Stable $Z$-strings with topological polarization in two Higgs doublet model

We introduce two $SU(2)$ doublets of Higgs scalar fields, $\Phi_{1}$ and $\Phi_{2}$, both with the hypercharge $Y=1$. The Lagrangian which describes the gauge and Higgs sectors can be written as

$\displaystyle{\mathcal{L}}=-\frac{1}{4}\left(Y_{\mu\nu}\right)^{2}-\frac{1}{4}\left(W_{\mu\nu}^{a}\right)^{2}+\left|D_{\mu}\Phi_{i}\right|^{2}-V(\Phi_{1},\Phi_{2}).$

(2.1)

Here, $Y_{\mu\nu}$ and $W^{a}_{\mu\nu}$ describe field strength tensors of the hypercharge and weak gauge interactions, respectively, with $\mu$ ($\nu$) and $a$ being Lorentz and weak iso-spin indices, respectively. $D_{\mu}$ represents the covariant derivative acting on the Higgs fields, and the index $i$ runs $i=1,2$. The most generic quartic potential $V(\Phi_{1},\Phi_{2})$ for the two Higgs doublets is given by

	$\displaystyle V(\Phi_{1},\Phi_{2})$	$\displaystyle=m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left(m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+{\rm h.c.}\right)$
		$\displaystyle+\frac{\beta_{1}}{2}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{\beta_{2}}{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$
		$\displaystyle+\beta_{3}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{2}^{\dagger}\Phi_{2}\right)+\beta_{4}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right)$
		$\displaystyle+\left\{\left(\frac{\beta_{5}}{2}\Phi_{1}^{\dagger}\Phi_{2}+\beta_{6}|\Phi_{1}|^{2}+\beta_{7}|\Phi_{2}|^{2}\right)\Phi_{1}^{\dagger}\Phi_{2}+{\rm h.c.}\right\}\,.$		(2.2)

In this paper, we assume that both the Higgs fields develop real vacuum expectation values (VEVs) as ¹¹1 Note that this convention is different by the factor “$\sqrt{2}$” from the usual notation in the literature.

$$\Phi_{1}=\begin{pmatrix}0\\ v_{1}\end{pmatrix},\quad\Phi_{2}=\begin{pmatrix}0\\ v_{2}\end{pmatrix}.$$

(2.3)

Then the electroweak scale, $v_{\mathrm{EW}}$ ($\simeq$ 246 GeV), can be expressed by these VEVs as $v_{\rm EW}^{2}=2v_{\rm sum}^{2}\equiv 2(v_{1}^{2}+v_{2}^{2})$. The masses of the gauge bosons are given by

$$m_{W}^{2}=\frac{g^{2}v_{\mathrm{EW}}^{2}}{4},\quad m_{Z}^{2}=\frac{g^{2}v_{\mathrm{EW}}^{2}}{4\cos^{2}\theta_{W}}$$

(2.4)

with the standard definitions $\cos\theta_{W}\equiv\frac{g}{\sqrt{g^{2}+g^{\prime 2}}}$, $Z_{\mu}\equiv W_{\mu}^{3}\cos\theta_{W}-Y_{\mu}\sin\theta_{W}$ and $A_{\mu}\equiv W_{\mu}^{3}\sin\theta_{W}+Y_{\mu}\cos\theta_{W}$. The $W^{\pm}$ bosons are defined as $W_{\mu}^{\pm}\equiv(W_{\mu}^{1}\pm iW_{\mu}^{2})/\sqrt{2}$.

For later use, we rewrite the Higgs fields in a two-by-two matrix form Grzadkowski:2010dj , $H$, defined by

$$H=\left(i\sigma_{2}\Phi_{1}^{*},\ \Phi_{2}\right).$$

(2.5)

The matrix scalar field $H$ transforms under the electroweak $SU(2)_{W}\times U(1)_{Y}$ symmetry as

$$H\to\exp\left[\frac{i}{2}\theta_{a}(x)\sigma_{a}\right]H~{}\exp\left[-\frac{i}{2}\theta_{Y}(x)\sigma_{3}\right],$$

(2.6)

where the group element acting from the left belongs to $SU(2)_{W}$ and the other element acting from the right beongs to $U(1)_{Y}$. Therefore the covariant derivative on $H$ can be expressed as

$$D_{\mu}H=\partial_{\mu}H-i\frac{g}{2}\sigma_{a}W_{\mu}^{a}H+i\frac{g^{\prime}}{2}H\sigma_{3}Y_{\mu}.$$

(2.7)

The VEV of $H$ is expressed by a diagonal matrix $\langle H\rangle=\mathrm{diag}(v_{1},v_{2})$, and the Higgs potential can be rewritten by using $H$ as follows:

