Bags, junctions, and networks of BPS and non-BPS defects

PHYSICAL REVIEW D, VOLUME 61, 105019

Bags, junctions, and networks of BPS and non-BPS defects

D. Bazeia and F. A. BritoDepartamento de Fı´sica, Universidade Federal da Paraı´ba, Caixa Postal 5008, 58051-970 Joa˜o Pessoa, Paraı´ba, Brazil

~Received 9 December 1999; published 26 April 2000!

We investigate several models of coupled scalar fields that present discreteZ2 ,Z23Z2 ,Z3 and other sym-metries. These models support topological domain wall solutions of the BPS and non-BPS type. The BPSsolutions are stable, but the stability of the non-BPS solutions may depend on the parameters that specify themodels. The BPS and non-BPS states give rise to bags, and also to three-junctions that may allow the presenceof networks of topological defects. In particular, we show that the non-BPS defects of a specific model thatengenders theZ3 symmetry give rise to a stable regular hexagonal network of domain walls.

PACS number~s!: 11.27.1d, 11.30.Er, 11.30.Pb

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I. INTRODUCTION

Bogomol’nyi, and Prasad and Sommerfield~BPS! @1#,have presented a way to find topological solutions that srate the lower bound in energy. These BPS states areposed to play some role in the development of modern pticle physics, because there are supersymmetric mowhere they are shorter multiplets, stable, and no continuvariation of parameters can modify these BPS states. R@2–4# investigated some models of coupled real scalar fiein bidimensional space-time. These models present pecfeatures, such as the presence of the Bogomol’nyi bouwith the corresponding appearance of first-order differenequations that solve the equations of motion in the casstatic configurations. These Refs.@2–4# have found topologi-cal solitons in specific models by taking advantage oftrial-orbit method first introduced by Rajaraman in Ref.@5#.These investigations provide a concrete way of finding Bstates and suggest several other studies, in particular onsubject of defects inside defects, as presented for instanRefs.@6–8# and more recently in@9–15#. The investigationsof defects inside defects presented in Refs.@11,12,14,15# arenatural extensions of the former papers@2–4#.

In supersymmetric models with chiral superfields tpresence of discrete symmetry may produce BPS andBPS walls. The non-BPS walls are solutions of the equatiof motion, and they do not necessarily saturate the lobound in energy. But the BPS walls saturate the lower boin energy, and lie in shorter multiplets, in a way such ththey preserve the supersymmetry only partially@16–18#.There are BPS states that preserve 1/2 and 1/4 of the susymmetry@16–23#, and in the present work we investigaseveral explicit examples. We start by first offering geneconsiderations in Sec. II. We examine specific modelsSec. III, where we find several solutions, of the BPS anon-BPS type. See also the more recent work@24–26#,where one can find other investigations concerning the Bdomain wall junctions.

Our work continues in Sec. IV, where we investigate sbility of several BPS and non-BPS solutions. This is donot only to show stability, but also to see how the bosomatter behaves in the background of BPS and non-BPSfects. The behavior of the fermionic matter in the extendsupersymmetric models can be investigated by following

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lines of Refs.@14,27,28#. In Sec. V we investigate two otheissues. The first in Sec. V A, where we consider the entrment of the other field, as in bag models@29–31#. This pos-sibility may be implemented in the first, second and thmodels, and the bag may be of the BPS or non-BPS tydepending of the particular model under consideration.examine the second issue in Sec. V B, where we investigfusion of defects. Here we briefly review Ref.@20#, whichintroduces investigations valid within the supersymmetcontext, and after we offer another possibility of showifusion of defects and stability of junctions, valid beyond tcontext of supersymmetry.

In Sec. VI we show how one can generate planar nworks of BPS and non-BPS defects. There we show thatthree-junction that appears in supersymmetric models mbe stable, but cannot generate a BPS network of defeHowever, when we give up supersymmetry we find a mowhere domain wall junctions are stable, the three-junctstrictly obey the triangle inequality and generate a stablework of defects. We summarize the main results in Sec. Vwhere we end the present work.

The subject of this work may find direct applicationsdiverse branches of physics. Besides the recent interestforward in Refs.@21–23#, we recall for instance that theZ3group is the center of SU(3), andthis may guide us towardapplications involving strong interactions. Some exampcan be found in Refs.@32,33#. Another possibility is relatedto the Z23Z3 symmetry, in an effort to entrap thZ3-symmetric portion of the model inside the topologicdefect generated by theZ2 symmetry. Other applications include issues related to cosmology@34# and magnetic materi-als @35#, and to pattern formation in systems of condensmatter @36#. The investigations that we present are doneflat space-time, in the (d,1) dimensional case. We use natral units, taking\5c51, and the metric tensor presents sinature in the form (1,2, . . . ,2). In the present work weare mainly concerned with the casesd51, 2, and 3. Wespecify the spatial dimension along with the interest ofsubject under investigation.

