Interactions between several types of cosmic strings

We study the interaction of several types of static straight cosmic strings, including local strings, global strings, and bosonic superconducting strings with and without magnetic currents. First, we evaluate the interaction energy of two widely separated cosmic strings using the point source formalism and show that the most dominant contribution to the interaction energy comes from the excitation of the lightest mediator particles in a underlying theory. The interaction energy at arbitrary separation distances is then analyzed numerically by the gradient flow method. It turns out that an additional scalar field introduced in the bosonic superconducting string becomes an additional source of attraction. For such a bosonic superconducting string, we find that a string with two winding numbers is energetically favorable compared to two strings with a single winding number in a certain parameter region. Our analysis reveals that a phase structure of bosonic superconducting strings is richer than that of local and global strings and that the formation of bound states at intersections of bosonic superconducting strings is favored.


Introduction
In the early universe, cosmological thermal phase transitions associated with spontaneous symmetry breaking can take place. If the topology of the vacuum manifold is not simply connected, then after phase transitions, macroscopic one-dimensional line-like topological defects called cosmic strings can be produced through the Kibble-Zurek mechanism [1,2]. (See Ref. [3] for the review of cosmic strings.) A typical example of a (local) cosmic string solution is the Abrikosov-Nielsen-Olesen (ANO) string [4,5] realized in the Abelian-Higgs model, where the condensation of the Higgs field takes place. Further introduction of a new U (1) gauge group and a new Higgs field leads to a localized condensate of the new Higgs inside the string in a certain parameter region, showing persistent current-carrying superconductivity [6]. In addition, if the theory includes Yukawa coupling between the stringforming Higgs and new fermions, there also exists a current-carrying cosmic string solution [6,7]. Depending on the spin statistics of the charge carriers, such current-carrying string solutions are conventionally called bosonic or fermionic superconducting strings, and their dynamics and phenomena have been investigated in many literatures [8][9][10][11][12][13][14][15][16][17][18][19][20].
After the formation of cosmic strings, they constitute a web-like structure called a string network system. During the evolution of a string network system, cosmic strings can intersect many times, and the dynamics of these intersections can affect the fate of the string network system. In particular, when two cosmic string segments intersect, it is conceivable that they generically snap and reconnect with their partners. This process is called reconnection (or intercommutation), and occurs dominantly for simple string solutions such as local strings and global strings, unless the relative velocities are sufficiently fast [21][22][23][24]. Interestingly, if the interaction between two string segments is attractive and strong enough, Y-shaped junctions (Y-junctions) can form after string intersection [25]. The formation of Y-junctions is discussed in Refs. [26][27][28][29][30][31][32]. Characteristic phenomena of Y-junctions such as distinctive gravitational lensing [33,34], gravitational wave bursts due to cusps and kinks [35][36][37] have also been discussed.
As a first step toward clarifying the dynamics of string network systems, including the formation of Y-junctions, it is very important to reveal the interaction between two static, straight cosmic string segments. (Here, "static" means the velocity of the string is zero.) Analytical studies of the interaction between two static, parallel global and local strings are initiated in Refs [21,38] under the assumption that the whole field configurations can be approximated by the superposition of each string. After these works, Speight proposed a novel method, called point source formalism, to calculate the interaction energy of two local strings [39]. It is shown that the asymptotic field configurations of a local string can be well represented by line-like static sources that couple linearly to the fields in the underlying theory. When two strings are sufficiently separated, the total external sources of the twostring system are given by the superposition of each string's source. The interaction energy can be then evaluated in linearized field theories using the standard Green's function method. This formulation can be easily extended to the calculation of interaction energies for more complicated cosmic string solutions.
On the other hand, when two static cosmic string segments are sufficiently close together, the point source formalism is not suitable to estimate the interaction energy. Clearly, in such a case, the non-linear effects caused by internal structure of cosmic strings on interaction energy cannot be ignored. Therefore, when two strings are very close together, it is very challenging to compute interaction energy analytically, which results in the need for numerical calculations that do not rely on the point source formalism. The interaction energy of two local strings at arbitrary separation distances have been accurately evaluated using the variational approach in Ref. [40] and the gradient flow method in Ref. [41]. (See also Refs. [42,43] for related works.) While the interaction energy and the dynamics of intersections of a simple two-localstring system have been clarified, cosmic string solutions with more complicated field configurations, such as bosonic superconducting strings, have not been fully understood yet. Ref. [44] has studied the interaction energy of bosonic superconducting strings without current, using the point source formalism that takes into account the effect of another scalar field condensing inside the string. However, there have been neither analytical studies of the interaction energy of bosonic current-carrying strings using the point source formalism nor numerical studies of bosonic superconducting strings that include the effects of nonlinear internal structure. 1 One of the main purposes of the present paper is to determine the interaction energy of two static, parallel bosonic superconducting strings using analytic calculations with the point source formalism and numerical calculations with the gradient flow method. In addition, for completeness, we will also apply this to local(ANO) and global strings.
For bosonic superconducting strings, we consider both the cases with and without magnetic currents associated with newly introducedŨ (1) gauge group. For the case without magnetic currents, we show that the interaction sourced by the Higgs field in the fundamental representation of U (1) is always attractive, which gives a dominant contribution to the interaction energy in a certain parameter space. Especially, the short-range attraction due to the present of this Higgs field results in a non-trivial phase structure of the interaction energy of the bosonic superconducting strings. We argue that this short-range attraction cannot be seen in the point source formalism and can only be captured by a full numerical calculation. In the case with magnetic currents, we find that the contribution triggered by the U (1) gauge field is related to the directions of the currents. But its effect is only important in the long-distance limit due to the effect of the back-reaction on U (1) Higgs field (current quenching effect), which allows only small values of currents. Meanwhile, the short-range attraction is found to be weaker with the presence of magnetic currents as another result of current quenching effect. Consequently, this attraction is maximal when the magnetic current is zero.
This paper is organized as follows. In Sec. 2, we review several static cosmic string solutions, including local(ANO) strings, global strings, bosonic superconducting strings. We then show that asymptotic configurations can be obtained in the linearized field theory by introducing certain external sources. In Sec. 3, the interaction energies of various string segments are estimated using the point source formalism. In Sec. 4, we present our numerical results using the relaxation method (gradient flow method). Sec. 5 is devoted to conclusions and discussion. Some numerical details are given in Appendices.

