PHYSICAL REVIEW D 101, 105020 (2020)
Exhausting all exact solutions of BPS domain wall networks in arbitrary dimensions
† ‡ Minoru Eto ,1,2,* Masaki Kawaguchi ,1, Muneto Nitta,2,3, and Ryotaro Sasaki1 1Department of Physics, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata, Yamagata 990-8560, Japan 2Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 3Department of Physics, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
(Received 15 April 2020; accepted 15 May 2020; published 28 May 2020)
We obtain full moduli parameters for generic nonplanar Bogomol’nyi-Prasad-Sommerfield networks of domain walls in an extended Abelian-Higgs model with N complex scalar fields and exhaust all exact solutions in the corresponding CPN−1 model. We develop a convenient description by grid diagrams which are polytopes determined by mass parameters of the model. To illustrate the validity of our method, we work out nonplanar domain wall networks for lower N in 3 þ 1 dimensions. In general, the networks can have compact vacuum bubbles, which are finite vacuum regions surrounded by domain walls, when the polytopes of the grid diagrams have inner vertices, and the size of bubbles can be controlled by moduli parameters. We also construct domain wall networks with bubbles in the shapes of the Platonic, Archimedean, Catalan, and Kepler-Poinsot solids.
DOI: 10.1103/PhysRevD.101.105020
I. INTRODUCTION equations called BPS equations. In such cases, one can often embed the theories to supersymmetric (SUSY) It sometimes happens that systems have multiple discrete vacua or ground states, which is inevitable when a discrete theories by appropriately adding fermion superpartners, symmetry is spontaneously broken. In such a case, there in which BPS solitons preserve some fractions of SUSY. appear domain walls (or kinks) in general [1–3] which are Their topological charges are central (or tensorial) charges inevitably created during second order phase transitions of corresponding SUSY algebras. The BPS domain walls 3 1 [4–7]. They are the simplest topological solitons appearing in þ dimensions were studied extensively in field N 1 – N 2 in various condensed matter systems such as magnets [8], theories with both ¼ SUSY [20 35] and ¼ graphenes [9], carbon nanotubes, superconductors [10], SUSY [36–56]; see Refs. [57–60] as a review. They 1 atomic Bose-Einstein condensates [11], and helium super- preserve a half of SUSY and thereby are called 2 BPS fluids [6,7,12], as well as high density nuclear matter states, accompanied by SUSY central (tensorial) charges m [13,14], quark matter [15], and early Universe [4,16]. Zm ðm ¼ 1, 2; labeling spatial coordinates x perpen- In cosmology, cosmological domain wall networks are dicular to the domain wall) as domain wall topological suggested as a candidate of dark matter and/or dark charges [22,24,61]. In general, if several domain walls meet energy [17]. along a line, it forms a planar domain wall junction. In As the cases of other topological solitons, domain walls SUSY models, the planar domain wall junctions preserve a can become Bogomol’nyi-Prasad-Sommerfield (BPS) 1 quarter SUSY [62–64], therefore are called 4 BPS states, states [18,19], attaining the minimum energy for a fixed accompanied by a junction topological charge Y in addition boundary condition and satisfy first order differential 1 to Zm (m ¼ 1, 2). The 4 BPS domain wall junctions have been studied in theories with N ¼ 1 SUSY [32,65–73] and *[email protected] † N ¼ 2 SUSY [74–83]. In the N ¼ 2 SUSY gauge models, [email protected] ‡ not only planar domain wall junctions, but also planar [email protected] 1 domain wall networks as 4 BPS states were constructed Published by the American Physical Society under the terms of [75,76]. The low-energy effective action for normalizable the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to modes within the networks was obtained [78] and applied the author(s) and the published article’s title, journal citation, to study of low-energy dynamics [79]. Non-BPS planer and DOI. Funded by SCOAP3. domain wall networks were also studied in Refs. [84,85].
