)}^{2}}{g\left[{f}_{n}^{2}\left(r\right)\right]\phantom{\rule{0.166667em}{0ex}}{f}_{n}^{2}\left(r\right)}={\left({c}_{2k+2}^{(n,k)}\right)}^{2}\frac{{(2k+2)}^{2}}{g\left(0\right)}{r}^{2k+1}+O\left({r}^{2k+3}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$$
- Asymptotic behavior of the function ${s}_{nk}\left(r\right)$: For large values of r, the modulus of the scalar complex field tends to a constant value $f\left(r\right)\to {f}_{\infty}>0$ that belongs to the vacuum circle $\mathcal{M}$, whereas the radial profile of the vector field tends to one: $\beta \left(r\right)\to 1$. Bearing this asymptotic behavior in mind, we see that at large r, the ODE equation (33) reduces to the modified Bessel differential equation:$$-{r}^{2}\frac{{d}^{2}{s}_{nk}\left(r\right)}{d{r}^{2}}-r\frac{d{s}_{nk}\left(r\right)}{dr}+\left[{(1+k-n)}^{2}+{r}^{2}\phantom{\rule{0.166667em}{0ex}}{f}_{\infty}^{2}\phantom{\rule{0.166667em}{0ex}}g\left({f}_{\infty}\right)\right]{s}_{nk}\left(r\right)=0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$$The general solution of this second-order ODE is well known:$$\begin{array}{ccc}\hfill {s}_{nk}\left(r\right)& \simeq & {C}_{1}\phantom{\rule{0.166667em}{0ex}}{I}_{1+k-n}\left[{f}_{\infty}\sqrt{g\left({f}_{\infty}\right)}\phantom{\rule{0.166667em}{0ex}}r\right]+{C}_{2}\phantom{\rule{0.166667em}{0ex}}{K}_{1+k-n}\left[{f}_{\infty}\sqrt{g\left({f}_{\infty}\right)}\phantom{\rule{0.166667em}{0ex}}r\right]\hfill \\ & \simeq & {\overline{C}}_{1}\phantom{\rule{0.166667em}{0ex}}\frac{1}{\sqrt{r}}\phantom{\rule{0.166667em}{0ex}}{e}^{{f}_{\infty}\sqrt{g\left({f}_{\infty}\right)}r}+{\overline{C}}_{2}\phantom{\rule{0.166667em}{0ex}}\frac{1}{\sqrt{r}}\phantom{\rule{0.166667em}{0ex}}{e}^{-{f}_{\infty}\sqrt{g\left({f}_{\infty}\right)}r}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$
- Intermediate regime: After describing analytically the eigenfunctions in the kernel of $\mathcal{D}$ near and far away from the vortex center, the Sturm–Liouville theory guarantees the existence of a regular solution at the origin $r=0$ of Equation (33) for every $k=0,1,\cdots ,n-1$, which has a decreasing exponential tail by simply tuning the values of the constants ${c}_{0}^{(n,k)}$ and ${c}_{2k+2}^{(n,k)}$ in order to obtain a solution with the adequate asymptotic behavior. In conclusion, there exists n zero modes $\xi (\overrightarrow{x};k)$ of the generic form (29) whose radial profiles ${s}_{nr}\left(r\right)$ and ${t}_{nr}\left(r\right)$ are solutions of the linear first-order ODE system (30). Moreover, all of these zero modes characterized by the wave number k are linearly independent. Integration in the angular variable shows that these eigenfunctions are orthogonal:$${\int}_{0}^{2\pi}\phantom{\rule{0.166667em}{0ex}}d\mathsf{\theta}\phantom{\rule{0.166667em}{0ex}}{\xi}^{T}(r,\mathsf{\theta};{k}_{1})\xb7\xi (r,\mathsf{\theta};{k}_{2})={\delta}_{{k}_{1}{k}_{2}}\xb7{F}^{T}(r;{k}_{1})F(r;{k}_{2})\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}$$Together with their corresponding orthogonal partners ${\xi}^{\perp}(\overrightarrow{x};k)$, this whole set of $2n$ zero modes forms a basis in the tangent space to the moduli space of BPS vortices.Sturm–Liouville theory is enough to ensure the existence of these null eigenfunctions in the intermediate range between a neighborhood of the origin and another one close to the infinite point. Nevertheless, there is no way of analytically finding the vortex solutions at intermediate range. It is possible, however, to gather good information about the BPS vortex zero mode profiles by using numerical methods. In this sense, it is better than directly attacking Equation (33) for ${s}_{nk}\left(r\right)$ simply to solve by numerical procedures the simpler equation in terms of the function ${h}_{nk}\left(r\right)$. Plugging:$${s}_{nk}\left(r\right)={r}^{n-k-1}{h}_{nk}\left(r\right)$$$$\begin{array}{ccc}& & \phantom{\rule{-14.22636pt}{0ex}}-rg\left[{f}_{n}^{2}\left(r\right)\right]\frac{{d}^{2}{h}_{nk}\left(r\right)}{d{r}^{2}}+\left[g\left[{f}_{n}^{2}\left(r\right)\right](1+2k-2n{\beta}_{n}\left(r\right))+2n{f}_{n}^{2}\left(r\right){g}^{\prime}\left[{f}_{n}^{2}\left(r\right)\right](1-{\beta}_{n}\left(r\right))\right]\frac{d{h}_{nk}\left(r\right)}{dr}+\hfill \\ & & \phantom{\rule{14.22636pt}{0ex}}+r{f}_{n}^{2}\left(r\right){\left[g\left({f}_{n}^{2}\left(r\right)\right)\right]}^{2}h\left(r\right)=0\hfill \end{array}$$

