Baryonic handles: Skyrmions as open vortex strings on a domain wall

We consider the BEC-Skyrme model which is based on a Skyrme-type model with a potential motivated by Bose-Einstein condensates (BECs) and, in particular, we study the Skyrmions in proximity of a domain wall that inhabits the theory. The theory turns out to have a rich flora of Skyrmion solutions that manifest themselves as twisted vortex rings or vortons in the bulk and vortex handles attached to the domain wall. The latter are linked open vortex strings. We further study the interaction between the domain wall and the Skyrmion and between the vortex handles themselves as well as between the vortex handle and the vortex ring in the bulk. We find that the domain wall provides a large binding energy for the solitons and it is energetically preferred to stay as close to the domain wall as possible; other configurations sticking into the bulk are metastable. We find that the most stable 2-Skyrmion is a torus-shaped braided string junction ending on the domain wall, which is produced by a collision of two vortex handles on the wall, but there is also a metastable configuration which is a doubly twisted vortex handle produced by a collision of a vortex handle on the wall and a vortex ring from the bulk.

A convenient parametrization of the Skyrme field (chiral Lagrangian field) is to write the O(4) vector field as a two component complex scalar field, φ 1,2 , with the constraint |φ 1 | 2 + |φ 2 | 2 = 1. The BEC-inspired potential takes the form V = M 2 |φ 1 | 2 |φ 2 | 2 and thus there are two different vacua: either φ 1 or φ 2 must vanish in the vacuum. In the vacuum where one component vanishes, the vacuum manifold S 1 is parametrized by the phase of the other component. For instance, in the vacuum where φ 2 vanishes, |φ 1 | = 1 and vice versa.
Hence, there exists a global vortex having logarithmically divergent energy [21][22][23]. In the vortex core, the other component is confined; for instance, in the core of a vortex having a winding in the φ 1 component, the φ 2 component is confined which yields a U(1) modulus. This is a global analog of the superconducting cosmic string [24]. These properties are also shared by a model with the potential V = M 2 |φ 1 | 2 [25]. A Skyrmion in the BEC-Skyrme model takes the form of a twisted vortex ring or a vorton, i.e. a vortex ring along which the phase of the confined component winds from 0 to 2π. If it is twisted B times (i.e. it winds from 0 to 2πB), then it carries baryon number B. This was actually first found in two-component BECs [9][10][11][12][13][14][15][16][17][18][19], and is shared by this model as well as the above mentioned model with potential V = M 2 |φ 1 | 2 [25].
The BEC-Skyrme model also admits a domain wall interpolating between the two vacua.
Vortices can terminate on the domain wall, just like in two-component BECs [19,[26][27][28] (see also Ref. [29]). This configuration resembles a D-brane in string theory, thereby called a D-brane soliton [30][31][32][33]. Vortex endpoints are also called boojums, since they resemble those in Helium-3 superfluids [34,35]. One question is what happens if the two ends of a vortex terminate on the same domain wall. A Skyrme model with the potential term V = M 2 ( (φ 1 )) 2 also admits a domain wall, and Skyrmions are absorbed into the wall becoming lumps or baby-Skyrmions on the wall [36][37][38]. Thus a further question is what happens to the Skyrmions in the presence of a domain wall in the BEC-Skyrme model. We will answer this question with the results of this paper.
In this paper, we will study all the physics in the proximity of the above-mentioned domain wall. If we start in the bulk and place a Skyrmion, it is a twisted vortex ring, as mentioned above. If the distance to the domain wall is not too large, an attractive force will pull the Skyrmion into the domain wall and it will be absorbed. The vortex ring is then connected to the vacuum on the other side of the domain wall and it thus manifests itself as a handle sitting on the domain wall, sticking into the bulk of the side it came from. This description sounds asymmetric and, in fact, we shall discover in this paper that the true energy minimizer symmetrizes itself to become symmetric under the exchange of φ 1 and φ 2 ; it becomes a link of two vortex handles. This is in stark contrast to the above-mentioned model with the potential V = M 2 ( (φ 1 )) 2 , in which case a Skyrmion absorbed into the wall becomes a baby-Skyrmion and no complicated structure appears [36][37][38]. The attraction between the vortex ring and the domain wall is also a result of a large reduction of the static energy by absorption into the domain wall. We perform an explicit calculation and find that there is a large binding energy for the vortex ring to gain at the cost of transforming itself into a handle sitting on the domain wall.
We shall also study the interactions between two vortex handles on the domain wall.
Since, as we already spoiled, the handle configuration will turn out to be symmetric, it does not matter on which side of the domain wall the handle sits. We find that there is an attractive channel and a repulsive channel between the two vortex handles. If we place the handles in the attractive channel, they will combine themselves into a braided string junction of a toroidal shape -a 2-Skyrmion absorbed into the domain wall.
We also investigate the interaction between the vortex handle and the vortex ring in the bulk and find that they always attract each other (or quickly rotate themselves into the attractive channel). Next the vortex ring goes through a string reconnection mechanism to transform the configuration into a doubly twisted vortex handle, which is the analog of a vortex ring that is twisted 2 times (hence baryon number two) being absorbed into the domain wall.
Finally, we compare the energies of the two Skyrmion configurations with baryon number two and find that the braided string junction has less energy than the doubly twisted vortex handle.
This paper is organized as follows. In Sec. II we will briefly review the BEC-Skyrme model and its symmetry and vacuum properties. Sec. III A reviews the domain wall. In Sec. III B, we construct the boojum for the first time in the BEC-Skyrme model; it is a semiinfinite string attached to the domain wall. In Sec. IV A the vortex handle is constructed.
In Sec. IV B the interaction between the domain wall and the vortex ring is studied. In Sec. IV C we add a twist to the modulus of the vortex handle, producing a 2-Skyrmion. In Sec. IV D we study the interactions between two vortex handles and find a new 2-Skyrmion: the braided string junction of toroidal shape. Sec. IV E considers the interaction between the vortex handle on a wall and the vortex ring in the bulk, which reproduces the doubly twisted vortex handle found in Sec. IV C. In Sec. IV F the energies of the two 2-Skyrmions are compared. Sec. IV G considers the construction of higher-charged configurations. Finally, we conclude with a discussion in Sec. V.