2470-0010=2020=101(10)=105020(22) 105020-1 Published by the American Physical Society ETO, KAWAGUCHI, NITTA, and SASAKI PHYS. REV. D 101, 105020 (2020)
Recently, the present authors proposed a model, a D þ 1- II. THE MODEL 1 dimensional Uð Þ gauge theory [86] admitting novel We study a Uð1Þ gauge theory with N charged complex analytic solutions of the BPS single nonplanar domain scalar fields HA (A ¼ 1; 2; …;N) and N0 real scalar fields wall junctions. This model cannot be made supersymmetric 0 ΣA (A0 ¼ 1; 2; …;N0)inD þ 1-dimensional spacetime. but still admits stable BPS states so that we can use the The Lagrangian is given by well-known Bogomol’nyi completion technique to derive BPS equations. The model consists of N charged complex 1 1 XN0 0 1 μν A0 μ A0 scalar fields and N neutral scalar fields coupled to the Uð Þ L ¼ − FμνF þ ∂μΣ ∂ Σ 4e2 2e2 gauge field. In Ref. [86], we restricted ourself to the special A0¼1 numbers N − 1 ¼ N0 ¼ D and imposed the invariance μ † − S 1 þ DμHðD HÞ V ð2:1Þ under the symmetric group Dþ1 of the rank D þ , which are the symmetry groups of the regular D simplex. 1 XN0 In this paper, we investigate generic nonplanar networks V ¼ Y2 þ ðΣA0 H − HMA0 ÞðΣA0 H − HMA0 Þ†; 2e2 of BPS domain walls in D þ 1 dimensions. We consider the A0¼1 ≥ 1 generic case of N D þ imposing no discrete symmetry ð2:2Þ and exhaust all exact solutions with full moduli of generic BPS nonplanar networks of domain walls in the infinite where H is an N component row vector made of HA, Uð1Þ gauge coupling limit in which the model reduces to N−1 the CP model. These are the first exact solutions of H ¼ðH1;H2; …;HNÞ; ð2:3Þ nonplanar domain wall networks in D dimensions (D ≥ 3). We first derive the BPS equations for generic Abelian Y is a scalar quantity defined by gauge theories in D þ 1 dimensions. Then, we partially solve them by the moduli matrix formalism [58] and find all Y ¼ e2ðv2 − HH†Þ; ð2:4Þ moduli parameters of the generic domain wall network solutions. We then demonstrate several concrete nonplanar and MA0 (A0 ¼ 1; …;N0) are N by N real diagonal mass −1 networks in the CPN model in D ¼ 3 for N ¼ 4,5,6.In matrices defined by the case of N ¼ 4, the solution has only one junction at A0 which four vacua meet. Network structures appear for M ¼ diagðmA0;1;mA0;2; …;mA0;NÞ: ð2:5Þ N>4. In the case of N ¼ 5, we show two different types of networks exist in general. The first type has a vacuum The spacetime index μ runs from 0 to D, and Fμν is an Uð1Þ bubble (a compact vacuum domain) surrounded by semi- gauge field strength. The coupling constants in the infinite vacuum domains. Instead, the second type does not Lagrangian in Eq. (2.1) are taken to be the so-called have any bubbles but all the vacuum domains are semi- Bogomol’nyi limit. For later use, let us define mA by an infinitely extended. In the N ¼ 6 case, there are three N0 vector whose components are the Ath diagonal elements different types according to the number of the vacuum of MA0 s, namely, bubbles, two, one, or zero. Finally, we find a connection to the well-known polyhedra known from ancient times. mA ¼ðm1;A;m2;A; …;mN0;AÞ: ð2:6Þ Indeed, we find the vacuum bubbles which are congruent with five Platonic solids. In addition, the Archimedean and In the following, we will mostly consider generic masses: the Catalan solids appear as the vacuum bubbles. We also construct the Kepler-Poinsot star solids as domain wall mA ≠ mB; if A ≠ B: ð2:7Þ networks. This paper is organized as follows. In Sec. II,we Since the scalar potential V is positive semidefinite, a introduce our model. In