Electroweak strings with dark scalar condensates and their (in)stability

The stability of"visible"electroweak-type cosmic strings is investigated in an extension of the Standard Model (SM) by a minimal dark sector, consisting of a U(1) gauge field, broken spontaneously by a scalar. The"visible"and dark sectors are coupled through a Higgs-portal and a gauge-kinetic mixing term. It is found that strings whose core is"filled"with a dark scalar condensate exhibit better stability properties than their analogues in the SM, when the electroweak mixing angle is close to $\theta_{\rm W}=\pi/2$. They become unstable as one lets $\theta_{\rm W}$ approach its physical value. The instability mechanism appears to be a $W$-boson condensation mechanism found in previous studies on the stability of electroweak strings.

has been considered in the case of the semilocal model coupled to a dark sector (DS), and a significant stabilizing effect has been found due to the Higgs portal coupling [22,23] and to gauge kinetic mixing (GKM) [24]. In the present paper, we shall extend this study to the full GSW model coupled to a DS.
The model of dark matter (DM) we shall consider here is the unified dark matter model put forward in Refs. [25,26], in which it is assumed that in the DS there are gauge interactions, the gauge group contains a U(1) factor, which is broken by a dark sector (DS) Higgs field. The DS and the visible sector (VS) interact via the Higgs portal coupling [22,23] and the gauge kinetic mixing (GKM) [24]. A subset of this model is the scalar phantom dark matter [22,23], in which dark matter is scalar, and there is no DS gauge field; in this case the dark scalar may have a zero vacuum expectation value (VEV). The parameters of the latter model are strongly restricted by observations [27,28,29]. In the present paper, we consider the case of a non-zero dark scalar VEV. For information on experimental constraints on dark matter, see Ref. [30].
In the model considered, there exist dark string solutions, i.e., string solutions where the flux is of the U(1) interaction in the DS, and the dark scalar has a non-zero winding [31,32,33,34,35,36,37,38,39]. Similar solutions in an U(1) × U(1) theory for higher windings have been considered in Refs. [40,41], and an earlier work on string solutions in a portal type theory is Ref. [42]. In these works, the strings have a non-zero winding in the DS. Dark strings in these models are stable, however, their interactions with the VS and their string tension is determined by the (yet unknown) parameters of the DS.
The complementary case, in which the flux is in the VS, and the role of the DM is to stabilise the string, yields a string tension determined by the electroweak scale, and interactions mostly determined by the electroweak parameters.
The semilocal limit of the theory is a generalisation of the Witten model [43], and the string solutions considered in Ref. [21] are embeddings of the solutions previously found in Refs. [44,45,46,47]. (Similar and quite interesting string solutions were found in a condensed matter setting, in Refs. [48,49].) Other mechanisms for the stabilisation of electroweak strings have been considered recently. We shall mention additional scalar fields bound in the string [50], special couplings (of the dilatonic type) [51], quantum fluctuations of an additional heavy fermion doublet coupled to the string [52,53], and interaction with a thermal photon bath [54].
In the present paper, we demonstrate that a stabilising effect acting on electroweak strings arises when the GSW theory is coupled to a DS through the GKM and the Higgs portal. We obtain the domain of stability of electroweak-dark strings for various parameter combinations, as well as the dependence of the strength of the instability on the parameters of the DS and the strength of the DS-VS couplings. We heavily use the analysis of the model (and its dark string solutions) in Ref. [32] and that of the stability of electroweak strings in Ref. [17].
Our results show that the Higgs portal coupling has a stabilising effect if the dark scalar is lighter than the VS Higgs particle, and that both stabilising effect of the Higgs portal coupling and of the GKM are suppressed for values of the Weinberg angle far from the semilocal limit, π/2. Close to the semilocal limit, we present data showing that EWD strings are stable with M H /M Z > 1 (up to M H /M Z ∼ 1.4), in contrast to electroweak strings.
The plan of the paper is as follows: we summarise the main characteristics of the model considered in Sec. 1. Electroweak-dark strings are introduced in Sec. 2, their stability analysis is performed in Sec. 3, and we conclude in Sec. 4. The particle content of the model is summarised in Appendices B and C, based on Ref. [32]. Some details of the calculations are relegated to Appendix E.