#### 3.2. Deformations of BPS Cylindrically-Symmetric Vortices of the Two Species by Their Zero Mode Fluctuations

## 4. Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BPS | Bogomolny–Prasad–Sommerfield |

AHM | Abelian Higgs model |

ANO | Abrikosov–Nielsen–Olesen |

PDE | Partial differential equation |

ODE | Ordinary differential equation |

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**Figure 1.**Cross-sections of the cylindrically-symmetric self-dual three-vortices in the south chart (

**a**,

**b**) and in the north chart (

**c**,

**d**). Graphics of the scalar field $\mathsf{\varphi}\left(\overrightarrow{x}\right)$ (

**a**,

**c**) and the vector field $V\left(\overrightarrow{x}\right)$ (

**b**,

**d**) profiles are depicted by means of superimposed Mathematica vector and density plots.

**Figure 2.**Graphical representations of the scalar and vector components of the south class three-vortex zero mode fluctuations ${\xi}^{S}(\overrightarrow{x},3,k)$ (displayed in the first and third columns, respectively) and the perturbed scalar and vector fields ${\tilde{\mathsf{\psi}}}^{S}(\overrightarrow{x},3,k)$ and ${\tilde{V}}^{S}(\overrightarrow{x},3,k)$ (displayed in the second and fourth columns, respectively) for the values $k=2$ (first row), $k=1$ (second row) and $k=0$ (third row).

**Figure 3.**Graphical representations of the scalar and vector components of the north class three-vortex zero mode fluctuations ${\xi}^{N}(\overrightarrow{x},3,k)$ (displayed in the first and third columns, respectively) and the perturbed scalar and vector fields ${\tilde{\mathsf{\psi}}}^{N}(\overrightarrow{x},3,k)$ and ${\tilde{V}}^{N}(\overrightarrow{x},3,k)$ (displayed in the second and fourth columns, respectively) for the values $k=2$ (first row), $k=1$ (second row) and $k=0$ (third row).

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Alonso-Izquierdo, A.; Mateos-Guilarte, J.
Higgs Phase in a Gauge U(1) Non-Linear CP^{1}-Model. Two Species of BPS Vortices and Their Zero Modes. *Symmetry* **2016**, *8*, 91.
https://doi.org/10.3390/sym8090091

**AMA Style**

Alonso-Izquierdo A, Mateos-Guilarte J.
Higgs Phase in a Gauge U(1) Non-Linear CP^{1}-Model. Two Species of BPS Vortices and Their Zero Modes. *Symmetry*. 2016; 8(9):91.
https://doi.org/10.3390/sym8090091

**Chicago/Turabian Style**

Alonso-Izquierdo, Alberto, and Juan Mateos-Guilarte.
2016. "Higgs Phase in a Gauge U(1) Non-Linear CP^{1}-Model. Two Species of BPS Vortices and Their Zero Modes" *Symmetry* 8, no. 9: 91.
https://doi.org/10.3390/sym8090091