Sec. III, we derive the BPS classical vacuum of the theory is determined by V ¼ 0: equations and clarify the moduli space of the BPS HH† ¼ v2, and ΣA0 H − HMA0 ¼ 0. In the generic case of solutions. In Sec. IV, we first consider the infinite Uð1Þ Eq. (2.7), there are N discrete vacua. The Ath vacua which gauge coupling limit in which the model reduces to the we will denote by hAi is given by HB ¼ vδB, and −1 A CPN B0 massive nonlinear sigma model. We then exhaust Σ ¼ mB0;A. Simply, the vacua can be identified to the all exact solutions with full moduli parameters for the BPS discrete points determined by the mass vectors fmAg in the equations. We further give several examples of nonplanar Σ space, networks in D ¼ 3. In Sec. V, we study relations between 3 the domain wall networks in D ¼ and the classic solids, hAi∶Σ ¼ mA: ð2:8Þ like the Platonic, Archimedean, Catalan, and Kepler- Poinsot solids. Finally, we summarize our results and give The Lagrangian (2.1) is motivated by supersymmetry. In 0 2 N 2 a discussion in Sec. VI. fact, when NF ¼ , it is a bosonic part of an ¼
105020-2 EXHAUSTING ALL EXACT SOLUTIONS OF BPS DOMAIN … PHYS. REV. D 101, 105020 (2020) supersymmetric theory with eight supercharges in four direction to the domain wall interpolating the vacua hAi and dimensions. hBi is given by
Z v2 m − m : 3:6 III. SOLVING BPS EQUATIONS FOR DOMAIN j j¼ j A Bj ð Þ WALL NETWORKS On the other hand, the domain wall junction charge can A. Derivation of the BPS equations be defined from the topological charge density Ymn as L Z From now on, we investigate BPS states of in Eq. (2.1) ξ ξ in the case that the number N0 of flavors for real adjoint ξ ξ m nY − m n ≤ 0 Ymn ¼ m n dx dx mn ¼ 2 Smn ; ð3:7Þ scalars is equal to the number of the spatial dimensions D.1 e In what follows, the Roman index m stands not only for the spacial index as m ¼ 1; 2; …;Dbut also for the index of N0 negatively contributing to the BPS energy. Here, we have (m ≡ A0). Then, the standard Bogomol’nyi completion for defined this system goes as follows: Z ∂mΣm ∂mΣn S ≡ dxmdxn det : ð3:8Þ 1 X mn ∂ Σ ∂ Σ E ¼ fF2 þð∂ Σ − ξ ξ ∂ Σ Þ2g n m n n 2e2 mn m n m n n m m>n xm − xn X 2 The integrand is a Jacobian of a map from the whole 1 Σm − Σn þ ξ ∂ Σ − Y plane to a region in the plane defined by the function 2e2 m m m m Σ m n Σ m n k X ð mðx ;x Þ; nðx ;x ÞÞ with all other coordinates x (k ≠ m, n) fixed. S can be either positive or negative, þ fDmH þ ξmðΣmH − HMmÞg mn m and its absolute value is the area of the image. Nevertheless, † Ymn is always negative since Smn is accompanied with × fDmH þ ξmðΣmH − HMmÞg X X X ð−ξmξnÞ, and so it should be understood as a sort of binding þ ξmZm þ ξmξnYmn þ ∂mJ m; ð3:1Þ energy among domain walls [67–69]. m m>n m Since the first three terms of Eq. (3.1) are positive semidefinite, the Bogomol’nyi energy bound is given by with ξm ¼ 1, and we have defined the domain wall Z X X X topological charge density m, the domain wall junction E ≥ ξ Z ξ ξ Y ∂ J Y J m m þ m n mn þ m m; ð3:9Þ charge density mn, and m by m m>n m
2 Zm ¼ v ∂mΣm; ð3:2Þ and it is saturated by the BPS states satisfying the BPS equations, 1 ∂mΣm ∂mΣn Y ¼ − det ; ð3:3Þ mn 2 F 0; : e ∂nΣm ∂nΣn mn ¼ ð3 10Þ † ξ ∂ Σ − ξ ∂ Σ 0 J m ¼ −ξmðΣmH − HMmÞH ; ð3:4Þ n m n m n m ¼ ; ð3:11Þ J respectively. The contribution by m vanishes under the ξmDmH þ ΣmH − HMm ¼ 0; ð3:12Þ space integrations since it is asymptotically zero because of X Σ − 0 the vacuum condition mH HMm ¼ . ξ ∂ Σ − Y ¼ 0; ð3:13Þ m m m m The domain wall tension Zm measured along the x m direction can be defined from Zm by Z where m; n ¼ 1; 