II. THE BEC-SKYRME MODEL
The model that we will consider in this paper is the generalized Skyrme model, consisting of the kinetic term, the Skyrme term [1,2], the BPS-Skyrme term [39,40] and finally the BEC-inspired potential [21][22][23] where σ a are the Pauli matrices. The vector φ ≡ (φ 1 (x), φ 2 (x)) T is a complex 2-vector field, the spacetime indices µ, ν, ρ, σ run over 0 through 3, the flat Minkowski metric is taken to be of the mostly positive signature, and finally, the nonlinear sigma model constraint is imposed as φ † φ = |φ 1 | 2 + |φ 2 | 2 = 1. The relation of the complex vector field φ, or equivalently the two complex fields φ 1,2 , to the usual chiral Lagrangian field used in the Skyrme model is given by and thus the nonlinear sigma model constraint reads det U = |φ 1 | 2 + |φ 2 | 2 = 1.
The topological degree of the map φ is B ∈ π 3 (S 3 ) and can be calculated as Once we turn on a nonvanishing potential M > 0, the Skyrmions survive, but the vacuum of the theory and the physics of the solitons change.
The two vacua of the model with nonvanishing potential V in (5), are : φ = (e iα , 0) T , which by the nonlinear sigma-model constraint yield the other component to be at its maximum.
The symmetry of the model with the potential V in Eq. (5) is explicitly broken from O(4) down to where the group is defined by the symmetries and U(1) 3 acts on Z 2 in such a way that they define a semi-direct product denoted by .
The unbroken symmetry groups in the vacua (9), are thus The target space (vacuum manifold) is thus given by the coset group and the nontrivial homotopy groups of this manifold read The theory thus supports both domain walls and vortices in addition to the Skyrmions.
Although we have included both the Skyrme term, L 4 , and the BPS-Skyrme term, L 6 , in the model, we will only use either of the terms as follows 2 + 4 model : c 4 = 1, c 6 = 0, 2 + 6 model : It turns out that the two models give qualitatively the same results, so we will only show some of the results for both models in the next section.
In Ref. [23] we studied the domain wall, the vortices and the Skyrmions in one vacuum (the H ⊗ vacuum).
In this paper we study the theory in the presence of the domain wall of Ref. [23] with Skyrmions as closed or open vortex strings in proximity of the domain wall.