	$\displaystyle V(H)$	$\displaystyle=-m_{1}^{2}~{}\mathrm{Tr}|H|^{2}-m_{2}^{2}~{}\mathrm{Tr}\left(|H|^{2}\sigma_{3}\right)-\left(m_{3}^{2}\det H+\mathrm{h.c.}\right)$
		$\displaystyle+\alpha_{1}~{}\mathrm{Tr}|H|^{4}+\alpha_{2}~{}\left(\mathrm{Tr}|H|^{2}\right)^{2}+\alpha_{3}~{}\mathrm{Tr}\left(|H|^{2}\sigma_{3}|H|^{2}\sigma_{3}\right)$
		$\displaystyle+\alpha_{4}~{}\mathrm{Tr}\left(|H|^{2}\sigma_{3}|H|^{2}\right)$
		$\displaystyle+\left\{\left(\alpha_{5}\det H+\alpha_{6}\mathrm{Tr}|H|^{2}+\alpha_{7}\mathrm{Tr}\left(|H|^{2}\sigma_{3}\right)\right)\det H+\mathrm{h.c.}\right\},$		(2.8)

with $|H|^{2}\equiv H^{\dagger}H$. The parameters $m_{3}$, $\alpha_{5}$, $\alpha_{6}$, and $\alpha_{7}$ are complex in general. The parameter sets in the Higgs potential in Eqs. (2.1) and (2.1) are related as

	$\displaystyle m_{11}^{2}=-m_{1}^{2}-m_{2}^{2},\hskip 20.00003ptm_{22}^{2}=-m_{1}^{2}+m_{2}^{2},\hskip 20.00003ptm_{12}^{2}=m_{3}^{2},$
	$\displaystyle\beta_{1}=2(\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}),\hskip 20.00003pt\beta_{2}=2(\alpha_{1}+\alpha_{2}+\alpha_{3}-\alpha_{4}),$
	$\displaystyle\beta_{3}=2(\alpha_{1}+\alpha_{2}-\alpha_{3}),\hskip 20.00003pt\beta_{4}=2(\alpha_{3}-\alpha_{1}),\hskip 20.00003pt\beta_{5}=2\alpha_{5},$
	$\displaystyle\beta_{6}=\alpha_{6}+\alpha_{7},\hskip 20.00003pt\beta_{7}=\alpha_{6}-\alpha_{7}\,.$		(2.9)

By the electroweak precision measurement, the electroweak $\rho$ parameter, defined by $\rho\equiv m_{W}^{2}/m_{Z}^{2}\cos^{2}\theta_{W}$, must be close to unity ParticleDataGroup:2020ssz . To satisfy this experimental constraint, we impose a global $SU(2)_{L}\times SU(2)_{R}$ symmetry on the Higgs potential. It is convenient to introduce the following two matrices:

$$H_{1}\equiv\left(i\sigma_{2}\Phi_{1}^{*},\ \Phi_{1}\right),\quad H_{2}\equiv\left(i\sigma_{2}\Phi_{2}^{*},\ \Phi_{2}\right)\,,$$

(2.10)

which transform under $SU(2)_{L}\times SU(2)_{R}$ as

$$H_{1}\to LH_{1}R^{\dagger},\quad H_{2}\to LH_{2}R^{\dagger},\hskip 20.00003ptL,R\in SU(2)_{L,R}\,.$$

(2.11)

The potential $V(\Phi_{1},\Phi_{2})$ given in Eq. (2.1) is invariant under this transformation when

$$\beta_{4}=\beta_{5},\quad m_{12}^{2},\beta_{5},\beta_{6},\beta_{7}\in\mathbb{R}\,,$$

(2.12)

which, in the notation in Eq. (2.1), is equivalent to

$$\alpha_{3}-\alpha_{1}=\alpha_{5},\quad m_{3},\alpha_{5},\alpha_{6},\alpha_{7}\in\mathbb{R}\,.$$

(2.13)

By the Higgs VEVs, $SU(2)_{L}\times SU(2)_{R}$ symmetry is spontaneously broken, but its diagonal subgroup

$$H_{1}\to UH_{1}U^{\dagger},\quad H_{2}\to UH_{2}U^{\dagger},\hskip 20.00003ptU\in SU(2)_{\mathrm{C}}\,$$

(2.14)

remains if $v_{1},v_{2}\in\mathbb{R}$. This residual symmetry is called the custodial $SU(2)_{\mathrm{C}}$ symmetry. This symmetry forces the CP-odd Higgs boson and the charged Higgs boson to be degenerate, so that the $\rho$ parameter is protected from receiving their large loop corrections. Note that the kinetic term of $H$ cannot be invariant under this transformation because of the presence of the $U(1)_{Y}$ gauge field. Thus the custodial symmetry is not an exact symmetry of the theory but is explicitly broken by the $U(1)_{Y}$ gauge interaction.

