Interactions of electrical and magnetic charges and dark topological defects

We consider a $\mathrm{U}(1)_{\mathrm{QED}}\times\mathrm{SU}(2)_{\mathrm{D}}$ gauge theory where the two sectors are coupled through higher dimensional operators:³³3We take the spacetime metric as $(g_{\mu\nu})=(-1,1,1,1)$.

	$\displaystyle\mathcal{L}$	$\displaystyle=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}{F^{a}_{\mathrm{D}}}_{\mu\nu}F_{\mathrm{D}}^{a\mu\nu}-\frac{1}{2}D_{\mu}\phi^{a}D^{\mu}\phi^{a}-\frac{1}{2}D_{\mu}\eta^{a}D^{\mu}\eta^{a}-V(\phi,\eta)+\mathcal{L}_{\theta}+\mathcal{L}_{\mathrm{mix}},$		(1)
	$\displaystyle\mathcal{L}_{\theta}$	$\displaystyle=-\frac{e^{2}\theta}{32\pi^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{e_{\mathrm{D}}^{2}\theta_{\mathrm{D}}}{32\pi^{2}}F_{\mathrm{D}\mu\nu}^{a}\tilde{F}_{\mathrm{D}}^{a\mu\nu}$		(2)
	$\displaystyle\mathcal{L}_{\mathrm{mix}}$	$\displaystyle=-\frac{c_{1}\phi^{a}}{2\Lambda}{F_{\mathrm{D}}}^{a}_{\mu\nu}F^{\mu\nu}-\frac{c_{2}\phi^{a}}{16\pi^{2}\Lambda}{F_{\mathrm{D}}}^{a}_{\mu\nu}\tilde{F}^{\mu\nu}\ .$		(3)

Here, $F_{\mu\nu}$ and $F^{a}_{\mathrm{D}\mu\nu}$ ($a=1,2,3$) are the field strengths of the $\mathrm{U(1)}_{\mathrm{QED}}$ and SU$(2)_{\mathrm{D}}$ gauge fields, $A_{\mu}$ and $A^{a}_{\mathrm{D}\mu}$, respectively. Their hodge duals are given by $\tilde{F}_{(\mathrm{D})\mu\nu}=\epsilon_{\mu\nu\rho\sigma}F_{(\mathrm{D})}^{\rho\sigma}/2$.⁴⁴4We adopt the convention $\epsilon_{0123}=1$. The three dimensional anti-symmetric tensor is $\epsilon_{ijk}=\epsilon^{ijk}=\epsilon_{0ijk}=\epsilon_{0}{}^{ijk}$. We also define electromagnetic fields as $E^{i}=F^{0i}=F_{i0}$ and $B^{i}=\epsilon^{ijk}F^{jk}/2$. We introduced two SU$(2)_{\mathrm{D}}$ adjoint scalar fields $\phi^{a}$ and $\eta^{a}$ ($a=1,2,3$). We call the $\mathrm{U(1)}_{\mathrm{QED}}$ and $\mathrm{SU}(2)_{\mathrm{D}}$ gauge coupling constants $e$ and $e_{\mathrm{D}}$. The covariant derivatives of $\phi$ and $\eta$ are given by,

	$\displaystyle D_{\mu}\phi^{a}$	$\displaystyle=\partial_{\mu}\phi^{a}+e_{\mathrm{D}}\epsilon^{abc}A_{\mathrm{D}\mu}^{b}\phi^{c}\ ,$		(4)
	$\displaystyle D_{\mu}\eta^{a}$	$\displaystyle=\partial_{\mu}\eta^{a}+e_{\mathrm{D}}\epsilon^{abc}A_{\mathrm{D}\mu}^{b}\eta^{c}\ .$		(5)

The higher dimensional operators with coefficients $c_{1,2}$ suppressed by the UV cutoff $\Lambda$ result in effective mixing parameters between QED photons and dark photons [19]. We take $\Lambda\gg v_{1}$, so that the effective mixing parameters are small. Throughout this paper, we assume that no SU(2)_D charged fields have U(1)_QED charge, although our discussion can be generalized.