Cosmic string solutions
In this section, we review various types of cosmic string solutions, including local, global, bosonic superconducting cosmic strings. We derive the asymptotic configurations of these strings analytically in the long-distance limit. It is shown that the asymptotic field configurations can be obtained by introducing corresponding external sources that linearly couple to fields in the free field theory.

Local strings
Let us first consider the simplest cosmic string solution realized in the Abelian-Higgs model, known as the Abrikosov-Nielsen-Olsen (ANO) string [4,5]. The Lagrangian density of the Abelian-Higgs model is given by 2 (2.1) where F µν = ∂ µ A ν − ∂ ν A µ and D µ ≡ ∂ µ − ieA µ are the field strength and the covariant derivative of the U (1) gauge field, respectively. The variational principle leads to the following equations of motion: Here, J µ is the Noether current associated with local U (1) symmetry given by The ANO string solution is obtained by considering the boundary conditions with nontrivial windings. Assuming that the cosmic string configuration is static, straight (here and hereafter we use "straight" to indicate translational symmetric in the string direction) and circularly symmetric on the plane perpendicular to the string axis direction, we can use the following ansatz: Here, (r, θ, z) is cylindrical coordinate, where r, θ and z is the radial distance, the angle and the height, respectively. ϕ r (r) and A θ (r) are real functions, and n is a non-zero integer number called winding number. Boundary conditions of ϕ and A µ at infinity are determined by finiteness of the energy, that is, the scalar potential, V (ϕ), and the covariant derivative D µ ϕ must vanish at infinity: Regularities of |ϕ| and A θ at the origin require the following boundary conditions, Here, A ≡ A µ dx µ while the integration is taken over the circle enclosing the origin in the two-dimensional plane which is perpendicular to the straight ANO string. Note that the first equality only holds at infinity. Let us now focus on the asymptotic solutions. We work in the different field parameterizations defined by where σ(x) is the radial component of the ϕ field and π(x) is the Nambu-Goldstone boson (NG-boson). Although the above field parameterizations are ill-defined at the origin ϕ = 0, they are convenient to seek asymptotic field configurations of the ANO string at infinity, as we will see below. With these field parameterizations, the Lagrangian density Eq. (2.1) can be rewritten as where m ϕ ≡ λ ϕ η and m e ≡ √ 2eη are masses of the scalar and U (1) gauge field evaluated at infinity, respectively, and Let us find asymptotic solutions for the scalar and the gauge fields. Circular symmetry restricts forms of field configurations as σ(x) = σ(r) and U µ (x) = U θ (r)δ µ θ . Since σ(r) ≃ 0 due to the boundary conditions Eq. (2.5), we can neglect terms that depend on σ(r) at infinity. Then the equation of motion for U θ (r) becomes linear and is given by Therefore, we can obtain an analytical asymptotic solution of U θ (r) under boundary condition Eq. (2.5) as where K m (x) is the modified Bessel function of the second kind order m, while k e is a numerical constant. It is worth noting that the modified Bessel functions have the following properties, K 0 (r) = π 2r e −r , K 1 (r) = π 2r e −r , (r → ∞). (2.12) Since K 1 (x) is exponentially suppressed at large distance x ≫ 1, we can safely neglect nonlinear terms of U θ , and then, the equation of motion for σ(r) is simplified as Here, we have assumed that the quadratic term of the gauge field (U θ ) 2 is negligible compared to σ(r), which is justified only for λ < 8e 2 [50]. The solution consistent with the boundary condition Eq. (2.5) is given by where k ϕ is a constant. For the parameter region λ > 8e 2 , the quadratic term (U θ ) 2 is not negligible, and thus, the asymptotic expression Eq. (2.14) is no longer justified [50]. In this case, the (U θ ) 2 term has to be included to derive the correct asymptotic configurations. Numerical constants k ϕ and k e are determined by the requirement that exact numerical solutions of Eq. (2.2) in the long-distance limit should coincide with the asymptotic solutions. For the ANO string configuration, this procedure is explicitly examined in Ref. [39]. For a special case λ/2e 2 = 1 called the critical coupling or Bogomol'nyi-Prasad-Sommerfield (BPS) limit, analytic expressions have been found [38].
Let us next find external sources that represent the presence of an ANO string with given constants k ϕ and k e . Following Ref. [39], we introduce external sources J σ and j µ in the linearized field theory as At infinity, since string configurations are described by Eqs. (2.11) and (2.14), we can read off the explicit forms of external sources: Here, we change the two-dimensional coordinate from (r, θ) to the two-dimensional Cartesian coordinate x ≡ (x, y) for later convenience. The Laplace operator is defined by ∇ 2 ≡ ∂ 2 x + ∂ 2 y , while ϵ ij is the two-dimensional Levi-Civita symbol. δ (2) (x) is the two-dimensional delta function. Following relations are used to derive Eq. (2.17) The above result implies that the ANO string configurations σ(r) and U θ (r) can be successfully represented by point-like external sources in the long-distance limit where the internal structure of the string is coarse-grained. As argued in Ref. [39], J σ and j µ represent a scalar monopole with charge −2πk ϕ and a magnetic dipole with moment −2πk e /m e whose direction is parallel or antiparallel to the z-direction depending on the sign of the winding number, respectively. In particular, the existence of magnetic dipole moments follows from the presence of the magnetic flux inside the string given by Eq. (2.7). Analogous to a charged electron linearly coupled to a massless photon, J σ and j µ couple to a massive scalar σ and a gauge field A µ , respectively, for a local string. This picture is very useful when calculating the interaction between two strings, as seen in Sec. 3.