The bosonic Lagrangian ${\mathcal{L}}$ in Eq. (2.1) is invariant under a $\mathbb{Z}_{2}$ transformation defined by

$\displaystyle\begin{cases}H_{i}\to i\sigma_{2}H_{i}(i\sigma_{2})^{\dagger}\\ W_{\mu}\to i\sigma_{2}W_{\mu}(i\sigma_{2})^{\dagger}\\ Y_{\mu}\to-Y_{\mu}\,,\end{cases}$

(2.15)

for $i=1,2$ when the parameters of the potential are real:

$$\text{CP conserving condition}:\quad m_{3},\alpha_{5},\alpha_{6},\alpha_{7}\in\mathbb{R}\,.$$

(2.16)

This is nothing but the CP symmetry (more precisely, C symmetry for the gauge fields) and acts on $H_{i}$ as a subgroup of the custodial $SU(2)_{\mathrm{C}}$ transformation since $i\sigma_{2}\in SU(2)_{\mathrm{C}}$. (Correspondingly, Eq. (2.16) is a necessary condition of Eq. (2.13).) Thus the custodial $SU(2)_{\mathrm{C}}$ symmetric Higgs potential is automatically invariant under the CP symmetry. In the doublet notation, the CP transformation acting on the Higgs fields is expressed as the complex conjugation:

$\displaystyle\begin{cases}\Phi_{1}\to\Phi_{1}^{\ast}\\ \Phi_{2}\to\Phi_{2}^{\ast}\,,\end{cases}$

(2.17)

and the CP symmetry is preserved in the vacuum if $v_{1},v_{2}\in\mathbb{R}$.

For later use, we move to another basis of the Higgs fields. It is known that 2HDM Lagrangian without the Yukawa couplings has an ambiguity corresponding to the $U(2)$ basis transformation of the Higgs doublets (see, e.g., Refs. Davidson:2005cw ; Haber:2006ue ; Grzadkowski:2010dj ; Haber:2010bw and Ref. Branco:2011iw for a review):

$$\Phi_{i}\to\Phi_{i}^{\prime}=\sum_{j=1}^{2}M_{ij}\Phi_{j},\quad M\in U(2)\quad(i=1,2)\,.$$

(2.18)

All Higgs potentials that are related by this basis transformation are physically equivalent and predict the same physics. Let us take the matrix $M_{ij}$ as

$$M=\frac{1}{2}\begin{pmatrix}1-i&1+i\\ 1+i&1-i\end{pmatrix}\,.$$

(2.19)

By this basis transformation in Eq. (2.18), the Higgs potential can be rewritten in terms of the new doublets $\Phi_{1}^{\prime}$ and $\Phi_{2}^{\prime}$ as

	$\displaystyle V(\Phi_{1},\Phi_{2})=$	$\displaystyle-m_{1}^{\prime 2}~{}\mathrm{Tr}|H^{\prime}|^{2}-m_{2}^{\prime 2}~{}\mathrm{Tr}\left(|H^{\prime}|^{2}\sigma_{3}\right)-\left(m_{3}^{\prime 2}\det H^{\prime}+\mathrm{h.c.}\right)$
		$\displaystyle+\alpha_{1}^{\prime}~{}\mathrm{Tr}|H^{\prime}|^{4}+\alpha_{2}^{\prime}~{}\left(\mathrm{Tr}|H^{\prime}|^{2}\right)^{2}+\alpha_{3}^{\prime}~{}\mathrm{Tr}\left(|H^{\prime}|^{2}\sigma_{3}|H^{\prime}|^{2}\sigma_{3}\right)$
		$\displaystyle+\alpha_{4}^{\prime}~{}\mathrm{Tr}\left(|H^{\prime}|^{2}\sigma_{3}|H^{\prime}|^{2}\right)$
		$\displaystyle+\left\{\left(\alpha_{5}^{\prime}\det H^{\prime}+\alpha_{6}^{\prime}\mathrm{Tr}|H^{\prime}|^{2}+\alpha_{7}^{\prime}\mathrm{Tr}\left(|H^{\prime}|^{2}\sigma_{3}\right)\right)\det H^{\prime}+\mathrm{h.c.}\right\},$		(2.20)

with $H^{\prime}=\left(i\sigma_{2}\Phi_{1}^{\prime*},\ \Phi_{2}^{\prime}\right)$. Here we have put the prime ^′ on the parameters to distinguish them from ones in the old basis in Eq. (2.1). The kinetic term of $H$ is simply $\mathrm{Tr}|D_{\mu}H^{\prime}|^{2}$ since the new doublets have the same gauge charges as those of the old ones. One should note that the parameters $m_{3}^{\prime},\alpha_{5}^{\prime},\alpha_{6}^{\prime},\alpha_{7}^{\prime}$ are not real in the new basis even when the CP symmetry is imposed in the old basis in Eq. (2.16). The explicit relation between the parameters in the old Higgs potential in Eq. (2.1) and the new one in Eq. (2.20) is given as