The scalar potential of $\phi^{a}$ and $\eta^{a}$ is assumed to be

$$V(\phi,\eta)=\frac{\lambda_{1}}{4}(\phi\cdot\phi-v_{1}^{2})^{2}+\frac{\lambda_{2}}{4}(\eta\cdot\eta-v_{2}^{2})^{2}+\frac{\kappa}{2}(\phi\cdot\eta)^{2}\ ,$$

(6)

where $\phi\cdot\phi=\phi^{a}\phi^{a}$ etc. For simplicity, we omit terms such as $(\phi\cdot\phi)(\eta\cdot\eta)$. The dimensionless coupling constants $\lambda_{1},\lambda_{2}$ and $\kappa$ are taken to be positive. We also take the mass scales to be hierarchical, i.e., $v_{1}\gg v_{2}$. At the vacuum, $\phi^{a}$ takes the trivial configuration, i.e. the vacuum expectation value (VEV),

$$\langle{\phi^{a}}\rangle=v_{1}\delta^{a3}\ ,$$

(7)

with which $\mathrm{SU}(2)_{\mathrm{D}}$ is broken down to $\mathrm{U}(1)_{\mathrm{D}}$. The remaining $\mathrm{U}(1)_{\mathrm{D}}$ symmetry corresponds to the SO(2) symmetry around the $a=3$ axis of $\mathrm{SO}(3)\simeq\mathrm{SU}(2)_{\mathrm{D}}$ vectors $\phi^{a}$ and $\eta^{a}$.

Below the $\mathrm{SU}(2)_{\mathrm{D}}$ breaking scale, a U(1)_D charged field $\chi$ can be formed out of $\eta^{a}$ as

$$\chi=\frac{1}{\sqrt{2}}(\eta^{1}-i\eta^{2})\ .$$

(8)

For $\kappa>0$, the last term of the potential (6) lifts the $a=3$ component of $\eta$ and the VEV of $\eta^{a}$ is required to be orthogonal to $\langle\phi^{a}\rangle$. As a result, $\langle\eta^{a}\rangle$ takes a value in the $(\eta^{1},\eta^{2})$ plane, i.e.,

$$\langle\eta^{a}\rangle=v_{2}\delta^{a1}\ ,$$

(9)

or

$$\langle\chi\rangle=\frac{1}{\sqrt{2}}v_{2}\ ,$$

(10)

which breaks $\mathrm{U}(1)_{\mathrm{D}}$ spontaneously. In this way, successive symmetry breaking $\mathrm{SU}(2)_{\mathrm{D}}\to\mathrm{U}(1)_{\mathrm{D}}\to\mathbb{Z}_{2}$ is achieved. The $\mathbb{Z}_{2}$ symmetry is the center of $\mathrm{SU}(2)_{\mathrm{D}}$.

For later use, we describe the effective U(1)${}_{\mathrm{QED}}\times$ U(1)_D theory for $\expectationvalue{\phi}\neq 0$. The effective Lagrangian is given by

	$\displaystyle\mathcal{L}=$	$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}{F_{\mathrm{D}}}_{\mu\nu}F_{\mathrm{D}}^{\mu\nu}+\frac{\epsilon}{2}F_{\mu\nu}F_{\mathrm{D}}^{\mu\nu}-\frac{\theta_{\mathrm{mix}}}{16\pi^{2}}F_{\mu\nu}\tilde{F}_{\mathrm{D}}{}^{\mu\nu}-\frac{e^{2}\theta}{32\pi^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{e_{\mathrm{D}}^{2}\theta_{\mathrm{D}}}{32\pi^{2}}F_{\mathrm{D}\mu\nu}\tilde{F}_{\mathrm{D}}^{\mu\nu}$		(11)
		$\displaystyle+eA_{\mu}J_{\mathrm{QED}}^{\mu}+e_{\mathrm{D}}A_{\mathrm{D}\mu}J_{\mathrm{D}}^{\mu}-D_{\mu}\chi D^{\mu}\chi^{*}-V(\chi)\ ,$		(12)

where $F_{\mathrm{D}\mu\nu}=\phi^{a}F^{a}_{\mathrm{D}\mu\nu}/v_{1}$ represents the $\mathrm{U(1)}_{\mathrm{D}}$ gauge field strength and $A_{\mathrm{D}\mu}$ the corresponding gauge field. We call the gauge field $A_{\mathrm{D}\mu}$ as the dark photon. Note that in the presence of monopoles/dyons, the effective theory is well-defined only far enough from them (so that $\absolutevalue{\phi}=v_{1}$) and $A_{\mathrm{D}\mu}$ can be defined only locally. We also explicitly displayed the currents $J_{\mathrm{QED}}^{\mu}$ and $J_{\mathrm{D}}^{\mu}$ coupled to the gauge fields, which were omitted in Eq. (1).