Global strings
In this subsection, we derive solutions for global strings and show that the existence of global strings can be approximately described by external currents associated with scalar and topological charges. The scalar and topological charges are linearly coupled to a massive scalar field and a two-index antisymmetric tensor field, respectively. The Lagrangian density of the Goldstone model with a complex scalar field is described by where V (ϕ) is defined as same as Eq. (2.1). With the scalar field parametrization given by Eq. (2.8), above Lagrangian can be rewritten as By using the ansatz of the static, straight string solution given by Eq. (2.4), we obtain the equations of motion of σ and π fields as Assuming the power law dependence, σ(r) ∝ r −a (a > 0), at infinity and inserting π = nθ into above expression, we obtain an asymptotic solution of σ(r) at the leading order, (2.23) We have confirmed that this result is in agreement with the numerical one. We first focus on the external source describing the massive excitation parameterized by σ(r). We introduce the external source as J G is also determined by the requirement that it reproduces the scalar configuration given by Eq. (2.23). Substituting the asymptotic solution in Eq. (2.23), to the leading order, one obtains From the above expression, it is obvious that J G is not a localized source, which is different from the point-like J σ in the ANO string. This stems from the fact that a contribution from the NG-boson, which is a massless degree of freedom originated from the term (∂ µ π) 2 in Eq. (2.21), is dominant at infinity. For the local string, this contribution is canceled by the gauge field configuration. We now turn our attention to external sources describing the massless excitations of the global string. Since a non-trivial winding at the center of the string implies the existence of a magnetic flux associated with a NG-boson, it is appropriate to introduce a dual field of NG bosons [51][52][53]. The dual field of the massless real scalar field π(x) is a two-index antisymmetric tensor field B µν satisfying the following equation: This equation makes sense when |ϕ| ̸ = 0 and π(x) is equivalent to B µν in Minkowski spacetime as long as the on-shell condition Eq. (2.26) is satisfied [51]. Then the Lagrangian density in terms of B µν can be written as where we have omitted the terms only depending on σ because they do not play an important role in the following discussion. H µνρ is the field strength tensor of B µν .
In the dual picture, the interaction of NG-boson and its magnetic charge can be equivalently expressed by introducing the electric source of the dual field, Here, J µν B is the external source of B µν . The equation of motion of the dual field is given by Due to the antisymmetric property of ϵ µνρσ , the right-hand side of the above equation is zero, except at the origin, |ϕ| = 0. At the origin, π is not well-defined, therefore, one cannot conclude that J µν B vanishes at the origin. The presence of the magnetic charge of the NGboson can be seen by integrating over the two-dimensional plane, which is perpendicular to the straight string axis, as We see that J µν B is zero except at the origin, and integrating over a two-dimensional plane gives a nonzero finite value. This is exactly the definition of a delta function, and consequently we have Since this external source represents the winding number of the global string, the interaction Eq. (2.29) with the external source Eq. (2.33) can be understood as a topological coupling between the NG-boson and the global string.

Bosonic Superconducting strings
In this subsection, we review the bosonic superconducting string and derive its asymptotic solutions. The simplest model that accommodates bosonic superconducting solutions possesses two gauge symmetries with U (1) × U (1) and the Lagrangian density is defined by [6] the field strength and the covariant derivative of the U (1), and U (1) gauge field, respectively. Here, we simply assume λ ϕ > 0, λφ > 0 and β > 0. The bosonic superconducting solution is realized when U (1) symmetry is spontaneously broken and gives rise to the ANO string configurations, as we discussed in Sec. 2.1, while U (1) symmetry is only broken in the string interior by the localizedφ condensation. Thus |ϕ| = 0 and |φ| ≃ ηφ at the string center, and |ϕ| = η ϕ and |φ| = 0 at infinity are realized. We shall discuss the parameter region leading to the bosonic superconducting cosmic string solution. For an illustrative purpose, we here neglect theŨ (1) gauge field and the non-trivial z-dependence of the phase ofφ, which will be included later. A presence of the biquadratic interaction of the form β|ϕ| 2 |φ| 2 contributes to the effective mass term ofφ as U (1) symmetry is unbroken well outside the string when the square-mass ofφ is positive, that is, ϕ is required to avoid the metastability at the minimum of potential energy, |ϕ| = η ϕ and |φ| = 0, corresponding to the string exterior. Near the center of the string, |ϕ| ≃ 0, condensation |φ| ̸ = 0 occur around the center of the string due to the instability m 2 ϕ < 0 atφ = 0. In this argument, however, we disregard the contribution from the gradient energy ofφ. It is shown in Refs. [6,54,55] that the localized condensation ofφ is energetically favorable for β ≲ λ 2 ϕ η 4 ϕ /(4λ ϕ η 4 ϕ ) including the gradient energy under the approximation that ϕ(r) ≃ r n around r = 0 and the back-reaction ofφ on ϕ is negligible. As a result, conditions to form the bosonic superconducting string without effects ofŨ (1) gauge field are summarized as In this class of models, the cosmic string can carry a current flowing along the string and exhibit superconductivity when the phase ofφ has a non-trivial z-dependence. For a static straight string, the general ansatz ofφ andÃ µ are given by [56] where s and α are real functions. This parameterization is not a mere gauge transformation fromφ =φ r andÃ µ = 0 unless α(z) is independent of z. Therefore it describes the real physical excitation. This ansatz is equivalent to the following expression up to the gauge transformation,φ When s(r) is not independent of r, the static solution of this equation is simply given by α(z) = ωz where ω is a constant. With ϕ and A µ following the same ansatz as the ANO string in Eq. (2.4), the equations of motion of ϕ r , A θ ,φ r and s are given by Here, we use the normalizations(r) = ωs(r) for convenience. Boundary conditions at infinity are determined by the finiteness of the energy, and are given by The first and the second conditions are the same as those of the ANO string in Eq. (2.5).
The regularity at the origin requires As in the case of the ANO string, no exact analytic solution is known. The numerical solutions of Eqs. (2.40) are shown in App. A. Let us now seek the asymptotic solution for the current-carrying superconducting string. Assumingφ r falls faster thans(r) 2 at large r, we neglect the third term and the non-linear terms ofφ r in the equation of motion forφ in Eq. (2.40). Noting thats(r → ∞) ≃s(r = 0) contributes to the effective mass ofφ, but we do not know the value ofs at infinity, Eq. (2.40) can be approximated by 43) and its solution at infinity is given bỹ where kφ is a numerical constant. Sinceφ r is exponentially suppressed at long distance, thesφ 2 r term in Eq. (2.40) can be ignored. As a result, the equation becomes the standard Laplace equation in two dimensions without angular dependence, whose solution is given bỹ Here, k s and k s0 are numerical constants. δ is the typical width of string core. We confirm that these asymptotic solutions in Eqs. (2.44) and (2.45) agree with our exact numerical calculations. Two numerical constants kφ and k s are then determined by the fitting of numerical calculations shown in table 1. We now briefly explain the effect of U (1) current induced by the condensation ofφ. The total amount of the current associated with U (1) symmetry flowing along the straight string is defined by Here, J z is the z-component of the Noether current associated with U (1). Since ϕ r condensation is strongly localized around the string, J z (r) is trapped inside the string. Very interestingly, one cannot make J tot arbitrarily large due to the following reason. It is clear that the effective mass term ofφ defined by Eq. (2.35) is now modified by the non-zeros as For a large value ofs(r = 0), the effective mass term at the origin becomes positive even withφ r = 0 near the string center, which results in symmetry restoration of U (1) and no condensation ofφ. Hence, for a large amplitude ofs(r = 0), a back-reaction toφ is significant which makes the strength of the condensation weaker. Consequently, J tot begins to get smaller ass(r = 0) is sufficiently large to have significant back rection onφ r . This phenomenology is known to be current quenching [6]. On the other hand, J tot also becomes smaller for a smallers(r = 0) when it is small enough that the back-reaction froms onφ r is a negligible amount. Therefore, there exists a maximum of the total current flowing inside the string, at least in the case of the magnetic current.
Next, let us find external sources that represent the bosonic superconducting string. As in the case of the ANO string, we introduce currents Jφ andj µ that reproduce the asymptotic field configurationsφ sol r andÃ sol µ ⇔s sol in the linear theory, In this expression, L AH is defined by Eq. (2.15). Substituting the asymptotic solutions into the equations of motion derived from Eq. (2.48), one can read off corresponding sources as (2.49) Here, k A = k s /g is a numerical constant. As is the case in the ANO string, a non-trivial configuration ofφ r can be expressed by Jφ regarded as the monopole charge of −2πkφ.
Similarly, a non-trivial configuration of the gauge fieldÃ z is sourced by the static magnetic current flowing along the string. One should note that k A can be expressed as k A = 2π J tot , as is evident from the Gauss's law. Therefore, unlike other numerical constants, k ϕ , k e and kφ, there is a strong restriction on k A due to the current quenching effect at least for the magnetic current.