III. VORTICES AND THE DOMAIN WALL
We will now consider that the 3-dimensional space has a domain wall separating two phases with vacua ⊗ and , respectively. Without loss of generality, we will consider the vortices in the ⊗ phase; the results apply to the phase by interchanging the complex scalar fields, φ 1 ↔ φ 2 .

A. The domain wall
As we will place everything in this paper in the presence of the domain wall, we will make a short review of the domain wall solution [21][22][23] here. Since the domain wall is a codimension-1 soliton, only the potential (5) and the kinetic term, L 2 , contribute to its The solution is thus with χ and ϑ being constant phase parameters. We will choose the upper sign throughout this paper and hence the ⊗ vacuum is always at z > 0 (up) and the vacuum is at z < 0 (down). Furthermore, we will set z 0 = 0 from now on. This is the translational modulus of the domain wall and we can always adjust our coordinate system such that the domain wall is at z = 0.

B. The boojum or D-brane soliton
We will now consider attaching a single (infinitely long) open vortex string to the domain wall from the side of the ⊗ phase. The energy of such a system is thus divergent for three different reasons: the domain wall carries an infinite energy, the semi-infinite vortex string has infinite energy and finally, the fact that it is a global vortex implies that the energy in the direction transverse to the string in the ⊗ phase diverges logarithmically due to the winding contribution to the kinetic term. This solution serves mostly as an illustration.
It will be convenient to parametrize the fields as follows The vortex is a solution that winds in φ 2 (since we are in the ⊗ phase) which thus means that ϑ is a winding phase. The "profile function" of the (global) vortex is sin f and due to the winding phase, it must vanish at the vortex center; we thus choose f = 0 there.
Asymptotically, f tends to its vacuum value in the ⊗ phase, which is f = π 2 . An initial condition for the numerical calculation of the boojum can thus be constructed by combining the domain wall solution and the vortex Ansatz where F(ρ) is a suitable guess for the vortex profile function obeying F(0) = 0 and F(∞) = 1.  The way this soliton becomes a Skyrmion is by turning on a twisting of its U(1) modulus, which lives in the string world volume. More precisely, we will let the phase field χ wind from 0 to π on the way out and away from the domain wall, and from π to 2π on the way back to the domain wall. This extra "winding" will make the string handle cover the 3-sphere (the target space) and hence comprise a unit baryon charge.
The above description is quite idealistic, however, the vortex string will have a tension which will tend to shorten the string as much as possible. For this reason, the unit charge Skyrmion or the 1-Skyrmion as a single vortex string handle attached to the domain wall, will be quite short.
There is a simple way to extend the above construction to a B-Skyrmion, i.e. by twisting the phase function χ not 1 time, but B times. More precisely, we can twist the function χ from 0 to πB on the way into the bulk and from πB to 2πB on the way back to the domain wall. This can produce a vortex handle with baryon charge or topological degree B. We will see shortly that the configuration becomes more complicated for higher twists, than what we described here.
We are now ready to present the numerical solution of the single vortex string handle attached to the domain wall. The numerical solution is shown in Fig. 2. Fig. 2(a) shows the energy isosurfaces in transparent blue and the baryon charge isosurfaces with an unconventional color scheme that we will describe shortly. Fig. 2 Large positive values of Q are plotted with red and large negative values are plotted with blue; green is zero and other colors are interpolation values between red and blue. The brackets around the spatial indices indicate that the indices are antisymmetrized and finally, is an ad-hoc small number that is regularizing the quantity at the vortex cores. Q is used solely for the intent of clarifying which vortices are vortices and which are antivortices.
We will now explain the color scheme utilized for coloring in the baryon charge isosurface that shows the Skyrmion configuration in Fig. 2(a). The color scheme is based on the parametrization (21) as The color is defined as a map from χ to the color circle (the hue), such that χ = 0 is red, χ = 2π/3 is green, χ = 4π/3 is blue, χ = π/3 is yellow, χ = π is cyan and χ = 5π/3 is magenta.
There is a surprise, that is not obvious from the construction we described above; it turns out that there is a dual string, meaning a string in the φ 1 field in addition to the string in the φ 2 field that we pictured so far. The nature of the Skyrmion, covering the entire 3-sphere (target space), implies that the closed string in the (⊗) bulk will have a dual string (of φ 1 ) piercing through the Skyrmion. When the Skyrmion is attached to the domain wall, the vortex ring (of φ 2 ) becomes a handle, but the dual string piercing the Skyrmion becomes a dual handle. Therefore, the 1-Skyrmion is actually symmetric between the two phases. Had we described everything in terms of strings in the phase, the vortex would be a string in the φ 1 field and it also becomes a handle attached to the domain wall, albeit from the other side, see Fig. 2(b). The vortex zero in φ 2 is depicted by a blue isosurface and the vortex zero in φ 1 is shown with a magenta isosurface. The two vortices (blue and magenta) link each other once (if we include their respective vacua).