We refer to the interactions with the couplings $\epsilon$ and $\theta_{\mathrm{mix}}$ as the kinetic and magnetic mixing. They arise from the higher dimensional operators (3), where the couplings are related to the underlying model parameters by

$$\epsilon=\frac{c_{1}v_{1}}{\Lambda}\ ,\quad\theta_{\mathrm{mix}}=\frac{c_{2}v_{1}}{\Lambda}\ .$$

(13)

As we assume $\Lambda\gg v_{1}$, these parameters are tiny.⁵⁵5 The parameter $\theta_{\mathrm{mix}}$ is related to $\theta_{12}$ in Ref. [19] via $\theta_{12}=ee_{\mathrm{D}}\theta_{\mathrm{mix}}$.

In the effective U$(1)_{\mathrm{D}}$ theory, only $\chi$ is relevant as the other components become heavy for $\kappa>0$. The covariant derivative of $\chi$ is given by

$$D_{\mu}\chi=(\partial_{\mu}-ie_{\mathrm{D}}A_{\mathrm{D}\mu})\chi\ .$$

(14)

The scalar potential $V(\chi)$ is obtained by substituting Eqs. (7), (8), and (9) into Eq. (6):

$$V(\chi)=\frac{\lambda}{4}(|\chi^{2}|-v^{2})^{2}\ ,$$

(15)

where $\lambda=\lambda_{2}/2$ and $v=v_{2}/\sqrt{2}$. At the vacuum, $\chi$ obtains a VEV $\langle\chi\rangle=v$, which spontaneously breaks the $\mathrm{U}(1)_{\mathrm{D}}$ symmetry, as in the previous subsection.

Let us consider the effective U$(1)_{\mathrm{QED}}\times$U$(1)_{\mathrm{D}}$ theory (11) assuming the trivial vacuum (7) with charged particles in $J_{\mathrm{QED}}^{\mu}$ and $J_{\mathrm{D}}^{\mu}$. The equations of motion for the field strengths can be written as

$$\mathcal{K}\partial_{\mu}\mathcal{F}^{\mu\nu}=-\mathcal{J}^{\nu}\ ,$$

(16)

where

$\displaystyle\mathcal{A}^{\mu}:=\pmqty{undef},\,\,\mathcal{F}^{\mu\nu}:=\partial^{\mu}\mathcal{A}^{\nu}-\partial^{\nu}\mathcal{A}^{\mu},\,\,\mathcal{K}:=\pmqty{undef},\,\,\mathcal{J}^{\mu}:=\pmqty{undef}.$

(17)

Note that $\theta_{\mathrm{mix}}$ does not appear here, as the magnetic mixing is a total derivative in the effective theory. For a point charge, $\mathcal{J}^{\mu}(x)=\mathcal{Q}\delta^{\mu}_{0}\delta^{3}(\mathbf{x})$ where $\mathcal{Q}=\pqty{e{n^{e}_{\mathrm{QED}}},e_{\mathrm{D}}{n^{e}_{\mathrm{D}}}}^{\top}$.⁶⁶6In the dark photon model in Sec. II.1, we assume no SU(2)_D charged fields have U(1)_QED charge, and hence, $n_{\mathrm{QED}}^{e}=0$ or $n_{\mathrm{D}}^{e}=0$ in the basis of Eq. (11). However, the interaction energy can be defined for more general cases. The static solution in the Coulomb gauge, $\nabla\cdot\vec{\mathcal{A}}=0$, is

$$\mathcal{A}^{0}=\frac{1}{4\pi r}\mathcal{K}^{-1}\mathcal{Q},\quad\vec{\mathcal{A}}=0\ ,$$

(18)

where $r$ denotes the distance from the point charge. Therefore, the electric potential energy between two point charges $\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ is given by

$$E_{\mathrm{int}}:=\mathcal{Q}^{\prime\top}\int_{r}^{\infty}\differential{x}^{i}\mathcal{F}^{0i}=\mathcal{Q}^{\prime\top}\mathcal{A}^{0}=\frac{1}{4\pi r}\mathcal{Q}^{\prime\top}\mathcal{K}^{-1}\mathcal{Q}\ ,$$

(19)

where $r$ is the distance between the charges. Here, the electric field is defined by $\mathcal{E}^{i}=\mathcal{F}^{0i}=-\mathcal{F}_{0i}$.