The model considered
We shall consider here string solutions in the GSW model coupled via gauge kinetic mixing [24] and the Higgs portal [22,23] to a dark sector (DS). The dark matter (DM) sector shall be considered in the unified dark matter model of Ref. [25,26,31]. From the full SM Lagrangian, the terms corresponding to the field that assume non-trivial values in the solutions considered are the electroweak (GSW) and DS Abelian gauge terms, where W , Y and C denote the VS non-Abelian, Abelian, and the DS gauge field strenghts. The scalar sector of the theory consists of the electroweak and the DS Higgs scalars, coupled to their respective gauge fields, where D µ andD µ denote the gauge covariant derivatives and the potential is For more details, and the definitions of the field strength tensors and the gauge covariant derivatives, see Appendix A. The (relevant) parameters of the theory are as follows: g is the non-Abelian [SU(2)] electroweak coupling, g ′ is the Abelian [U(1)] electroweak coupling,ĝ is the DS Abelian [U(1)] coupling, ε is the gauge-kinetic mixing (GKM), λ 1,2 the scalar self-couplings, λ ′ is the Higgs portal coupling. The theory considered is the same as in references [32,33,34], with the exception of the sign of the GKM term, which has been chosen in accord with Ref. [21].
The Weinberg angle is defined by tan θ W = g ′ /g, and the limit, in which g → 0, and thus θ W → π/2 is referred to as the semilocal limit; in which the VS and the DS Abelian gauge fields and the scalar sectors are retained, and the non-Abelian gauge field decouples and the SU(2) symmetry becomes global. The semilocal limit of the GSW model is often referred to as the SU(2) symmetric semilocal model.
Fields corresponding to physical particles are found by fixing the gauge, expanding about the vacuum Φ = (0, η 1 ) and χ = η 2 , and diagonalising the quadratic terms (see Appendix B and Ref. [32]). The particle properties which we use to determine the parameters in the Lagrangian are the masses of the W and Z bosons, M W and M Z , the electric charge e, and the Higgs mass M H in the VS (which are, by now, all determined experimentally to a very good accuracy [30]), and the dark gauge vector boson mass M X , dark sector scalar mass M S and the scalar mixing angle θ s (the latter being largely unconstrained as long as the DS particles are heavy enough [25,26]).
For the purposes of the electroweak-dark string Ansatz, let us remark that the fields Y µ , W 3 µ , and C µ are mixed into the (physical) fields A µ (electromagnetism), Z µ (weak) and X µ (dark U(1)) gauge fields. In our solutions, the lower Higgs component and the dark scalar fields assume non-trivial values, and the gauge fields Z µ and X µ .
For details of the physical fields in the gauge sector, and their couplings to the scalars, see Ref. [32], and Appendix B. Also in Appendix B, we consider rescaling of the model to non-dimensionalise the fields, and introduce β 1,2 and β ′ as the coefficients of the quartic terms in the rescaled potential. These quantities shall play somewhat analogous roles in the radial equations of cylindrically symmetric strings as the ratio of the scalar and the vector masses in the Abelian Higgs and semilocal models, and the ratio of the Higgs and Z boson masses in the GSW model, β = M H /M Z , which we shall refer to as the Ginzburg-Landau (GL) parameter. For the physical fields in the scalar sector, see Appendix C.