	$\displaystyle m_{11}^{2}$	$\displaystyle=-m_{1}^{\prime 2}+\mathrm{Im}\,m_{3}^{\prime 2}\,,~{}$
	$\displaystyle m_{22}^{2}$	$\displaystyle=-m_{1}^{\prime 2}-\mathrm{Im}\,m_{3}^{\prime 2}\,,$
	$\displaystyle m_{12}^{2}$	$\displaystyle=im_{2}^{\prime 2}+\mathrm{Re}\,m_{3}^{\prime 2}\,,$
	$\displaystyle\beta_{1}$	$\displaystyle=\alpha_{1}^{\prime}+2\alpha_{2}^{\prime}+\alpha_{3}^{\prime}-\mathrm{Re}\,\alpha_{5}^{\prime}-2\,\mathrm{Im}\,\alpha_{6}^{\prime}\,,$
	$\displaystyle\beta_{2}$	$\displaystyle=\alpha_{1}^{\prime}+2\alpha_{2}^{\prime}+\alpha_{3}^{\prime}-\mathrm{Re}\,\alpha_{5}^{\prime}+2\,\mathrm{Im}\,\alpha_{6}^{\prime}\,,$
	$\displaystyle\beta_{3}$	$\displaystyle=3\alpha_{1}^{\prime}+2\alpha_{2}^{\prime}-\alpha_{3}^{\prime}+\mathrm{Re}\,\alpha_{5}^{\prime}\,,$
	$\displaystyle\beta_{4}$	$\displaystyle=-\alpha_{1}^{\prime}+3\alpha_{3}^{\prime}+\mathrm{Re}\,\alpha_{5}^{\prime}\,,$
	$\displaystyle\beta_{5}$	$\displaystyle=-\alpha_{1}^{\prime}-\alpha_{3}^{\prime}+\mathrm{Re}\,\alpha_{5}^{\prime}+2i\,\mathrm{Re}\,\alpha_{7}^{\prime}\,,$
	$\displaystyle\beta_{6}$	$\displaystyle=i\alpha_{4}^{\prime}-\mathrm{Im}\,\alpha_{5}^{\prime}+\mathrm{Re}\,\alpha_{6}^{\prime}-i\,\mathrm{Im}\,\alpha_{7}^{\prime}\,,$
	$\displaystyle\beta_{7}$	$\displaystyle=i\alpha_{4}^{\prime}+\mathrm{Im}\,\alpha_{5}^{\prime}+\mathrm{Re}\,\alpha_{6}^{\prime}+i\,\mathrm{Im}\,\alpha_{7}^{\prime}\,.$		(2.21)

In this new basis, the $SU(2)_{L}\times SU(2)_{R}$ symmetry can be expressed as Grzadkowski:2010dj ; Pomarol:1993mu

$$H^{\prime}\to LH^{\prime}R^{\dagger},\quad L,R\in SU(2)_{L,R}\,.$$

(2.22)

The Higgs potential $V(H^{\prime})$ (Eq. (2.20)) is invariant under this $SU(2)_{L}\times SU(2)_{R}$ symmetry when

$$SU(2)_{L}\times SU(2)_{R}:\quad m_{2}^{\prime}=\alpha_{3}^{\prime}=\alpha_{4}^{\prime}=\alpha_{7}^{\prime}=0.$$

(2.23)

Note that this condition of the $SU(2)_{L}\times SU(2)_{R}$ symmetry seems different from Eq (2.13) although they are physically equivalent Davidson:2005cw ; Haber:2006ue ; Grzadkowski:2010dj ; Haber:2010bw . This symmetry is spontaneously broken into its subgroup $SU(2)_{\mathrm{C}}$,

$$H^{\prime}\to U^{\prime}H^{\prime}U^{\prime\dagger},\hskip 20.00003ptU^{\prime}\in SU(2)_{\mathrm{C}}\,$$

(2.24)

if $v_{1}^{\prime\ast}=v_{2}^{\prime}$. This is the custodial $SU(2)_{\mathrm{C}}$ symmetry in the new basis and equivalent to that in the old basis. This corresponds to the case II studied by Pomarol and Vega in Ref. Pomarol:1993mu . The CP symmetry, which acts on $H^{\prime}$ as a $\mathbb{Z}_{2}$ subgroup of $SU(2)_{\mathrm{C}}$, can be expressed in this basis as

$\displaystyle\begin{cases}H^{\prime}\to i\sigma_{2}H^{\prime}(i\sigma_{2})^{\dagger}\\ W_{\mu}\to i\sigma_{2}W_{\mu}(i\sigma_{2})^{\dagger}\\ Y_{\mu}\to-Y_{\mu}\,,\end{cases}$

(2.25)

or, in terms of the doublet notation for the Higgs field,

$\displaystyle\begin{cases}\Phi_{1}^{\prime}\to\Phi_{2}^{\prime\ast}\\ \Phi_{2}^{\prime}\to\Phi_{1}^{\prime\ast}\,.\end{cases}$

(2.26)

Thus, in this basis, the CP transformation is a combination of the complex conjugation and the exchange of the doublets. Imposing the CP symmetry on the Lagrangian reads

$$\text{CP conserving condition}:\quad m_{2}^{\prime}=\alpha_{4}^{\prime}=\alpha_{7}^{\prime}=0.$$

(2.27)

Note that the CP symmetry is not spontaneously broken in the vacuum if $v_{1}^{\prime\ast}=v_{2}^{\prime}$.