To see the effect of the kinetic mixing on the electric potential energy, let us first consider the case of two QED electric charges. Plugging in $\mathcal{Q}=(e{n^{e}_{\mathrm{QED}}},0)^{\top}$ and $\mathcal{Q}^{\prime}=(e{n^{e\,\prime}_{\mathrm{QED}}},0)^{\top}$, Eq. (19) leads to

$$E_{\mathrm{int}}=\frac{e^{2}}{1-\epsilon^{2}}\times\frac{{n^{e}_{\mathrm{QED}}}{n^{e\,\prime}_{\mathrm{QED}}}}{4\pi r}\ .$$

(20)

This is familiar Coulomb’s law, except that $e^{2}$ is replaced with $e^{2}/(1-\epsilon^{2})$. This deviation is due to the interaction between QED charges via dark photon exchange.

For a dark electric charge and a QED test particle, i.e., $\mathcal{Q}=(0,e_{\mathrm{D}}{n^{e}_{\mathrm{D}}})^{\top}$ and $\mathcal{Q}^{\prime}=(e{n^{e}_{\mathrm{QED}}},0)^{\top}$, we have

$$E_{\mathrm{int}}=\frac{\epsilon ee_{\mathrm{D}}}{1-\epsilon^{2}}\times\frac{{n^{e}_{\mathrm{QED}}}{n^{e}_{\mathrm{D}}}}{4\pi r}\ .$$

(21)

Physically, this indicates that the QED test charged particle feels Coulomb force from the dark electric charged particle as if it has QED electric charge $\epsilon{n^{e}_{\mathrm{D}}}e_{\mathrm{D}}/e$.

Note that the definition of the charges depends on the basis of the U(1) gauge fields. That is, the redefinition $\mathcal{A}\to\mathcal{S}\mathcal{A}$ with a 2$\times 2$ regular matrix $\mathcal{S}$ transforms $\mathcal{Q}$ to $\mathcal{S}^{-1\top}\mathcal{Q}$. The interaction energy $E_{\text{int}}$ is, on the other hand, independent of the basis, since it is a physical observable. Indeed, the field redefinition also changes $\mathcal{K}$ to $\mathcal{S}^{-1\top}\mathcal{K}\mathcal{S}^{-1}$, and hence, the interaction energy (19) is intact.

Next, we move on to the case with dark magnetic monopoles. At the phase transition, SU$(2)_{\mathrm{D}}\to\text{U}(1)_{\mathrm{D}}$, the ’t Hooft-Polyakov monopole can appear [25, 26]. In the absence of the kinetic and magnetic mixing terms, the static configuration of the monopole at the origin is given by

$\displaystyle\phi^{a}=v_{1}H(r)\displaystyle{\frac{x^{a}}{r}}\ ,\quad A_{\mathrm{D}0}^{a}=0\ ,\quad A^{a}_{\mathrm{D}i}=\frac{1}{e_{\mathrm{D}}}\displaystyle{\frac{\epsilon^{aij}x^{j}}{r^{2}}}F(r)\ ,\quad(i,j=1,2,3)\ ,$

(22)

where $r=\sqrt{x^{2}+y^{2}+z^{2}}$. The profile functions $H(r)$ and $F(r)$ satisfy the boundary conditions

	$\displaystyle H(r)$	$\displaystyle\to\text{const.}\times r\ ,\,(r\to 0)\ ,$	$\displaystyle\qquad H(r)$	$\displaystyle\to 1\ ,\,(r\to\infty)\ ,$		(23)
	$\displaystyle F(r)$	$\displaystyle\to\text{const.}\times r^{2}\ ,\,(r\to 0)\ ,$	$\displaystyle\qquad F(r)$	$\displaystyle\to 1\ ,\,(r\to\infty)\ ,$		(24)

where they approach their asymptotic values exponentially at $r\to\infty$.

To see the magnetic field, it is convenient to define the effective U(1)_D field strength as

$$F_{\mathrm{D}\mu\nu}:=\frac{1}{v_{1}}\phi^{a}F^{a}_{\mathrm{D}\mu\nu}$$

(25)

(see e.g. Ref. [27]). The only non-vanishing components of $F_{\mathrm{D}}^{\mu\nu}$ are

$$F_{\mathrm{D}}^{ij}=-\frac{1}{e_{\mathrm{D}}}\frac{\epsilon^{ijk}x^{k}}{r^{3}}(2F-F^{2})H\ ,\quad(i,j=1,2,3)\ .$$

(26)

Hence, the dark magnetic charge of the monopole solution is given by

$$Q_{\mathrm{D}}^{m}:=\int_{r\to\infty}\differential[2]S_{i}B_{\mathrm{D}}^{i}=-\frac{4\pi}{e_{\mathrm{D}}}\ ,$$

(27)

where $\differential[2]{S}_{i}$ is the surface element of a two dimensional sphere surrounding the monopole.