Electroweak-dark strings
The Abrikosov-Nielsen-Olesen (ANO) string [10,11] is a well-known cylindrically symmetric solution of the Abelian Higgs model, in which the scalar field has a winding number n, the gauge field has a nonvanishing radial component, and the resulting string or flux tube contains n flux quanta.
The ANO string can be embedded in the GSW theory by assuming that the component of the Higgs field having non-zero VEV in the vacuum has a winding, and the flux is in the Z field. Using cylindrical coordinates r, ϑ, z, the Ansatz describes a cylindrically symmetric vortex string (or flux tube) centred on the z-axis, with n flux quanta [2,3,9].
The unified dark matter model [25,26] extends the GSW model with a dark sector, containing a Higgs field χ and an additional U(1) gauge field. The Ansatz (4) is accordingly extended, preserving cylindrical symmetry, as where now the fields Z and X are the physical fields combined of Y and X.
In the Abelian Higgs model, ANO strings for n = 1 are stable, and topological. |n| > 1 vortices are stable for β < 1 [55]. It is important to note, that for embeddings of ANO strings, new instabilities may arise which excite the additional fields, therefore, for embedded vortices, new stability analyses are necessary.
String solutions in the semilocal model within the Ansatz (4) are termed semilocal strings; they are embedded ANO strings, and their stability depends on the GL parameter also for n = 1. For β < 1, semilocal strings are stable, and unstable for β > 1 [19,20,9]. The mechanism of the instability is that a condensate of the other Higgs component, φ 1 forms in the core of the string, and eventually dilutes the flux.
In the GSW model, strings within the Ansatz (4) are referred to as electroweak strings or Z-strings. Their stability depends on the parameters of the model; they are stable for β < 1, and a sufficiently large value of the Weinberg angle, which depends on the GL parameter, β = M H /M Z , i.e., they are stable for β < 1 and rather close to the semilocal limit [14,15,16,17]. The mechanism of the instability is unwinding through the condensation of Higgs and W bosons in the string core.
In the model outlined above, and its semilocal limit, string solutions with winding in the dark sector have been considered in Refs. [31,37,38,39]. These strings are stable topological strings. Their energy scale is determined by the scale of the symmetry breaking in the dark sector, which is presently to a large extent unconstrained by measurement.
The scale of strings in the visible sector, within the Ansatz (4) and (5) is the electroweak scale. This is the main motivation behind the search for mechanisms stabilising electroweak strings. Besides, as the mechanism behind the instability is the formation of condensates in the string core, the idea arises naturally to look for other fields which may fill up the core, thus preventing the instability. In Ref. [21], this idea has been considered in the semilocal limit of the EWD model, i.e., in the semilocal model extended with a scalar and another U(1) gauge field in the DS. There two cases, termed 1VEV and 2VEV have been considered, depending on whether only the VS Higgs, or both the VS and the DS scalar field obtain a VEV. Relevant to the dark matter model of Refs. [25,26] it the latter case. In both case it has been found in Ref. [21], that the stabilising effect is significant, semilocal-dark strings may exist for β significantly above unity.
Semilocal-dark strings in the case with no GKM may be considered embeddings of string solutions in non-symmetric extended Abelian Higgs models considered in Refs. [45,46]. Also, in the 1VEV case, semilocal and semilocal-dark strings may coexist, and their stability is considered separately. The energy of semilocal-dark strings is lower, and they stabilize for a larger set of parameters.  Here, we consider electroweak-dark strings, i.e., solutions within the Ansatz (4), (5) within the full GSW model coupled to a dark sector containing an U(1) gauge field and a scalar.
The resulting radial equations are given in Appendix E.1. The solutions are found using the shooting to a fitting point method [56], and an example is displayed in Fig. 1. SM parameters are set to physical values, and DS parameters are set to such values, that they are heavier than their visible counterparts. Additionally to the profile functions f , f d , z and x, the energy within a radius is shown. (For SM parameter values, see Ref. [30]).

Stability analysis
We analyse the stability of the electroweak-dark strings by linearising the field equations around them. An essential part of this procedure is a choice of gauge for the perturbations, for which the background field gauge is highly practical [55,17]. The known instabilities of electroweak strings occur in the sector containing the upper Higgs component and perturbations of W ± [17,2,3], which is decoupled from other fields. The perturbations are expanded in partial waves, following the procedure of Refs. [55,17,57,45,46,21]. The resulting equations assume the form of an eigenvalue problem for a set of coupled ordinary differential equations for each partial wave ℓ = 0, 1, . . . , where Ψ ℓ is a vector of the radial functions of the perturbations. Electroweak strings with unit flux have instabilities in the ℓ = 0 sector, therefore, we shall focus on that. An eigenvalue Ω 2 < 0 signals instability (an exponentially growing mode). The radial equations (6) have been solved with the shooting to a fitting point method [56], as were the radial equations of the background vortex, Eq. (23). Our numerical methods were found to be stable for M S ∼ M H .