B. Interactions between wall and a closed vortex string
In this section, we will start from a 1-Skyrmion that can stably exist in the bulk [23] and put it near the domain wall. The 1-Skyrmion in the ⊗ bulk exists as a vortex ring (closed vortex string) in the φ 2 field (blue). If we were to place it in the phase instead, the 1-Skyrmion would be a vortex ring in the φ 1 field (magenta). If the 1-Skyrmion is placed far way from the domain wall, the force between them is exponentially suppressed and it will take a long time before an attraction will accelerate the 1-Skyrmion towards the domain wall. Therefore, we will place the 1-Skyrmion in the bulk in near proximity to the domain wall and the attraction happens quite rapidly.
The simulation is made not with a relativistic kinetic term, but with the relaxation method, which is dissipative. If the dynamics was made with a real relativistic kinetic term, the interaction would be oscillating many times and only come to a final fixed point when all the excess energy has been radiated away. Instead we will evolve the dynamics of the simulation with a first-order kinetic term, which is dissipative as mentioned already. This means that the energy is not conserved and the configuration will quickly approach the fixed point losing the excess energy to the dissipative term in the evolution.
The numerical calculation is shown in Fig. 4. From this simulation, we can also see the origin of the dual handle, i.e. the vortex handle that exists in the φ 1 field. In the ⊗ bulk it was just a string that pierced through the 1-Skyrmion making it into a torus as claimed in Ref. [23]. This piercing string connects the vacua on both sides of the torus and energy-wise we find it more intuitive to think about the 1-Skyrmion as being a wrapped-up vortex ring in the φ 2 field. However, once the vortex ring in the φ 2 field is absorbed into the wall and becomes a vortex handle, the dual string

(otherwise it would become an anti-Skyrmion).
Now that we have understood the 1-Skyrmion absorbed into the domain wall from two different perspectives, let us consider the 2-Skyrmions in the next section.