Now, let us consider the effect of the kinetic and magnetic mixings. As we assume those parameters to be tiny, their effects on the configuration (22) can be safely neglected. (For the stability of the topological defects in the presence of the mixing terms, see the Appendix B.) The equation of motion for $A_{\mu}$ in the $\mathrm{U(1)}_{\mathrm{QED}}\times\mathrm{SU}(2)_{\mathrm{D}}$ theory is

$$\partial_{\mu}F^{\mu\nu}-\epsilon\partial_{\mu}F_{\mathrm{D}}^{\mu\nu}+\frac{\theta_{\mathrm{mix}}}{8\pi^{2}}\partial_{\mu}\tilde{F}_{\mathrm{D}}^{\mu\nu}=0\ .$$

(28)

The third term vanishes at $r\gg(e_{\mathrm{D}}v_{1})^{-1}$ due to the Bianchi identity of the effective U(1)_D theory. In the vicinity of the monopole $r\sim\order{(e_{\mathrm{D}}v_{1})^{-1}}$, on the other hand, it does not vanish where

$$\partial_{\mu}\tilde{F}_{\mathrm{D}}^{\mu\nu}=\frac{1}{v_{1}}\partial_{\mu}(\phi^{a}\tilde{F}^{a\mu\nu}_{\mathrm{D}})=\frac{1}{v_{1}}(D_{\mu}\phi)^{a}\tilde{F}^{a\mu\nu}_{\mathrm{D}}\neq 0\ .$$

(29)

In the last equality, we used the Bianchi identity of SU$(2)_{\mathrm{D}}$, i.e., $D_{\mu}\tilde{F}_{\mathrm{D}}^{a\mu\nu}=0$. Besides, the effective field strength $F_{\mathrm{D}}^{\mu\nu}$ satisfies $\partial_{\mu}F_{\mathrm{D}}^{\mu\nu}=0$ even at $r\to 0$, and hence, the second term in Eq. (28) vanishes.

As a result, the equation of motion for the QED electric field is given by

$$\partial_{i}E^{i}=\frac{\theta_{\mathrm{mix}}}{8\pi^{2}}\partial_{i}B_{\mathrm{D}}^{i}.$$

(30)

Thus, we find the solution of Eq. (28) in the Coulomb gauge to be

$$A^{0}\simeq-\frac{\theta_{\mathrm{mix}}Q_{\mathrm{D}}^{m}}{8\pi^{2}}\times\frac{1}{4\pi r}=\frac{\theta_{\mathrm{mix}}}{8\pi^{2}e_{\mathrm{D}}}\times\frac{1}{r}\ ,\quad\vec{A}=0\ ,$$

(31)

for $r\gg(e_{\mathrm{D}}v_{1})^{-1}$ to the leading order of the mixing parameters.

Accordingly, the interaction energy between a QED test particle with $\mathcal{Q}=({n^{e}_{\mathrm{QED}}},0)^{\top}$ and a dark monopole is given by,

$$E_{\mathrm{int}}=-\frac{e\theta_{\mathrm{mix}}Q_{\mathrm{D}}^{m}{n^{e}_{\mathrm{QED}}}}{8\pi^{2}}\times\frac{1}{4\pi r}\ ,$$

(32)

for $r\gg(e_{\mathrm{D}}v_{1})^{-1}$ to the leading order of the mixing parameters. This shows that the dark magnetic monopole exerts Coulomb force to QED particles through the magnetic mixing, whereas the kinetic mixing induces no interactions between them [19].

The $\mathrm{SU}(2)_{\mathrm{D}}$ sector admits dyons, magnetic monopoles that also have electric charge [28]. The dyon solution is described by Eq. (22) but with $A_{\mathrm{D}0}^{a}$ replaced by

$$A^{a}_{\mathrm{D}0}=\frac{1}{e_{\mathrm{D}}}\frac{x^{a}}{r^{2}}J(r)\ .$$

(33)

The boundary conditions for $J(r)$ are

$$J(r)\to\text{const.}\times r^{2}\ ,\,(r\to 0)\ ,\qquad J(r)\to Mr+b\ ,\,(r\to\infty)\ ,$$

(34)

where $M$ and $b$ are the parameters with mass dimensions one and zero, respectively.