Hereafter, we work in this new basis. Because the old and new basis correspond to the cases I and II studied by Pomarol and Vega Pomarol:1993mu , we call them “PV-I basis” and “PV-II basis”, respectively. For a simple notation, we will omit the prime on the parameters and the Higgs fields even in the PV-II basis. Note that, in Refs. Eto:2019hhf ; Eto:2020hjb ; Eto:2020opf , Eqs. (2.25) and (2.26) (the CP symmetry in the PV-II basis) is called “$(\mathbb{Z}_{2})_{\mathrm{C}}$ symmetry” and the CP symmetry for the Higgs fields is defined in terms of only a charge conjugation instead of Eqs. (2.25) and (2.26). Correspondingly, some terminology is different from this paper. See Appendix A for summary of their relations.

In Secs. 3, 4 and 5, we impose on the Higgs potential in the PV-II basis a global $U(1)$ symmetry, which is called the $U(1)_{a}$ symmetry and defined by a rotation of the relative phase of the two doublets:

$$H\to e^{i\alpha}H\quad(0\leq\alpha<2\pi)$$

(2.28)

or, equivalently,

$$\Phi_{1}\to e^{-i\alpha}\Phi_{1},\quad\Phi_{2}\to e^{i\alpha}\Phi_{2}\,.$$

(2.29)

The Lagrangian is invariant under the $U(1)_{a}$ transformation, when

$$\text{$U(1)_{a}$ condition}:\quad m_{3}=\alpha_{5}=\alpha_{6}=\alpha_{7}=0.$$

(2.30)

Note that, because the $U(1)_{a}$ transformation does not commute with the basis transformation in Eq. (2.18), one should take care about which basis the $U(1)_{a}$ symmetry is imposed in. As stated above, we work in the PV-II basis throughout this paper.

After $H$ gets the VEV, this $U(1)_{a}$ symmetry is spontaneously broken and the corresponding Nambu-Goldstone (NG) boson appears, which we call the CP-even Higgs boson ($H^{0}$).²²2This is called CP-odd Higgs boson in Refs. Eto:2019hhf ; Eto:2020hjb ; Eto:2020opf , see Appendix A. Note that the CP symmetry is defined by Eq. (2.25) in this basis. As is shown later, the spontaneously broken $U(1)_{a}$ symmetry gives rise to non-trivial topological excitations.

Because an experimental lower bound on the mass of $H^{0}$ is typically $\mathcal{O}(100)$ GeV (which highly depends on how the doublets couple to the SM fermions), such a massless $H^{0}$ is phenomenologically disfavored. Therefore, in realistic models, we should break the $U(1)_{a}$ symmetry explicitly by switching on $m_{3}$, $\alpha_{5}$, $\alpha_{6}$ or $\alpha_{7}$ making $H^{0}$ massive.

We present the Higgs mass spectrum in the case with the $SU(2)_{L}\times SU(2)_{R}$ symmetry, i.e.,

$$SU(2)_{L}\times SU(2)_{R}:\quad m_{2}=\alpha_{3}=\alpha_{4}=\alpha_{7}=0\,$$

(2.31)

while the $U(1)_{a}$ symmetry is not imposed because it should be explicitly broken in phenomenologically viable cases. For simplicity, we suppose that all parameters are real, i.e.,

$$\mathrm{Im}\,m_{3}=\mathrm{Im}\,\alpha_{5}=\mathrm{Im}\,\alpha_{6}=0\,.$$

(2.32)

In this case, the Higgs VEVs become real values given by

$$v_{1}=v_{2}=\sqrt{\frac{m_{1}^{2}+m_{3}^{2}}{2(\alpha_{1}+2\alpha_{2}+\alpha_{5}+2\alpha_{6})}}\equiv v,$$

(2.33)

($\tan\beta\equiv v_{2}/v_{1}=1$) and the custodial symmetry $SU(2)_{\mathrm{C}}(\subset SU(2)_{L}\times SU(2)_{R})$ is preserved in the vacuum.