C. The handle with double twist
In this section, we will take the approach of adding a twist as described briefly in section IV A. The string in φ 2 that emanates from the domain wall into the ⊗ phase is twisted in the χ field from 0 to 2π and on the way back to the domain wall it further twists to 4π, yielding a total twist of 4π = 2πB; hence B = 2. As mentioned in section IV A, there is unavoidably a dual string in φ 1 in the Skyrmion and we will see that it is related to the number of twists. More precisely, once the twist is higher than one, the dual string is either multiple-wound or there are several dual strings. (blue) is linked with two individual φ 1 vortices (magenta).
In this section we study the interactions of two individual 1-Skyrmions both already absorbed into the domain wall. Under normal circumstances, this is also a way of creating multi-Skyrmions. The recipe is to find the attractive channel between the two 1-Skyrmions and let them combine into the (probably) optimal 2-Skyrmion. It turns out that the theory is much more complex when we have a domain wall, as we shall see shortly.
As mentioned above, it is well known that there is an attractive channel and a repulsive channel for Skyrmions, depending on their mutual orientations in field space (target space).
In this case, where we consider the interaction in the world volume of the domain wall, pairs that both attract each other. The two attractive forces win over the single repulsive force and the net force is attractive.
The resulting interaction attracts the two vortex handles and they combine to form a torus in the plane of the domain wall. The torus configuration in the standard Skyrme model is well known of course. It is, nevertheless, interesting to look at the Skyrmion-Skyrmion interactions in terms of the vortices in the two fields φ 1,2 . In the second column of Fig. 7, we can see the vortex lines (φ 1 is magenta and φ 2 is blue). In order to see the direction of the vortex, i.e. whether it is a vortex or an antivortex in the (x, y)-plane, we have to refer to columns 3 and 4 of Fig. 7. The interaction in the full 3-dimensional picture is somewhat more complicated than in the simplistic picture of the interactions seen only in the plane of the domain wall. The colliding vortex-vortex pair still occurs, of course, but what happens more precisely is that after the vortex-vortex collision in the plane has happened, a vortex string junction has been made that is left in the ⊗ bulk (out of the plane of the domain wall). After the creation of the string junction in the field φ 2 , there are still just two independent strings in φ 1 (magenta), but the Skyrmion configuration is quite oval and once it relaxes into a more symmetric (toroidal) shape, the two vortex strings in φ 1 also form the same string junction, but sitting in the bulk and from a top view, the position of the (anti)vortices on the domain wall are rotated by 45 degrees with respect to the φ 2 ones, see Fig. 7. It is a bit difficult to see the 3-dimensional structure of the configuration in the second column of Fig. 7, so we have duplicated these images, but from a different view point in Fig. 8. Now we can better see that the configuration has two string junctions with four vortices emanating (2 vortices and 2 antivortices) and the two junctions are thus braiding their four fingers.
The above example illustrates well what happens in the attractive channel. We will now consider the case shown in Fig. 9, where we have rotated both the Skyrmion handles in such a way that there are two repulsive interactions coming from a vortex-vortex pair and an antivortex-antivortex pair in the φ 2 field, which dominates over the vortex-antivortex attraction in φ 1 . We have only shown a single snapshot of the configuration, because what happens next is that they both run away from each other.

E. Interactions between handle and a closed vortex string
In this section, we will consider a different kind of interaction, namely between the vortex handle on the domain wall and the vortex ring in the (⊗) bulk.
The numerical calculation is shown in Fig. 10. vortex ring in the (⊗) bulk is not too important, because the vortex ring in the bulk will rotate into the attractive channel as chosen as the initial configuration (first row) in Fig. 10.
The vortex ring (in φ 2 , blue) in the ⊗ bulk is initially oriented perpendicularly to the plane of the domain wall. The first thing that happens in the interaction is that the vortex in φ 2 (blue) is pulled down towards the vortex handle on the domain wall and it reconnects such that the string becomes a longer handle, see rows 1-3 of Fig. 10. The beauty of the reconnection is that it automatically produces a vortex handle with double twist in the χ field (see the parametrization (21)). This can be seen from the fact that the vortex handle in φ 2 encloses (links with) two dual strings in φ 1 , see the third row of Fig. 10. The final phase in the relaxation just minimizes the energy by making the the doubly twisted vortex handle shorter and more compact, see the fourth row of Fig. 10.