The dark magnetic field $F_{\mathrm{D}ij}$ is not modified by Eq. (33). On the other hand, the dark electric field no longer vanishes:

$$F_{\mathrm{D}}^{0i}=\frac{1}{e_{\mathrm{D}}}\frac{x^{i}}{r}\derivative{r}\frac{J(r)}{r}\to-\frac{b}{e_{\mathrm{D}}}\frac{x^{i}}{r^{3}}\ .$$

(35)

Hence, the dark electric charge of the dyon is found to be

$${Q^{e}_{\mathrm{D}}}=-\frac{4\pi b}{e_{\mathrm{D}}}=bQ^{m}_{\mathrm{D}}$$

(36)

in the absence of the mixing to the QED sector.

By remembering how the dark electric charges and dark magnetic charges induce the Coulomb force on QED charged particles (see Eq. (28)), we find the interaction energy to be

$$E_{\mathrm{int}}=e{n^{e}_{\mathrm{QED}}}\pqty{-\frac{\theta_{\mathrm{mix}}}{8\pi^{2}}Q_{\mathrm{D}}^{m}+\epsilon{Q^{e}_{\mathrm{D}}}}\times\frac{1}{4\pi r}\ ,$$

(37)

to the leading order in the mixing parameters.

This concludes our analysis on the interactions between dark charge objects and QED charges in the $\mathrm{U(1)}_{\mathrm{D}}$ symmetric phase. Fig. 1 summarizes the results in this section.

The dark magnetic charge is quantized as it corresponds to the topological number $n_{\mathrm{D}}^{m}\in\mathbb{Z}$ of the configuration, with which $Q^{m}_{\mathrm{D}}=-4\pi n_{\mathrm{D}}^{m}/e_{\mathrm{D}}$. Its quantization is not affected by the mixings to the QED sector.

The dark electric charge is arbitrary at the classical level, as in Eq. (36). In a quantum theory, however, the dyon electric charge has to be quantized [29, 30]. To see this in our setup, let us consider the residual global $\mathrm{U(1)}_{\mathrm{D}}$ symmetry around $\phi$,

$\displaystyle\delta A^{a}_{\mathrm{D}\mu}=-\frac{1}{e_{\mathrm{D}}v_{1}}D_{\mu}\phi^{a}\ ,\quad\delta A_{\mu}=0\ ,\quad\delta\phi^{a}=0\ .$

(38)

As shown in Appendix A, the corresponding Noether charge is given by

$$N_{\text{U(1)}_{\mathrm{D}}}=\frac{1}{e_{\mathrm{D}}}{Q^{e}_{\mathrm{D}}}-\frac{\epsilon}{e_{\mathrm{D}}}{Q^{e}_{\mathrm{QED}}}-\frac{\theta_{\mathrm{D}}e_{\mathrm{D}}}{8\pi^{2}}Q^{m}_{\mathrm{D}}\ .$$

(39)

The electric and magnetic charges are measured by electric flux,

$\displaystyle\pmqty{undef}:=\int\differential[2]{S}_{i}\mathcal{E}^{i}\ ,$

(40)

and the magnetic flux (see Eq. (27)). Since $N_{\mathrm{U(1)}_{\mathrm{D}}}$ is one of the generators of global SO(3)${}_{\mathrm{D}}\simeq\,\,$SU(2)_D transformation, we find $N_{\text{U(1)}_{\mathrm{D}}}\in\mathbb{Z}$, which constrains ${Q^{e}_{\mathrm{D}}}$ of dyons [31, 32] (see also Ref. [33]). Note that this is the usual Witten effect in the absence of the mixings.

Let us also comment on the effects of the $\theta$-terms $\mathcal{L}_{\theta}$ to the equations of motion. In our formulation, the $\mathrm{U(1)}_{\mathrm{QED}}\times$SU(2)_D gauge potentials $A_{\mu}$ and $A_{\mathrm{D}\mu}^{a}$ are globally defined, and hence, $\mathcal{L}_{\theta}$ does not affect the equations of motion. In the U$(1)_{\mathrm{QED}}\times$U$(1)_{\mathrm{D}}$ formulation, on the other hand, it is also possible to introduce monopoles as a singularity [19]. In this treatment, $\mathcal{L}_{\theta}$ classically induces an electric field around a dark monopole (see also Ref. [34]).⁷⁷7 Strictly speaking, singularities in the dark sector obscure the boundary condition of the QED gauge potential. Our treatment based on the $\mathrm{U(1)}_{\mathrm{QED}}\times\mathrm{SU}(2)_{\mathrm{D}}$ theory does not have such subtleties.