In the matrix notation, fluctuations around the vacuum can be parametrized as

$\displaystyle H$

$\displaystyle=v{\bf 1}_{2}+\frac{1}{\sqrt{2}}\left(\chi^{A}+i\pi^{A}\right)\sigma^{A}\hskip 20.00003pt(A=0,\cdots,3)$

(2.34)

with $\sigma^{A}=(\bm{1},\sigma^{a})$ ($a=1,2,3$). Here $\pi^{0}$ is the CP-even neutral Higgs boson $H^{0}$. It becomes the massless Nambu-Goldstone (NG) boson in the case with the exact $U(1)_{a}$ symmetry under the condition (2.30). On the other hand, $\pi^{a}$’s are would-be NG bosons for $SU(2)_{W}\times U(1)_{Y}$ and eaten by the gauge bosons. In this setup, due to the custodial symmetry, the components $(\chi^{1},\chi^{2},\chi^{3})$ form a custodial triplet while $\chi^{0}$ is a custodial singlet. Taking linear combinations of $\chi^{1}$ and $\chi^{2}$, one obtains mass eigenstates with definite $U(1)_{\mathrm{EM}}$ charges, which are identified with the charged Higgs boson $H^{\pm}$. On the other hand, $\chi^{3}$ is neutral under $U(1)_{\mathrm{EM}}$ and has odd parity under the CP symmetry defined in Eq. (2.25). It is called the CP-odd neutral Higgs boson $A^{0}$.³³3In Refs. Eto:2019hhf ; Eto:2020hjb ; Eto:2020opf , this particle is called the CP-even Higgs boson, see Appendix A. Their masses are degenerate as

$$m_{H^{\pm}}^{2}=m_{A^{0}}^{2}=2m_{3}^{2}+4v^{2}(\alpha_{1}-\alpha_{5}-\alpha_{6})\,.$$

(2.35)

Note that $\chi^{0}$ and $H^{0}$ do not mix with each other at the tree level and are mass eigenstates since we set the all parameters as real. In addition, the CP-even Higgs $H^{0}$ does not have the $H^{0}VV$ coupling with the gauge bosons ($V=W$ and $Z$) at the tree level while $\chi^{0}$ has the same coupling with the gauge bosons as that of the SM Higgs boson. Thus $\chi^{0}$ must be identified with the SM-like Higgs boson $h^{0}$. Their mass eigenvalues are given by

$$m_{h^{0}}^{2}=4v^{2}(\alpha_{1}+2\alpha_{2}+\alpha_{5}+2\alpha_{6})$$

(2.36)

and

$$m_{H^{0}}^{2}=2m_{3}^{2}-4v^{2}(2\alpha_{5}+\alpha_{6})\,.$$

(2.37)

The parameter $m_{3}^{2}$ (or $m_{12}^{2}$) is often called the decoupling parameter because if $m_{3}^{2}$ is taken to infinity $m_{3}^{2}\to\infty$ keeping $v$, all particles other than the SM Higgs boson $h^{0}$ become infinitely heavy, $m_{H^{\pm}}^{2},m_{H^{0}}^{2},m_{A^{0}}^{2}\to\infty$, which means that they are decoupled and that the model reduces to the SM Higgs sector.

Before discussing $Z$-strings, let us point out the simplest string without a $Z$-flux. A global $U(1)_{a}$ string is the simplest string whose ansatz is given by

$\displaystyle H^{\text{global}}$

$\displaystyle=vf^{\text{global}}(r)e^{i\varphi}\,\bm{1}_{2\times 2},\quad Z_{i}=0,$

(3.1)

where $r\equiv\sqrt{x^{2}+y^{2}}$ and $\varphi$ is the azimuthal angle around the $z$-axis. The boundary conditions imposed on the profile function are $f^{\text{global}}(0)=0$ and $f^{\text{global}}(\infty)=1$. Thus, the asymptotic form of $H^{\text{global}}$ at $r\to\infty$ is $\sim v\,\exp[{i\varphi}]~{}\mathrm{diag}\left(1,1\right)$. This string is accompanied by no $Z$ flux. Since this is a global vortex, its tension is logarithmically divergent.

In the presence of the $U(1)_{a}$ symmetry breaking terms, the asymptotic potential of the ansatz (3.1), in which the azimuthal angle $\varphi$ is replaced with a monototicaly increasing function $\phi(\varphi)$ of it with the same range ($\phi(0)=0$ and $\phi(2\pi)=2\pi$), becomes

	$\displaystyle V$	$\displaystyle\sim(-m_{3}^{2}+2\alpha_{6}v^{2})\det H^{\text{global}}+\alpha_{5}(\det H^{\text{global}})^{2}+{\rm h.c.}$
		$\displaystyle=2(-m_{3}^{2}+2\alpha_{6}v^{2})v^{2}\cos 2\phi+2\alpha_{5}v^{2}\cos 4\phi.$		(3.2)

This is a variant of sine-Gordon (quadruple sine-Gordon) potential, and thus a single $U(1)_{a}$ string is attached by at most four domain walls Eto:2018hhg ; Eto:2018tnk . How many walls are attached to it depends on the parameters.