Cosmic string interactions: Analytic studies
In this section, we investigate the interaction energy between two strings using the method proposed in Ref. [39]. In the following analysis, we assume that two strings are straight and static, separated by a fixed distance, and that each string possesses a single winding number, n = ±1. We will argue that the most dominant contribution comes from the lightest field in the underlying theory.

Local strings interactions
In this subsection, we compute interaction energies of local(ANO) strings. When two strings are widely separated, effective descriptions of ANO strings with external sources in Eq. (2.15) are applicable. We assume the whole field configuration can be approximated by the superposition of each source, Here, we introduce the two-dimensional Cartesian coordinate x = (x 1 , x 2 ) which parameterizes the plane perpendicular to the string axes.
One can solve the above differential equations by the standard Green's function method: Putting above solutions into original Lagrangian given by Eq. (2.15), we obtain The interaction energy is then evaluated as In this calculation, we have used the fact that D is the two-dimensional Yukawa potential with mass M . Substituting the superposed external sources in Eq. (3.1) into Eq. (3.5), we find This result is in agreement with the previous studies [38,39]. The first term of Eq. (3.8) represents the interaction energy coming from the massive gauge field associated with the presence of the magnetic flux of the ANO strings, while the second term is the contribution from the massive scalar field. The signs of Eq. (3.8) simply reflect the fact that the gauge interaction through the magnetic flux is repulsive for winding numbers with the same signs, while the scalar interaction is always attractive since the charge of scalar field should always take the same sign. Due to the asymptotic forms of K 0 (x), the interaction energy is exponentially suppressed on length scales longer than the inverse mass of the underlying field. For 2e 2 /λ > 1 (m ϕ < m e ), the scalar attraction is always dominant at d → ∞. Conversely, if 2e 2 /λ < 1 (m ϕ > m e ), the repulsion sourced by the gauge field is dominant at d → ∞ if the winding number of two strings are opposite. These arguments are essentially independent of the precise values of the charges, k ϕ and k e , as long as they are finite values. Therefore, the interaction energy of the string at d → ∞ can be easily revealed by applying the point source formalism.
Once the values of λ ϕ and e 2 are specified, k ϕ and k e can be calculated explicitly as explained in Sec. 2.1. In this case, as in the original analysis [39], we can discuss which interactions are dominant at any d . We will do this in the next section and discuss the validity of the point source formalism by comparing the results obtained by the non-linear numerical calculations with the analytic ones.