Let us consider the case without monopoles, where the effective theory (11) is valid. At the trivial vacuum (10), the $\mathrm{U(1)}_{\mathrm{D}}\times\mathrm{U(1)}_{\mathrm{QED}}$ model is reduced to

	$\displaystyle\mathcal{L}=$	$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}{F_{\mathrm{D}}}_{\mu\nu}F_{\mathrm{D}}^{\mu\nu}-\frac{1}{2}m_{\mathrm{D}}^{2}A_{\mathrm{D}\mu}A^{\mu}_{\mathrm{D}}+\frac{\epsilon}{2}F_{\mu\nu}F_{\mathrm{D}}^{\mu\nu}$		(41)
		$\displaystyle-\frac{\theta_{\mathrm{mix}}}{16\pi^{2}}F_{\mu\nu}\tilde{F}_{\mathrm{D}}{}^{\mu\nu}-\frac{e^{2}\theta}{32\pi^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{e_{\mathrm{D}}^{2}\theta_{\mathrm{D}}}{32\pi^{2}}F_{\mathrm{D}\mu\nu}\tilde{F}_{\mathrm{D}}^{\mu\nu}$		(42)
		$\displaystyle+eA_{\mu}J_{\mathrm{QED}}^{\mu}+e_{\mathrm{D}}A_{\mathrm{D}\mu}J_{\mathrm{D}}^{\mu}\ ,$		(43)

where $m_{\mathrm{D}}^{2}=2e_{\mathrm{D}}^{2}v^{2}$.

In this case, it is most convenient to introduce a new basis

$\displaystyle\pmqty{undef}=:\pmqty{undef}\pmqty{undef}\ ,$

(44)

with which the equations of motion are given by

		$\displaystyle\partial_{\mu}F^{\prime\mu\nu}=eJ_{\mathrm{QED}}^{\nu}\ ,$		(45)
		$\displaystyle\partial_{\mu}F_{\mathrm{D}}^{\prime\mu\nu}-m_{\mathrm{D}}^{\prime 2}A_{\mathrm{D}}^{\prime\nu}=\frac{e_{\mathrm{D}}}{\sqrt{1-\epsilon^{2}}}J_{\mathrm{D}}^{\nu}+\frac{\epsilon e}{\sqrt{1-\epsilon^{2}}}J_{\mathrm{QED}}^{\nu}\ ,$		(46)

where $m_{\mathrm{D}}^{\prime 2}=m_{\mathrm{D}}^{2}/(1-\epsilon^{2})$. We refer to the bases $(A_{\mu},A_{\mathrm{D},\mu})$ and $(A^{\prime}_{\mu},A_{\mathrm{D}\mu}^{\prime})$ the original and decoupled bases, respectively.

Then the interaction energy between a dark electric charge and a QED test particle, i.e., $\mathcal{Q}=(0,e_{\mathrm{D}}{n^{e}_{\mathrm{D}}})^{\top}$ and $\mathcal{Q}^{\prime}=(e{n^{e}_{\mathrm{QED}}},0)^{\top}$ in the original basis, is suppressed by $e^{-m_{\mathrm{D}}^{\prime}r}$:

$$E_{\mathrm{int}}=\frac{\epsilon ee_{\mathrm{D}}}{1-\epsilon^{2}}\times\frac{{n^{e}_{\mathrm{QED}}}{n^{e}_{\mathrm{D}}}}{4\pi r}e^{-m_{\mathrm{D}}^{\prime}r}\ .$$

(47)

Note that the $\theta$-terms in Eq. (11) has no observable effect in this case.

Let us continue to assume the absence of monopoles. However, we now consider the vacuum configuration of $\mathrm{U(1)}_{\mathrm{D}}$ breaking associated with a string as discussed in Ref. [18]. We continue to use the decoupled basis. The static string solution along the $z$-axis is given by the form (see e.g., Ref. [35])