2; …;D. One can verify that all solutions ∞ mξ Z 2ξ Σ − Σ ≥ 0 of the above BPS equations solve the full equations of Zm ¼ dx m m ¼ v mð mjxm¼þ∞ mjxm¼−∞Þ ; 1 −∞ motion. This is the D-dimensional extension of the 4 BPS 2 ðno sum over mÞ: ð3:5Þ equations of the planar domain wall junction in D ¼ cases studied in Refs. [74,75,78,79]. In the previous work [86] by the present authors, we Note that Zm is always positive regardless of the choice of generalized the topological charge densities Z and Y ξm. Hence, the genuine tension measured along the normal m mn under the observation that Zm and Ymn are nothing but one- m → Σ 1 and two-dimensional Jacobians of maps x m and In Ref. [86], we considered the same model with restricted to m n → Σ Σ a special case in which the flavor number N is also related to the ðx ;x Þ ð m; nÞ, respectively. For a domain wall Z spatial dimensions, N ¼ D þ 1. Instead, this work studies the interpolating the hAi and hBi vacua, an integral of m case with generic Nð≥ D þ 1Þ. (with an appropriate rescale to make it dimensionless)
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1 measures a covering number of a map from R ð−∞ < ðS; H0Þ → VðS; H0Þ;V∈ C ; ð3:20Þ m x < ∞Þ onto the one-dimensional interval ðmm;A < Σm < mm;BÞ. The charge is topological and indeed takes values where V is constant. This is called the V transformation either þ1,0,or−1. Similar arguments hold for Ymn, and it under which any physical information is independent. is straightforward to generalize it in D dimensions as To solve the master equation, we first need to fix the moduli matrix H0. Once H0 is given, the boundary W ðm1;m2; …;m Þ condition at jxj → ∞ is automatically specified. Namely, d 0 d 1 ∂ Σ ∂ Σ ∂ Σ we should solve the master equation with the boundary m1 m1 m1 m2 m1 m B d C condition B ∂ Σ ∂ Σ ∂ Σ C B m2 m1 m2 m2 m2 md C 2ξ m † detB C; 3:14 Ω mMmx ¼ B . . . . C ð Þ lim ¼ H0e H0: ð3:21Þ @ . . . A jxj→∞ ∂ Σ ∂ Σ ∂ Σ md m1 md m2 md md As a trivial example, let us consider a homogeneous 1 1 1 ≤ ≤ ∈ 1 2 … α 1 2 … vacuum, say the first vacuum h i. The h i vacuum where d D, mα f ; ; ;Dg ( ¼ ; ; ;d), configuration is given by the moduli matrix and mα >mβ if α > β. We have W1ðmÞ ∝ Zm and W2 m; n ∝ Y D ð Þ mn. In general, the BPS solutions in H0 ¼ðh1; 0; …; 0Þ;h1 ∈ C : ð3:22Þ dimensions have d-dimensional substructures, and Wd provides us topological numbers associated with the The corresponding master equation becomes substructures.
1 m 2 2 2 −1 2 2ξ m 1x ∇ log Ω ¼ e v ð1 − Ω jh1j e m m; Þ; ð3:23Þ B. The moduli matrix method 2 Let us solve the BPS equations (3.10)–(3.13). To this which can be solved by end, we first introduce a complex scalar function SðxmÞ by m 2 2ξ m 1x Ω ¼jh1j e m m; : ð3:24Þ Am − iξmΣm ¼ −i∂m log S: ð3:15Þ
Plugging this into Eq. (3.18), we immediately find Σ ¼ m1. Then, Eqs. (3.10) and (3.11) are automatically satisfied. It is also straightforward to verify H ¼ðv; 0; …; 0Þ. Note Plugging this into Eq. (3.12), we have that the constant h1 in the moduli matrix does not play any role in the above example. This redundancy comes ∂ H ∂ log S H − ξ HM 0: 3:16 m þð m Þ m m ¼ ð Þ from the V transformation in Eq. (3.20). Indeed, we could, from the first point, fix the moduli matrix (3.22) as This can be solved by H0 ¼ðh1; 0; …; 0Þ → ð1; 0; …; 0Þ. −1 ξ M xm Nontrivial inhomogeneous solutions including a single H ¼ vS H0e m m ; ð3:17Þ domain wall connecting the hAi and hBi vacua are gen- erated by the moduli matrix with the nonzero constants where H0 is an arbitrary complex constant N vector. H0 is only in the Ath and Bth entries as called the moduli matrix because the elements of H0 are moduli (integration constants) of the BPS solutions. 