Domain of stability
As a validation of our code we have reproduced the domain of stability of Z-strings in the Salam-Weinberg model (electroweak strings) and compared it to the data of Ref. [17]. In our model, ǫ = θ s = 0 corresponds to the case of the electroweak strings (with the DS decoupled).
Our method was as follows: we set M Z , M W and e to their physical values [30], and initially, M H as well, and M 2 S = M 2 S ± 2000 GeV 2 , then we first lowered M H and M S keeping M S /M H fixed, and then approached the semilocal limit, i.e., increased θ W towards π/2 while keepingḡ, g, ǫ and the scalar potential parameters fixed, until Ω 2 = 0 was reached (i.e., as long as there was a negative eigenvalue).
Our results for the case of no GKM are summarised in Table 1, with data from Ref. [17] added for comparison 1 . There is an excellent agreement between our data, and that of Ref. [17].
The stability of electroweak strings is restricted to β 1 < 1 (i.e., a Higgs mass smaller than the Z boson mass), and close to the semilocal limit, θ W → π/2.
In Fig. 2, the effect of the Higgs portal coupling is shown. Motivation for this was the results for semilocal-dark strings in Ref. [21]. We have found that the Higgs portal coupling indeed has a stabilizing effect, however, in the experimentally undesirable parameter range, when the DS scalar is lighter than the Higgs. In the M S > M H case, we actually found a destabilising effect.
For an explanation, let us consider the potential for the perturbation function δφ 1 , which is most relevant in the semilocal limit [see Eq. (33)], and estimate its value at the origin. Here f (0) = 0, and we approximate the value of f d such that it minimizes the potential V of the theory when In the case of M H < M S , and θ W close to π/2, this is a negative contribution. It is found, that, quite remarkably, if the boundary curve of the domain of stability is plotted on the √ β 1sin 2 θ W plane (Fig. 3), the curves for different values of M S /M H coincide. The differences between the value of √ β 1 corresponding to the onset of instability between the cases considered is comparable to the numerical errors. For this reason, in what follows, when we consider the effects of other parameters, we shall only plot one curve, in this parametrisation.
In Table 1, we have collected some numerical data for reproducibility, and, for comparison, we have added the data points read off Fig. 1 Table 1: Some points on the boundary of the domain of stability; for comparison, we also show data read off of Fig. 1 of Ref. [17]. Another interaction, which is known to have a stabilising effect in the semilocal case is the GKM (see Ref. [21], where it is shown to lower the energy of semilocal-dark strings). Fig. 4 shows the effect of the GKM on the domain of stability. We have found that at the semilocal limit, the enhancement in the maximal value of the quartic potential coefficient β 1 is significant for a large GKM; however, this is rapidly reduced by tuning θ W away from π/2.
In Fig. 5, the effect of the mass of the dark gauge boson is shown. The sensitivity to the DS gauge boson mass is in contrast to the insensitivity in the case of stabilisation by the scalar potential (i.e., no GKM, Fig. 6).   Table 2. We have concluded, that the parameters with the largest influence are M S and θ s .

The behaviour of the eigenvalue
We have next varied M S > M H (so that dark Higgs decays do not exclude the considered parameter values) and θ s , in the range M 2 H ≤ M 2 S ≤ 2M 2 H and 0 ≤ θ s ≤ 1.5. We have found no stable solutions. The eigenvalue seems to depend most strongly on the parameters M 2 S and θ s . In Fig. 7 we present numerical data of the eigenvalue Ω 2 as a function of the two parameters that seem most relevant (i.e., they parametrise the scalar sector most directly), M S and θ s . Note, that the eigenvalue is always negative (signaling instability), and becomes more negative with larger values of the dark scalar mass M S .
In Fig. 8 Fig. 9 shows a typical Ω 2 -θ s curve in the parameter range studied. The curves in Fig. 8 and in Fig. 9 are cross sections of the surfaces in Fig. 7. In The data indicates clearly, that in the parameter range where the parameters of the dark sector are similar to those of the electroweak (visible) sector, no stable solutions exist. In this parameter range, we have found, that a larger dark scalar mass corresponds to stronger instability. On the other hand, for M S < M H , we do find a stabilising effect, and similarly, for large values of the GKM.
The fact that the stabilising effects are rather strong in the semilocal limit, and much weaker for smaller values of the Weinberg angle, are explained by the nature of the instability. In the semilocal model, the instability is due to the possibility of unwinding in the scalar (Φ) sector [19,20], however, in the full non-Abelian theory, the instability also involves the condensation of W bosons in the string core [14,15,16,17]; in the present model, the dark sector only couples to the Higgs scalar and the weak hypercharge U(1) fields, and does not influence the W fields other than slightly distorting the background vortex.