For later use, we here review the non-topological $Z$-string obtained by embedding of the $Z$-string in the SM Nambu:1977ag ; Vachaspati:1992fi into 2HDM.

The non-topological $Z$-string is the same form of the ANO vortex for the two Higgs doublets, whose ansatz is given as

	$\displaystyle H^{\text{non-top}}$	$\displaystyle=vf^{\text{non-top}}(r)e^{i\varphi\sigma_{3}}\,\bm{1}_{2\times 2}$		(3.3)
	$\displaystyle Z_{i}^{\text{non-top}}$	$\displaystyle=-\frac{2\cos\theta_{\mathrm{W}}}{g}\frac{\epsilon_{3ij}x^{j}}{r^{2}}\left(1-w^{\text{non-top}}(r)\right),$		(3.4)

with the polar coordinates $(r,\varphi)$. The boundary conditions imposed on the profile functions are $f^{\text{non-top}}(0)=w^{\text{non-top}}(\infty)=0$, $w^{\text{non-top}}(0)=f^{\text{non-top}}(\infty)=1$. Thus, the asymptotic form of $H^{\text{non-top}}$ at $r\to\infty$ is $\sim v\exp[{i\varphi}\sigma_{3}]~{}\mathrm{diag}\left(1,1\right)$. All winding phases of the Higgs field are canceled by the $Z$ gauge field at infinity. Thus, it is a local vortex whose tension is finite. The amount of the $Z$ flux is given by

$$\Phi_{Z}^{\text{non-top}}=\frac{4\pi\cos\theta_{\mathrm{W}}}{g}$$

(3.5)

along the $z$-axis. This is nothing but that of the non-topological $Z$-string in the SM Nambu:1977ag ; Vachaspati:1992fi .

This configuration solves the EOMs as the ANO vortex string. Note, however, that its stability is not ensured because it does not have any topological charge. The boundary $S^{1}$ at $r\to\infty$ is mapped onto the $U(1)_{Z}$ gauge orbit, but it can be unwound inside the whole $SU(2)_{W}\times U(1)_{Y}$ gauge orbits. Indeed, the $Z$-string in the SM is known to be unstable for experimental values of the SM Higgs mass, the $W$ boson mass and the Weinberg angle. See Appendix B. The non-topological $Z$-string in the 2HDM is unstable as well due to the same reason.

In Refs.Eto:2018tnk ; Eto:2018hhg ; Dvali:1994qf ; Dvali:1993sg , it is pointed out that, unlike in the SM, 2HDMs allow topologically stable strings to exist thanks to the global $U(1)_{a}$ symmetry. First, consider topological strings with the $Z$ flux (topological $Z$-strings). There are two types of topological $Z$-strings corresponding to which one of the two Higgs doublets has a winding phase. To see that, let us take $W_{\mu}^{\pm}=A_{\mu}=0$.

One of the topological $Z$-strings is called the $(1,0)$-string, whose ansatz located on the $z$ axis is given by

	$\displaystyle H^{(1,0)}$	$\displaystyle=v\begin{pmatrix}f^{(1,0)}(r)e^{i\varphi}&0\\ 0&h^{(1,0)}(r)\end{pmatrix},$		(3.6)
	$\displaystyle Z_{i}^{(1,0)}$	$\displaystyle=-\frac{\cos\theta_{\mathrm{W}}}{g}\frac{\epsilon_{3ij}x^{j}}{r^{2}}\left(1-w^{(1,0)}(r)\right)$		(3.7)

with the polar coordinates $(r,\varphi)$. The boundary conditions imposed on the profile functions are

$$f^{(1,0)}(0)={h^{(1,0)}}^{\prime}(0)=w^{(1,0)}(\infty)=0,\quad w^{(1,0)}(0)=f^{(1,0)}(\infty)=h^{(1,0)}(\infty)=1\,.$$

(3.8)

Thus, the asymptotic form of $H^{(1,0)}$ at $r\to\infty$ is

$$H^{(1,0)}\to v\exp\left[{\frac{i\varphi}{2}}\right]\exp\left[{\frac{i\varphi}{2}\sigma_{3}}\right]\,.$$

(3.9)

From this form, it is clear that this string is a hybrid of halves of a global $U(1)_{a}$ string in Eq. (3.1) and a non-topological $Z$-string in Eqs. (3.3) and (3.4). The profile functions $f^{(1,0)}$, $h^{(1,0)}$ and $w^{(1,0)}$ are determined by solving the EOMs. See Fig. 1 for the numerical solution. Note that $h(r)$ does not necessarily vanish at the center of the string, $r=0$, since it does not have a winding phase. Instead, it tends to obtain a non-zero condensation. In particular, its condensation value becomes larger when $m_{h^{0}}$ is larger than the other masses (right panel). This property will be important in later sections.