0 … 0 1 0 … 0 0 … 0 Finally, we are left with the fourth equation (3.13).To H0 ¼ð ; ; ; ; ; ; ;hB; ; ; Þ Σ solve this, let us express the neutral real scalar fields m in ∼ ð0; …; 0;hA; 0; …; 0; 1; 0; …; 0Þ: ð3:25Þ terms of S, The nonzero moduli parameters h and h are related by 1 A B Σ ξ ∂ Ω Ω ≡ 2 the V transformation as h h ¼ 1. m ¼ 2 m m log ; jSj ; ð3:18Þ A B Similarly, if we have H0 with three nonzero constants, where Ω is the gauge-invariant quantity. Using this, the we will have a domain wall junction dividing the corre- fourth BPS equation (3.13) is cast into the following sponding three vacua. The moduli matrix is given by Poisson equation in D dimensions: H0 ¼ð0;…;0;1;0;…;0;hB;0;…;0;hC;0;…;0Þ: ð3:26Þ 1 2 2 2 −1 2ξ m † ∇ Ω 1 − Ω mMmx 2 log ¼ e v ð H0e H0Þ: ð3:19Þ Maximally complex solutions dividing N vacuum domains in D dimensions are, then, obtained by the moduli We call this the master equation for the BPS states. matrix which has no zeros in any elements. Such compli- We note that the original fields Am, Σm, and H are intact cated extended objects in D dimensions are not easy for us under the following transformation: to handle without exact solutions. Unfortunately, the master
105020-4 EXHAUSTING ALL EXACT SOLUTIONS OF BPS DOMAIN … PHYS. REV. D 101, 105020 (2020) equation (3.19) does not seem analytically solvable, except In the next section, we will see that the position of the d for the very special cases possessing the highest discrete wall estimated by the weight is related to the generalized S W symmetry group Dþ1, the symmetric group of the degree topological charge d defined in Eq. (3.14). D þ 1, in the model with the special number of the flavor N D 1 and finely tuned parameters g, c, and m 0 ,as ¼ þ A ;A IV. EXHAUSTING ALL EXACT SOLUTIONS demonstrated in Ref. [86]. Even in the fine-tuned models, OF DOMAIN WALL NETWORKS the exact analytic solution were only found for a single IN THE CPN − 1 MODEL junction of the domain walls since N ¼ D þ 1 is the minimum number. However, as we show in the next A. General solutions section, the solutions include domain wall networks when There is a great simplification allowing us to obtain all N>Dþ 1. exact solutions for generic domain wall networks in D To see shapes of domain wall networks, let us define a dimensions. It is the infinite gauge coupling limit in which weight for each vacuum by we formally send the gauge coupling e to infinity in the Lagrangian (2.1). There, the kinetic terms of Aμ and Σm hAi m˜ ·xþa w ¼ e A A ; ðA ¼ 1; …;NÞ: ð3:27Þ vanish to become Lagrange multipliers. At the same time, the first term in the potential (2.2) forces the charged fields Since the weight is the exponential function of the spatial HA to take their values in the restricted region S2N−1 − 1 coordinate, only one weight dominates the rest N defined by HH† ¼ v2. Furthermore, the overall phase of x weights at each point . Suppose the weight of the hAi H is gauged. Therefore, the physical target space in the x A vacuum is dominant in vicinity of a point 0. There, we infinite gauge coupling limit is reduced to the complex hAi 2 have Ω ∼ ðw Þ , and then Σ reads from Eq. (3.18), projective space,