Discussion
In this paper, we have presented a study of electroweak-dark strings, complementing those of dark strings in Higgs portal models [31,32,33,34,37,38]. We have demonstrated the main properties of the equations describing these strings, and their numerical solutions. We have shown that these strings exists at the well known scale of electroweak strings, in contrast to the unknown scale of dark strings. We have also demonstrated, that the interaction with a dark sector has a stabilising effect on electroweak strings: gauge kinetic mixing has a significant effect, and also the Higgs portal coupling (at least for M S < M H ).

A Definitions and sign conventions
The gauge sector Lagrangian (1) was expressed with the gauge field strength tensors, which in turn, are expressed with the gauge vector potentials as follows: † denotes adjoint (transposed complex conjugate) and * complex conjugate, and τ a are the Pauli matrices in internal (isospin) space.
The fields W a µ , Y µ and C µ are referred to as VS SU (2), U(1) and DS U(1) gauge fields. The Lagrangians (1) and (2) expressed with these fields reflect the symmetries of the model in a manifest form. On the other hand, the particle content of the theory is better expressed with the so-called physical fields, for which, see Appendix B.
In what follows, we shall denote the middle line of the matrix M by α, i.e., In the new variables, the gauge Lagrangian can be recast as  where in Eq. (12) a = 1, 2 andW a µν = ∂ µ W 1 ν − ∂ ν W a µ (i.e., the linear part of the field strength tensor). The gauge covariant derivatives of the scalars become where Note, that g AS = 0, i.e., the dark scalar is indeed dark. The resulting particle masses are For more details, see Ref. [32].

C Particle content -scalars
In the scalar sector, the particles correspond to amplitude fluctuations of the Higgs field assuming a VEV (φ 2 ) and, similarly, amplitude fluctuations of the dark scalar χ [32]. Here we convert the formulae of Ref. [32] to our notations for convenience. The scalar mass matrix in the basis of the fields h = √ 2(|φ 1 | − η 1 ) and s = √ 2(|χ| − η 2 ) in the Lagrangian (2) is where m 2 H = 4λ 1 η 2 1 and m 2 S = 4λ 2 η 2 2 . The physical fields φ H , φ S are rotated at an angle φ s , and the scalar mixing angle is given as The corresponding eigenvalues (squared scalar masses) are For more details, see Ref. [32].

D Couplings
The couplings after diagonalisation are calculated in Ref. [32]; which are reproduced here with the replacement ǫ → −ǫ (for agreement with Ref. [21]):

E Details of the calculations
To obtain the solutions, and assess their stability, we start with the field equations derived from the gauge and scalar Lagrangians (12) and (2), where W µνa =W µνa + gε ab (W µb W ν3 − W νb W µ3 ), ε ab is antisymmetric and ε 12 = 1, the Abelian currents are given by and the non-Abelian one as where τ a denote the Pauli matrices. In Eqns. (21), (22), the a = 1, 2, and the notation g ZH = g Xφ 2 , g XH = g Zφ 2 , g Zφ + = g Xφ 1 and g Xφ + = g Xφ 1 is used (see Ref. [32]).

E.1 Radial equations of the vortex solutions
Plugging in the Ansatz (4),(5) into the field equations yields the radial equations, where ′ = d/dr, and r denotes the (rescaled) radial coordinate. Note, that without the dark sector, one would get the ANO vortex [10,11] embedded in the Z field. The energy density of a field configuration within the Ansatz (4), (5) is 2 ) with β i = 2λ i /g 2 ZH and β ′ = λ ′ /g 2 ZH , is the (rescaled) potential. The energy within a given radius is E(r) = 2π r 0 Erdr.