Global strings interactions
In this subsection, we estimate the interaction energy of two straight and static global strings. In the following discussion, we first focus on the interaction energy from the topological charge. The effective Lagrangian density of the NG-boson is given by Eq. (2.29) through its dual fields. If the two global strings are far apart, the external sources can be approximated by a superposition of their respective sources, where n 1 and n 2 are winding numbers of global strings whose axes placed at x = x 1 and x = x 2 , respectively. In the region sufficiently far from the string axis, one can use the approximation |ϕ| ≃ η ϕ . Then one can solve the equation of motion of the dual field B µν in Eq. (2.30) by using the standard Green's function method as Here, we fix a gauge by imposing ∂ µ B µν = 0, which is analogous to the Coulomb gauge in the classical electrodynamics. 3 We introduce infinitesimal mass ϵ of B µν to avoid infrared divergence, which will be taken to be zero at the end of calculation. Then the interaction energy becomes where D B is defined by two-dimensional massless propagator, The interaction energy is then evaluated as The approximation of modified Bessel function K 0 (ϵd) ≃ − log(ϵd) for ϵd ≪ 1 is used in the last equality. We redefine the origin of the interaction energy at d = m −1 η . This yields (3.14) The above logarithmic dependence at long distance, d → ∞, is in agreement with previous studies [21,38]. The massless two-index antisymmetric tensor field leads to the long-range force. The same-sign topological charges of two global strings can be regarded as the same magnetic charges of axions which lead to the repulsive force. Let us next estimate the interaction energy of the massive mode. Assuming that two strings are widely separated such that external sources can be approximated by the following form, Then we obtain the interaction energy of σ field as There exists divergent contribution at two string axes, x = x i (i = 1, 2). Obviously, effective descriptions by the external sources cannot be justified at these points. Therefore we introduce the cutoff which is of the same order of the string thickness δ = 1/(λ ϕ √ η ϕ ). Then one can evaluate the interaction energy as This interaction energy is suppressed by 1/d 2 in the long-distance limit. Thus, the interaction energy is dominated by the external source associated with the topological charge in (3.14) at d → ∞.

Bosonic superconducting strings interactions
In this subsection, we compute the interaction energy of bosonic superconducting strings. As discussed in Sec. 2.3, there are four external sources corresponding to non-trivial field configurations in the U (1)× U (1) model. Since the asymptotic behavior of the string configurations associated with the spontaneously broken U (1) is exactly the same as for the ANO string, the interaction energy originated from these fields is given by Eq. (3.8). Here, we estimate the interaction energy sourced by the U (1) Higgs field, and the current trapped inside the string. In the following discussion, we omit the contributions from the U (1) sector, but they are always implicitly present. An effective description of the widely separated bosonic superconducting strings is given by Eq. (2.48) with the following superimposed external sources,  Table 1 is used. Also shown is a plot ofĒ int in the region of 8 <d < 20 using the same benchmark point.
Calculation of the interaction energy is straightforward using the Green's function method.
(See e.g. Sec. 3.1 for detailed computations.) The resultant interaction energy from U (1) sector is expressed as Here, δ is the typical width of the strings. The first term is originated from the localized condensate scalar fieldφ, while the second term comes from the current trapped inside the string. The constant kφ does not depend on the sign of the winding number or the direction of the current. Regardless of the directions of the two strings, kφ 1 and kφ 2 should take the same signature. This implies that the first term is always attractive. The second term, on the other hand, depends on the direction of the static currents. If the direction of each current is the same (opposite), then it is an attractive (repulsive) force. It is worth noting that the first term and the contribution from the U (1) sector are exponentially suppressed on their mass scales, whereas the second logarithmic term is always dominant at infinity. This stems from the fact that the mass ofφ r is generically non-vanishing at infinity, while the U (1) gauge field is massless because the condensate ofφ r is localized inside the string. Figure 1 shows the dependence of the interaction energy of bosonic superconducting cosmic strings, the sum of Eq. (3.8) and Eq. (3.19), on the separation distance for the benchmark point A specified in the App. A. We used the normalized quantityd andĒ int in the figure according to the App. A. The charges k ϕ , k e , kφ and k A are shown in Table 1. As shown in the right panel of the figure, the logarithmic dependence ofĒ int ond is observed in the long-distance limit. k A is the charge associated with the U (1) current trapped inside the string, and is found to be one or two orders of magnitude smaller than the other charges due to current quenching effects, as explained in Sec. 2.3. Tiny values of k A have a small effect on E int compared to other contributions when d is comparable to or shorter than the mass scale of all fields in the underlying theory. Thus, the effect of non-zero currents is only important in the long-distance limit. This is a generic feature of bosonic superconducting strings and can also be observed for other parameter choices.
- 15 -In this section, the interaction energy of static and straight strings is evaluated numerically for arbitrary separation distances. In particular, the gradient flow method is used. This method was recently employed in Ref. [41] to reveal the interaction energy of two conventional local strings and two local strings with Coleman-Weinberg potential.
The action for numerical simulations is the same as Eq. (2.34): (4.1) If one neglects the part ofφ andÃ µ with β = 0 and g = 0, this action represents local strings, and if one further neglects the part of A µ with e = 0, it represents global strings. We use the normalization described in App. A.1 for local strings and superconducting strings. For global strings, we use another normalization described in App. A.2. We prepare a two-dimensional lattice plane and place the two axes of the strings at (x,ȳ) = (±d/2, 0), where (x,ȳ) is the two-dimensional Cartesian coordinate. Here, the position of string axis is defined as the point where the value of the scalar field,φ, vanishes, that is, |φ| = 0. In the following analysis, we will focus on the case where two strings having the unity winding numbers with the same signs. The results for two strings with opposite winding numbers are discussed with point source formalism in Chap. 3. Analysis of the interaction energy between cosmic strings with winding numbers more than unity is beyond the scope of the present paper. The number of lattice sites is 600 × 600 and the box length isL = 30 (corresponding to lattice spacing withā = 0.05), unless otherwise stated. The details of the simulation setup and the numerical recipe are explained in App. B.
Before presenting the numerical results, let us explain how the axes of the two strings are fixed in the numerical simulation. If we do not fix the angular mode ofφ but fix the string axes by simply imposing |φ| = 0 at (x,ȳ) = (±d/2, 0), the numerical simulation shows that four points ofφ ≃ 0 are observed at large values ofλ ϕ ≫ 2. Two of these points are located on the initial string axes (x,ȳ) = (±d/2, 0), but the other two should not appear. Therefore, only imposing the condition |φ| = 0 on the initial string axes may not guarantee a fixed separation distanced. We thus fix the angular dependence ofφ in the whole region of the simulation box, in addition to the condition |φ| = 0 at (x,ȳ) = (±d/2, 0) as in Ref. [40]. In this case, we confirm that the positions of the two string axes are maintained in the numerical simulation, and hence, the fixed separation distanced is well-defined. (See App. B for detailed procedures to fix the angular dependence ofφ.) For this reason, we fix the angular dependence ofφ for the entire simulation box in the following analysis.