1 2 −1 ˜ hAi 2 N Σjx∼x ¼ ∇ logðw Þ ¼ m : ð3:28Þ N−1 SUðNÞ S 0 2 A CP ≃ ≃ : ð4:1Þ SUðN − 1Þ × Uð1Þ S1 This implies that the region where the weight whAi is Indeed, if we eliminate the gauge field Aμ from Lj →∞,it dominant corresponds to the vacuum hAi. e reduces to the standard Lagrangian of the nonlinear CPN−1 Let us next consider a situation that the vacua hAi and model. Similarly, if we eliminate the neutral scalar fields hBi are next to each other. We can estimate where the two Σ vacua transit by comparing the weights of those vacua. The m, we get a nontrivial potential which lifts all the points of the CPN−1 target space leaving N discrete points as vacua. transition occurs at points on which the two weights are Σ equal. This condition determines a hyperplane which is a Here, we do not eliminate the auxiliary fields Aμ and m. The BPS equations in Eqs. (3.10)–(3.13) remain the same, subspace of codimension 1 in RD, and the first three equations (3.10)–(3.12) are solved by Eqs. (3.15) and (3.17). The fourth equation (3.13) rewritten hA; Bi∶whAi ¼ whBi ⇔ ðm˜ − m˜ Þ · x þ a − a ¼ 0: A B A B as Eq. (3.19) reduces to an algebraic equation, easily ð3:29Þ solved by
2ξ m † 2 Ω mMmx This is a straightforward generalization of D ¼ [75] to je→∞ ¼ H0e H0: ð4:2Þ generic D dimensions. This hyperplane is nothing but a domain wall interpolating the vacua hAi and hBi, and we Thus, we have completely solved the BPS equations for call it a 1-wall. arbitrary moduli matrix H0 in the infinite gauge coupling The three vacua, say hAi, hBi, and hCi, can happen to be limit. We would like to emphasize that this is the first adjacent at a hyperplane of codimension 2, which is solutions of the domain wall networks in D ≥ 3. conventionally called the domain wall junction. We call For later convenience, let us rewrite this in a more useful it a 2-wall. The position of the 2-wall corresponds to the form. First, let us denote H0 as region where the weights are equal as a1þib1 a2þib2 a þib H0 ¼ðe ;e ; …;e N N Þ; ð4:3Þ hA; B; Ci∶whAi ¼ whBi ¼ whCi: ð3:30Þ where faAg and fbAg are N realP parametersP which we 0 These can be naturally generalized to a d wall which is a restrict to satisfy the conditions A aA ¼ A bA ¼ by d codimensional intersection dividing d þ 1 vacua. The using the V transformation. Furthermore, let us define the position of the d wall is defined by new mass vectors
… ∶ hA1i hA2i hAdþ1i m˜ ξ ξ … ξ hA1;A2; ;Adþ1i w ¼ w ¼ ¼w : ð3:31Þ A ¼ð 1m1;A; 2m2;A; ; DmD;AÞ: ð4:4Þ
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Then, we have
XN 2 m˜ x Ω ð A· þaAÞ je→∞ ¼ e : ð4:5Þ A¼1 By using this, the BPS energy density can be simply expressed by 2 XN 2 v 2 2 m˜ x E ∇˜ Σ ∇ ð A· þaAÞ je→∞ ¼ v · ¼ 2 log e ; ð4:6Þ A¼1 where we have defined
˜ ˜ 2 2 ∇ ¼ðξ1∂1; ξ2∂2; …; ξD∂DÞ; ∇ ¼ ∇ ð4:7Þ 1 ˜ and used Σ ¼ 2 ∇ log Ω. From this expression, we can see that the N − 1 real parameters faAg are the moduli which relate to the shape of the networks in the real space. On the other hand, the other parameters fbAg are internal moduli of Uð1ÞN−1 associated with constituent domain walls. FIG. 1. The grid diagram (Σ space): the four vacua (red points) D 1 The domain walls for ¼ and the domain wall in N ¼ 4 model are shown. networks for D ¼ 2 have been studied very well in the literature; therefore, we will concentrate on D ¼ 3 in the following subsections. Ω ¼ wh1i þ wh2i, and the exact domain wall solution is given as B. N = 4: Tetrahedron: Single domain wall junction 2ðm˜ 1·xþaÞ 2ðm˜ 2·x−aÞ m1 1e m1 2e Let us start with a model with N ¼ 4. There are N ¼ 4 Σ ; þ ; 1 ¼ 2 m˜ x− 2 m˜ x− ; ð4:12Þ vacua which are the minimal numbers for a nonplanar e ð 1· aÞ þ e ð 2· aÞ domain wall junction to exist. For simplicity, let us set the 2ðm˜ 1·xþaÞ 2ðm˜ 2·x−aÞ m2;1e þ m2;2e four mass vectors mA to be four vertices of a regular Σ2 ¼ ; ð4:13Þ 2ðm˜ 1·x−aÞ 2ðm˜ 2·x−aÞ tetrahedron as e þ e