F. Energy comparison of the two B = 2 Skyrmions
It is interesting to see that the interaction between the two handles in the plane of the domain wall created a different 2-Skyrmion (a braided string junction torus) compared to the interaction of the vortex handle and the vortex ring in the bulk, which created the doubly twisted vortex handle that we constructed in Sec. IV C. In order to know which configuration is the stable one, we will compare their energies numerically. As the domain wall has an infinite energy in the infinite space, we will subtract off the domain wall energy  The results of the numerical calculations for the energies are shown in Tab. I. In particular, we first calculate the masses of the vortex handle for M = 3, 7 compared to the vortex ring and see that there is a strong binding energy for the Skyrmion to be gained by getting absorbed into the domain wall. Next we compare the braided string junction of Sec. IV D with the doubly twisted handles of Secs. IV C and IV E. The conclusion drawn from the energy measurements is clear; the braided string junction of toroidal shape has the far lowest energy of the two different 2-Skyrmions and thus is the stable one.

G. Higher-charged handles
As we have seen in the previous sections, the interaction dynamics is quite intricate and the number of ways of combining handles and rings with various numbers of twists is overwhelmingly large. Therefore, a complete study of higher-charged Skyrmions in this theory is beyond the scope of the paper. However, let us mention that there are in principle 3 different possibilities: multi-Skyrmions as multi-solitons in the domain wall (as e.g. the braided string junction as a 2-Skyrmion), multi-Skyrmions as higher-twisted handles sticking into the bulk and finally, a hybrid of the previous two options.
In this section, we only explore one possibility as the initial condition, but then use the numerical calculations and relax them to the nearest metastable configuration; that is, we consider a single open vortex handle with various twists as the initial guess.
The first numerical result is for B = 3, i.e. a single vortex in φ 2 that is twisted 3 times yielding a 3-Skyrmion, see Fig. 11. It turns out to be metastable and it has exactly one vortex handle in the field φ 2 and it is linked with 3 individual dual strings of φ 1 .
Next, we try to add a twist to the previous initial guess in order to create a 4-Skyrmion as a vortex handle on the domain wall. The result is shown in Fig. 12 and it also turns out to be metastable. A curiosity is that the 4 dual string are located near the top "hole" of the Skyrmion and the bottom "hole", but not near the middle "hole," which seems to appear due to a stereo cusp in the φ 2 vortex handle.
We now add two more twists to the initial guess and try to search for a 6-Skyrmion, but this time the metastability of the configuration was not found (or the guess was too far away from such a solution). The numerical result is shown in Fig. 13, where the vortex handle has collapsed into a single handle and a 5-Skyrmion that consists of a double string junction in the field φ 1 with complicated twists around it (dual strings). The last attempt at looking for a higher-charged handle is to add another twist to the former guess, yielding a total of 7 twists. The numerical result is shown in Fig. 14 and it also collapsed from a vortex handle into a double string junction, intricately braided with dual strings, yielding a 7-Skyrmion.
The true minimizers of the energy for B ≥ 3 may not be the solutions found here.

V. DISCUSSION AND OUTLOOK
In this paper we have considered a BEC-inspired potential in the Skyrme model and a sextic version of the Skyrme model, which due to the potential possesses vortex strings, and in particular we have studied the setting in the presence of a domain wall -which is also possessed by the theory with the BEC-inspired potential. The vortex strings contain a U (1) modulus, that once twisted by 2π yields a unit baryon charge. The first intuitive picture is that we wind the vortex string once to get a 1-Skyrmion in the bulk and when it is placed in the vicinity of the domain wall, we have found that attractive forces absorb the Skyrmion and it becomes a vortex handle sitting on the domain wall. It turns out that it inevitably  An obvious direction for further studies is to consider higher-charged Skyrmions, which may yield a large number of metastable configurations; in particular we have not necessarily found the true energy minimizers for B ≥ 3. This would require a large search for configurations based on many different initial guesses. We will leave this for future work.
An interesting observation is that all the Skyrmions of baryon number B seem to be composed of vortex zeros of φ 1 and of φ 2 that link each other B times. It could be interesting to study this fact further and investigate whether this is always the case.
An important development would be to see how many of the results in this model can be carried over to BECs and under what circumstances.