Local strings
In this subsection, we present numerical results for the interaction energy of two local strings, originally calculated using the variational method in Ref. [40]. Figure 2 shows the field configurations and the energy density of two local strings for λ ϕ = 2 with a fixed separation,d = 2, on aȳ = 0 slice, while Fig. 3 displays these surface plots with the same parameter set. We confirm that the configurations ofφ andĀ i can be approximated byφ =φ +φ− andĀ i =Ā i+ +Ā i− , whereφ ± andĀ i± (i = 1, 2) are the n = 1 local string solutions whose axes are fixed at (x,ȳ) = (±d/2, 0) for sufficiently larged. It is also confirmed that the n = 2 solution of local strings is reproduced in the limit ofd = 0 as it should be. Figure 4 displays the dependence of the energy density ond at different values ofλ ϕ . From this figure, it is easy to see that the dependence ofĒ int ond can be classified into three cases:λ ϕ ≷ 2,λ ϕ = 2. In the special case withλ ϕ = 2,Ē int does not depend ond. Forλ ϕ > 2, E int becomes smaller asd is increased. In this case, a stable solution is obtained in the long distance limit,d → ∞, but the n = 2 ANO solution is unstable against perturbations ofd. Conversely, forλ ϕ < 2,Ē int becomes smaller asd becomes smaller. Therefore, the n = 2 ANO solution is stable forλ ϕ < 2 in contrast to the case withλ ϕ > 2. It should be noted that, since the field configuration with n = 2 is a static solution to the equation of motion, the first derivative ofĒ int with respect tod must vanish atd = 0 for an arbitraryλ ϕ . The stability of local strings is generically difficult to capture with an analytic approach whend is small, except forλ ϕ = 2.0. However, as explicitly shown in Ref. [41], analytic estimation is possible by using perturbations around the BPS state ifλ ϕ is close to the BPS limitλ ϕ ≃ 2.
Finally, in Fig. 5, the results obtained by our full numerical calculation based on the gradient flow method are compared with those obtained by the point source formalism and the variational approach [40]. In the figure, we focus on two benchmark points used in Ref. [40]. We see that our results are in very good agreement with those of Ref. [40]. For short distances where the inverse of the lightest mass among the scalar and the gauge bosons is longer than d, the results of the point source formalism deviate from the full numerical calculation. In particular, analyses relying on the point source formalism indicate the existence of a saddle  point or a minimum ofĒ int with respect tod, suggesting a non-trivial phase structure of the local string. Our full numerical calculations, however, did not find such a non-trivial phase structure. This is because an effective description of the local string by the point source formalism cannot be justified for smalld.

Global strings
In this subsection, we present numerical results for the interaction energy of two straight global strings. A numerical study of the interaction energy of two global strings is given in Ref. [58]. Figure 6 shows the field configuration of |φ| and the energy density of two global strings onȳ = 0 slice forλ ϕ = 4 with the fixed separation,d = 2, while Fig. 7 is a surface plot for the same parameter set. We have numerically confirmed that the whole field configuration cannot be well approximated by the form of superposition ansatz,φ =φ +φ− , whereφ ± is the global string configuration with unit winding number. Their axes are placed at (x,ȳ) = (±d/2, 0), similarly to the local string case. The interaction energy densityĒ int (x,ȳ) has a kink around the string axes because the position of the string axes are fixed by hand. Figure 8 shows the dependence ofĒ int ond in the case of a global string withλ ϕ = 4. In the right panel of the figure, for clarity, the origin of the vacuum energy of the local string is shifted such that itsĒ int atd = 0 coincides with that of the global string. The logarithmic dependence ofĒ int ond is observed at larged, which is in good agreement with the results of the point source formalism and previous study [58]. When the distance from the string axis is closer than m −1 e , the angular mode ofφ is not canceled by the gauge field, so the behavior of the local string is almost identical to that of the global string. Thus, when d ≲ m −1 e , the behavior ofĒ int of the local string is the same as that of the global string. This feature can be seen from the right panel of Fig. 8. Therefore, the phase structure of the interaction energy of the global string can be understood as the limit of e → 0 (m −1 e → ∞).

Bosonic superconducting strings without current
In this subsection, we show our numerical results for the interaction energy of two bosonic superconducting strings without current. It can be seen that the introduction of an addi- tional scalar field whose condensate is localized inside the string has a dramatic effect on the interaction energy in a certain parameter region. Figure 9 shows field configurations and the energy density of two superconducting strings onȳ = 0 slice with fixed separation,d = 2.1, for the benchmark point withλ ϕ = 8,λφ = 80,β = 24 andηφ = 0.55. Figure 10 is a surface plot for the same parameter set. The whole field configurations can be approximated by the superposition ansatz,φ =φ +φ− ,Ā i = A i+ +Ā i− , andφ r =φ r+ +φ r− , whereφ ± ,Ā i± andφ r± are the configurations of each string whose axis is placed at (x,ȳ) = (±d/2, 0), respectively. The same benchmark point is taken as in Figs. 11 and 12 ford = 2. By comparing the condigurations atd = 2.1 andd = 2, we can see the coalescence ofφ happens atd = 2 where the magnitude ofφ enlarges and configurations of two string attach to each other. This coalescence cannot be captured by the point source formalism and leads to a characteristic phase structure of interaction energy, as we will discuss below. Figure 13 shows the dependence ofĒ int ond by the gradient flow method and by the point source formalism for the benchmark point withλ ϕ = 8,λφ = 80,β = 24 andηφ = 0.55. As can be seen from the figure, the results relying on the point source formalism are in good agreement with those of the gradient flow method at infinity. Figure 14 shows the dependence ofĒ int on the separation distanced for variousλ ϕ ,ηφ,λφ andβ in the parameter space which allows the formation of bosonic superconducting strings with the unit winding number. In the figure, the origin of the vacuum energy is adjusted such that all interaction energies coincide atd → ∞ for visibility. In order to understand the effect ofφ r onĒ int , we comparē E int of the bosonic superconducting string with that of the local string in the figures. These figures show that the dependence ofĒ int ond is almost the same as that of local strings. On the other hand, it is drastically different for smalld becauseĒ int is a decreasing function with respect tod. A remarkable feature is that the first derivative ofĒ int looks discontinuous at the transition point,d =d c , within the resolution of our lattice setup. We suppose that this kink might be smoothened as we improve the lattice resolution. We find that this kink behavior is triggered by the coalescence ofφ since the coalescence ofφ also happens atd =d c . The dependence of E int on d is roughly understood as follows. For bosonic superconducting strings (without current), there are three important length scales associated with the scalar field |ϕ|, the U (1) gauge field, andφ, namely, m ϕ , m e , and mφ evaluated at infinity. For all benchmark points taken in the Fig. 14 aforementioned coalescence ofφ r takes place, and the non-linear effect ofφ is important for 2m −1 ϕ ≳ d. In this region, the configuration ofφ r possesses approximate circular symmetry, and the gradient energy ofφ r gives rise to the attraction. This additional attraction due tõ ϕ r gives the bosonic superconducting string a richer phase structure than the conventional local string. The effects of ηφ, λφ, λ ϕ , and β on the phase structure of the interaction energy may be qualitatively understood by these arguments since these parameters are related to the mass parameters, m e , m ϕ and mφ. These qualitative features are numerically confirmed to be the same for other hierarchical mass structures.