	$\displaystyle\chi$	$\displaystyle=vh(\rho)e^{in\varphi_{A}}\ ,$		(48)
	$\displaystyle A^{\prime}_{\mathrm{D}i}$	$\displaystyle=-\frac{n}{e_{\mathrm{D}}^{\prime}}\frac{\epsilon_{ij}x^{j}}{\rho^{2}}f(\rho)\ ,~{}~{}~{}~{}(i,j=1,2)\ ,$		(49)
	$\displaystyle A^{\prime}_{\mathrm{D}0}$	$\displaystyle=A^{\prime}_{\mathrm{D}3}=0\ ,$		(50)

where $n\in\mathbb{Z}$ is the winding number of the string configuration, $h(\rho)$, $f(\rho)$ the profile functions, and $e^{\prime}_{\mathrm{D}}=e_{\mathrm{D}}/\sqrt{1-\epsilon^{2}}$. The cylindrical coordinate is given by $\varphi_{A}={\arctan}(y/x)$ and $\rho=\sqrt{x^{2}+y^{2}}$. The two-dimensional anti-symmetric tensor is defined by $\epsilon_{12}=1$.⁸⁸8Noting that $\differential{\varphi_{A}}=-\differential{x}^{i}\epsilon_{ij}x^{j}/\rho^{2}$, Eq. (49) can be rewritten by $A^{\prime}_{\mathrm{D}i}\differential{x}^{i}=n/e_{\mathrm{D}}^{\prime}\times f(\rho)\differential{\varphi_{A}}$. The boundary conditions for the profile functions are

	$\displaystyle h(\rho)\rightarrow 0\ ,~{}(\rho\rightarrow 0)$	$\displaystyle\ ,~{}~{}~{}~{}~{}h(\rho)\rightarrow 1\ ,~{}(\rho\rightarrow\infty)\ ,$		(51)
	$\displaystyle f(\rho)\rightarrow 0\ ,~{}(\rho\rightarrow 0)$	$\displaystyle\ ,~{}~{}~{}~{}~{}f(\rho)\rightarrow 1\ ,~{}(\rho\rightarrow\infty)\ .$		(52)

They approach unity for $\rho\gg(e^{\prime}_{\mathrm{D}}v)^{-1}$ exponentially. The winding number is related to the dark magnetic flux along the string core by

$\displaystyle\int\differential[2]{x}B_{\mathrm{D}3}^{\prime}=\oint_{\rho\rightarrow\infty}A^{\prime}_{\mathrm{D}i}\differential{x}^{i}=\frac{2\pi n}{e_{\mathrm{D}}^{\prime}}\ .$

(53)

In the decoupled basis, the absence of the kinetic mixing implies $A^{\prime}_{\mu}=0$. Nevertheless, QED test charges defined in the original basis feel the Aharonov-Bohm (AB) effect through $A_{\mu}\neq 0$. The corresponding AB phase around the string is given by [18]

$\displaystyle{n^{e}_{\mathrm{QED}}}W_{\mathrm{QED}}=\frac{{n^{e}_{\mathrm{QED}}}\epsilon e}{\sqrt{1-\epsilon^{2}}}\oint A^{\prime}_{\mathrm{D}\mu}\differential{x}^{\mu}=\frac{2\pi n{n^{e}_{\mathrm{QED}}}q\epsilon e}{e_{\mathrm{D}}}\ .$

(54)

As in the case of elementary dark charges, the $\theta$-terms do not affect the equations of motion because of the $\mathrm{U(1)}_{\mathrm{QED}}$ and $\mathrm{U(1)}_{\mathrm{D}}$ Bianchi identities. Thus, they do not modify the field configurations, and hence, the AB phases.

It is also instructive to see the dark string in the original basis. Substituting Eq. (49) into Eq. (44), we find

	$\displaystyle A_{i}$	$\displaystyle=-\frac{\epsilon n}{e_{\mathrm{D}}}\frac{\epsilon_{ij}x^{j}}{\rho^{2}}f(\rho)\ ,$		(55)
	$\displaystyle A_{\mathrm{D}i}$	$\displaystyle=-\frac{n}{e_{\mathrm{D}}}\frac{\epsilon_{ij}x^{j}}{\rho^{2}}f(\rho)$		(56)

for $i,j=1,2$. In this picture, $A_{i}$ is induced by the $\mathrm{U(1)}_{\mathrm{D}}$ current of $\chi$,

$\displaystyle J_{\mathrm{\chi}}^{i}=i\chi D^{i}\chi^{\dagger}-i\chi^{\dagger}D^{i}\chi=2v^{2}n\frac{\epsilon^{ij}x^{j}}{\rho^{2}}h^{2}(f-1)\ ,$

(57)

through the kinetic mixing. This expression allows us to interpret the AB effect on QED charges as a result of a solenoid around the string.