On the other hand, there is the other solution, called the $(0,1)$-string:

	$\displaystyle H^{(0,1)}$	$\displaystyle=v\begin{pmatrix}h^{(0,1)}(r)&0\\ 0&f^{(0,1)}(r)e^{i\varphi}\end{pmatrix},$		(3.10)
	$\displaystyle Z_{i}^{(0,1)}$	$\displaystyle=\frac{\cos\theta_{\mathrm{W}}}{g}\frac{\epsilon_{3ij}x^{j}}{r^{2}}\left(1-w^{(0,1)}(r)\right),$		(3.11)

with the asymptotic form of the Higgs fields,

$$H^{(0,1)}\to v\exp\left[{\frac{i\varphi}{2}}\right]\exp\left[{\frac{-i\varphi}{2}\sigma_{3}}\right]$$

(3.12)

for $r\to\infty$. The boundary conditions for $f^{(0,1)}$, $h^{(0,1)}$ and $w^{(0,1)}$ are the same as the $(1,0)$-string.

Note that, the two strings, $(1,0)$- and $(0,1)$ strings, are related to each other by the CP transformation Eq. (2.25). Since we are interested in the CP symmetric case, we have

$$f^{(1,0)}(r)=f^{(0,1)}(r)\equiv f(r),\hskip 20.00003pth^{(1,0)}(r)=h^{(0,1)}(r)\equiv h(r),$$

(3.13)

$$w^{(1,0)}(r)=w^{(0,1)}(r)\equiv w(r)\,,$$

(3.14)

and the two strings have degenerate tensions (energy per unit length).⁴⁴4 When the $U(1)_{Y}$ gauge coupling is turned off, these two strings are continuously degenerated parametrized by ${\mathbb{C}}P^{1}$ moduli associated with spontaneously broken custodial symmetry in the vicinity of the string Eto:2018tnk ; Eto:2018hhg . Such a string is called a non-Abelian string, see, e.g., Refs. Tong:2005un ; Eto:2006pg ; Shifman:2007ce ; Shifman:2009zz ; Eto:2013hoa as a review. Such a continuous degeneracy is lifted by the $U(1)_{Y}$ gauge coupling that explicitly breaks the custodial symmetry, leaving the (1,0) and (0,1) strings as strings with the minimum tension.

Looking at the asymptotic forms in Eqs. (3.9) and (3.12), it is clear that both the $(1,0)$- and $(0,1)$-strings have winding number $1/2$ for the global $U(1)_{a}$ symmetry, and thus they are topological vortex strings of the global type. This is easily seen by calculating the following topological current:⁵⁵5This normalization of $\mathcal{A}_{i}$ is different from that in Ref. Eto:2020hjb by a factor $8\pi$.

$$\mathcal{A}_{i}\equiv\frac{1}{8\pi v^{2}}\epsilon_{ijk}\partial^{j}J^{k}\,,$$

(3.15)

where $J_{i}$ is the Noether current of the $U(1)_{a}$ symmetry and defined as

$$J_{i}=-i\,\mathrm{Tr}\left[H^{\dagger}D_{i}H-(D_{i}H)^{\dagger}H\right].$$

(3.16)

This current is topologically conserved, $\partial_{i}\mathcal{A}_{i}=0$. For the $(1,0)$- and $(0,1)$-strings located along the $z$ axis, $\mathcal{A}_{i}$ is obtained as

	$\displaystyle\mathcal{A}_{3}$	$\displaystyle=\frac{1}{8\pi v^{2}}\left[\partial^{1}J^{2}-\partial^{2}J^{1}\right]$
		$\displaystyle=\frac{1}{8\pi r}\partial_{r}\left[2f(r)^{2}-(1-w(r))(f(r)^{2}-h(r)^{2})\right],$		(3.17)

which is half-quantized after integrating over the $x-y$ plane,

$$q_{a}\equiv 2\pi\int_{0}^{\infty}drr\mathcal{A}_{3}=\frac{1}{2}\,,$$

(3.18)

and the other components vanish, $\mathcal{A}_{1}=\mathcal{A}_{2}=0$. Therefore, the two strings are topologically stable and cannot be removed by any continuous deformation.

Similarly to standard global vortices, their tensions (masses per unit length) logarithmically diverge. It can be seen from the kinetic term of the Higgs field:

$$2\pi\int drr~{}\mathrm{Tr}|D_{i}H^{(1,0)}|^{2}\sim 2\pi\int drr~{}\mathrm{Tr}|D_{i}H^{(0,1)}|^{2}\sim\pi v^{2}\int\frac{dr}{r}$$

(3.19)

for $r\to\infty$. This is a quarter of that for a singly quantized global $U(1)_{a}$ vortex because of the half winding number for $U(1)_{a}$ Eto:2018tnk .