Bosonic superconducting strings with current
In this subsection, we discuss numerical results including the effect of non-zero currents associated with the U (1) gauge field. Because the massless U (1) gauge field has a long-range effect on another string, we expand the lattice simulation box is from 600×600 to 1000×1000 to see long-distance behavior and to have enough space to decouple two strings while keeping the lattice spaceā = 0.05. Figure 15 shows the field configurations and energy density of two static superconducting strings onȳ = 0 slice with fixed separation,d = 2, forλ ϕ = 8,λφ = 80,β = 24,ηφ = 0.55,s(0) = 0.4, while Fig. 16 is a surface plot for the same parameter set. We numerically confirm that the whole field configuration of U (1) gauge field can be approximated by a superposition of each string when two strings are far enough from each other. Figure 17 shows the dependence ofĒ int of superconducting strings on a separation distanced with various currents, including zero current. For clarity, all vacuum energies are matched atd = 12. The logarithmic growth of the U (1) gauge field can be seen in the long-distance limit, but it is very hard to see because of the small charge k A due to current quenching effects as explained in Sec. 2.3. In fact, the value of k A is tiny on the order of 10 −2 ∼ 10 −3 , while the other charges are order unity as shown in Table 1.
As we have shown for the case without currents, the gradient energy ofφ leads to an additional attraction at short range. If the amplitude of the U (1) gauge field,s, on the string axis is not zero, theφ r condensation becomes smaller, and consequently, the attraction becomes weaker than in the absence of current, as discussed in the previous section. As the initial value ofs increases, the strength of the attraction becomes even weaker. We can see this clearly from Fig. 17. This effect can be explained by the back-reaction onφ r , which is the same mechanism as that of current quenching. Thus, even if an additional attraction due to the U (1) gauge field is introduced, the total attraction at short range is still suppressed by the back-reaction effect. This implies that the attraction in the U (1) sector is maximized when the current is zero at a separation distance comparable to the string thickness.

Conclusions
In this paper, we investigated the interaction energy of two static, straight, various types of cosmic strings using the point source formalism proposed in Ref. [39] and numerical calculations using the gradient flow method. In the analysis based on the point source formalism, we analytically derived the asymptotic configurations of the cosmic strings and showed that the asymptotic field configurations can be realized by the linearized field theory introducing appropriate external sources. The interaction energy can then be calculated using the standard Green's function method under the assumption that the external source can be approximated by a superposition of each string. We confirmed that our results are consistent with those of previous studies on local [39] and global strings [58]. In the case of bosonic superconducting strings, the Higgs field in the fundamental representation under the new gauge group U (1) is an additional source of attraction. In addition to this, if the static magnetic current is trapped inside the string, it becomes a source of long-range attraction (repulsion) if the directions of the currents in the two strings are parallel (antiparallel). However, due to the current quenching effect, there is an upper bound on the current, which leads to the upper bound on the charge of the string. As a result, the U (1) gauge field cannot be the most significant source of interaction unless the strings are very far apart. This is the essential reason why the apparent discontinuity (kink behavior) of the first derivative ofĒ int at the transition point is triggered by the coalescence ofφ r rather than that of the U (1) gauge field, s.
We numerically evaluated the interaction energies of several types of cosmic strings at arbitrary separation distance using the relaxation method (gradient flow method), and compared the results with those obtained using the point source formalism. In general, analyses relying on the point source formalism are in good agreement with the full numerical results of the gradient flow method only for separation distances longer than the inverse of the lightest mass in the underlying theory evaluated at infinity. The results for the simple local string are in good agreement with the results of the previous study [41], and with those by the variational approach [40]. Long-range interactions due to NG-bosons are also found to exist in the case of global strings. For both local and global strings, the phase structure is simple in the sense that the interaction energy is a monotonic function of the separation distance. In the case of superconducting strings, an additional short-range attraction is induced by the Higgs field in the fundamental representation of U (1). This leads to the emergence of nontrivial phase structures of interaction energy (see e.g. Fig. 14), that cannot be captured by the point source formalism. Furthermore, when the effects of magnetic currents are included, a logarithmic dependence of the interaction energy on the separation distance on at large distance is found, which is consistent with the predictions of the point source formalism. However, the total strength of the attraction is weakened by the current quenching effect at a separation distance comparable to the string thickness. Therefore, we conclude that the attraction between two static bosonic superconducting strings is maximized when the total current associated with U (1) is zero.
Our new results on the interaction energy of bosonic superconducting strings indicate that there is a viable parameter region leading to an attraction force, at least when two strings are static and straight. Thus, the necessary condition for the bound state of the two strings is satisfied. Therefore, it would be very interesting to investigate the dynamical collision of the two-string system and to discuss the possibility of bound state formation of the two bosonic superconducting strings. Such analysis could include the effects of non-zero relative angles and velocities on the bound state formation of the strings. We leave it as a future study.

A Normalization and Numerical solutions
This appendix describes the normalization used in the numerical calculations and the numerical recipes for obtaining various types of cosmic string configurations.

A.1 Normalization of local and bosonic superconducting strings
We consider the following action: For convenience, we introduce the following dimensionless lengthx and various dimensionless fieldsφ,Ā µ ,φ andĀ µ as   Table 1). The vacuum energy is unified with local string case for demonstration purpose. Assuming a static configuration, the two-dimensional energy (tension) of the string can be extracted as follows, In this normalization,ηφ,λ ϕ ,λφ,β andḡ are treated as free parameters. Note that this normalization cannot be applied to global strings becausex becomes ill-defined for e = 0. Therefore, we will use a different normalization for global strings.

A.2 Normalization of global strings
In the case of global strings, the normalization defined by Eqs. (A.2) and (A.4) cannot take the limit e → 0, so the normalization of the complex scalar field and length scale must be changed. We then use the following normalization in the numerical computation of the global strings.φ With this normalization, the tension of the global string becomes Here, the signs of the time derivatives of the various fields are determined such that the energy of the system decreases with time evolution. First, we sets(r) = 0 and find the field configurations forφ,Ā θ andφ. We initially prepare functions for the fieldsφ r (t = 0,r),Ā θ (t = 0,r) andφ r (t = 0,r) that satisfy boundary conditions at origin and infinity, then evolve these functions numerically with the fictitious time. Specifically, forφ r andĀ θ , we prepare hyperbolic tangent configuration, and Gaussian configurations forφ r . When the time evolution converges to ∂ tφr = ∂ tĀθ = ∂ tφr = 0, a solution to the original differential equations is obtained. The time evolution converges quickly, consistent with the results of Ref. [60]. Now let us include the effect of the gauge field,s(r). In addition to the differential equations Eq. (A.9), we need to include the following differential equation, The differential equations Eqs. (A.9) and (A.10) are solved iteratively. To be precise, the following procedure is used for our numerical computation.
• (i) We first obtain the configuration ofφ r ,Ā θ andφ r by solving Eq. (A.9) without the gauge field,s(r) = 0, using the gradient flow method. • (ii) Then, we solve Eq. (A.10) with theφ r configuration evaluated in the first step (i) (or in the third step (iii)) using the standard initial value method. The initial value of s(r = 0) is set by hand.
• (iii) We use the currents(r) = 0 evaluated in the second step and solve Eq. (A.9) again to update the configuration ofφ r ,Ā θ andφ r . After this, we return to the second step (ii).
If the above iteration converges well, the whole configuration of the bosonic superconducting string, including the gauge field, can be obtained numerically. We have confirmed that the iteration converges well, at least for the benchmark points shown in Table. 1. The effective origin and infinity are set tor = 10 −10 andr = 40, respectively, and the finite size effect is found to be negligible. We take three benchmark points A, B and C shown in Table 1. The numerical constants k ϕ , k e , kφ and k A are determined such that the asymptotic solutions Eqs. (2.14) (2.11) (2.44) and (2.45) are matched with the numerical results at infinity. From Table. 1, it is easy to see that the total current calculated in Eq. (2.46) decreases ass(r = 0) is increased above the threshold value.
The configurations ofφ r ,Ā θ ,φ ands are shown in Fig. 18 for the benchmark point A with initial conditions = 0.2 (leading to the total current J tot ≃ 0.0277). As expected, the configurations ofφ r (r),Ā θ (r) andφ r are exponentially suppressed, whiles(r) increases logarithmically at long distance. The corresponding magnetic field defined asB θ = ∂rĀ z (r) is depicted in Fig. 19. It is clear from the figure that the magnetic field is proportional to  Figure 19. The magnetic fieldB θ = ∂rĀ z (r) for the benchmark point A with the initial conditioñ s(r = 0) = 0.2 is shown.
1/r outside the string but decays towards the center of string, where it decays to zero. The penetration depth of the magnetic field is approximately given by the mass of the gauge field acquired by the Higgs mechanism inside the string. This effect is known as the Meissner effect.

B Numerical calculation of the interaction energy of two cosmic strings
This appendix details the numerical setup for the interaction energy of two straight cosmic strings. In the system of two straight cosmic strings, the calculations are performed in Cartesian coordinates rather than cylindrical coordinates, since the cylindrical symmetry is explicitly broken unless the axes of the two strings coincide. To find the configuration, we apply the gradient flow method as in the case of a single straight string. In particular, we parameterize the fieldsφ,Ā µ ,φ ands as ϕ(t, x, y) =φ x (t, x, y) + iφ y (t, x, y) ,Ā µ (t, x, y) = (0,Ā x (t, x, y),Ā y (t, x, y), 0) , ϕ(t, x, y) =φ r (t, x, y) ,Ā z (t, x, y) = 1 gs (t, x, y) . (B.1) Since we work in the Coulomb gauge, an additional term L Coulomb = (∂ xĀx +∂ yĀy ) 2 /2 is added to the Lagrangian density defined by Eq (2.34). Using this parameterization and the normalization defined in Sec. 4, we numerically solve the following equations, ∂φ r ∂t =∆φ −s 2φ r −λφ 2φ r (φ 2 r −η 2 ϕ ) −β|φ| 2φ r , ∂s ∂t =∆s − 2ḡ 2sφ2 r .
To solve the above equations numerically, we prepare a two-dimensional simulation box with a length of L = 30 per side (the box size is increased to L = 50 for superconducting strings with current. The lattice spacing is a = 0.05 in units of eη ϕ = 1 for local and bosonic superconducting strings and a = 0.05 in units of η ϕ = 1 for global strings. The (fictitious) time evolution of all fields is evaluated fromt = 0 tot = 15 in time steps of δt = 5 × 10 −4 by the standard Euler method.