II. GENERAL CONSIDERATIONS

Let us start with two real scalar fieldsf andx. We con-sider models defined via the Lagrangian density

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D. BAZEIA AND F. A. BRITO PHYSICAL REVIEW D 61 105019

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This is valid in (d,1) space-time dimensions.V5V(f,x) isthe potential. It has the general form

V~f,x!51

2Wf

2 11

2Wx

2 . ~2!

W5W(f,x) is some smooth function of the fieldsf andx,andWf5(]W/]f), etc. This is the form of the~real bosonicportions of the! models we deal with in the present worThese models admit a supersymmetric extension, and insupersymmetric theory the functionW is the superpotential.

Models defined viaW(f,x) present several interestinproperties@2–4#. The equations of motion are

]2f

]t22¹2f1

]V

]f50, ~3!

]2x

]t22¹2x1

]V

]x50. ~4!

For static field configurations living in the (1,1) space-timthat is, for fieldsf5f(x) andx5x(x) they reduce to

d2f

dx25WfWff1WxWxf , ~5!

d2x

dx25WfWfx1WxWxx . ~6!

These equations are solved by first-order differential eqtions

df

dx5Wf , ~7!

dx

dx5Wx . ~8!

The energy of solutions that solve the first-order equatiare bounded to the valueEB

i j 5uDWi j u, where

DWi j 5W~f i ,x i !2W~f j ,x j !. ~9!

Here (f i ,x i) is the i th vacuum state of the model undconsideration. The reason for this is simple: the energystatic configurations is

E51

2E2`

`

dxF S df

dx D 2

1S dx

dxD 2

1Wf2 1Wx

2G . ~10!

It can be rewritten in the form

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2E2`

`

dxF S df

dx2WfD 2

1S dx

dx2WxD 2G

1E2`

`

dxS Wf

df

dx1Wx

dx

dxD . ~11!

We recall that

dW

dx5Wf

df

dx1Wx

dx

dx~12!

to see that the energy gets to the boundEB when the fieldconfigurations obey the first-order equations~7! and~8!. Thisis the Bogomol’nyi bound, and the corresponding solutioare BPS solutions. The BPS solutions are linearly or clacally stable—see Sec. IV.

The energy of BPS solutions can be also written as

EB5E2`

`

dxS S df

dx D 2

1S dx

dxD 2D ~13!

5E2`

`

dx~Wf2 1Wx

2!. ~14!

The above result shows that the gradient~g! and potential~p!portions of the energy

g5E2`

`

dx g~x!5E2`

`

dx1

2 F S df

dx D 2

1S dx

dxD 2G , ~15!

p5E2`

`

dx p~x!5E2`

`

dx1

2~Wf

2 1Wx2! ~16!

contribute evenly to the total energy of the solution. Athough this is shared by all the BPS states, it is not pecuto BPS states, since there are non-BPS states that alsoenergy evenly distributed in its potential and gradieportions—see below.

In the case of two scalar fields, there are other possarrangements to the energy~10!. We can, for instance, consider the possibility

E51

2E2`

`

dxF S df

dx2WxD 2

1S dx

dx2WfD 2G

1E2`

`

dxS Wf

dx

dx1Wx

df

dx D . ~17!

This expression is also minimized to obey Eqs.~13! and~14!when one sets

df

dx5Wx , ~18!

dx

dx5Wf . ~19!

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BAGS, JUNCTIONS, AND NETWORKS OF BPS AND . . . PHYSICAL REVIEW D61 105019

We can show that solutions to these equations~18! and~19!only solve the equations of motion when we imposeW(f,x) the extra condition

Wff5Wxx . ~20!

Unfortunately, however, this condition~20! factorizesW(f,x) into the sumW1(f1)1W2(f2) when one rotatesthe (f,x) plane to (f1 ,f2), with

f651

A2~x6f! ~21!

as first shown in Ref.@15#. The condition~20! turns thesystem into two decoupled systems.

We can consider another possibility to rewrite the ene~10!. We set

E51

2E2`

`

dxF S df

dx1WxD 2

1S dx

dx2WfD 2G

1E2`

`

dxS Wf

dx

dx2Wx

df

dx D . ~22!

This expression is also minimized to obey Eqs.~13! and~14!when we write

df

dx52Wx , ~23!

dx

dx5Wf . ~24!

These equations~23! and~24! solve the equations of motiowhen one imposes onW(f,x) the extra condition

Wff1Wxx50. ~25!

This result is interesting. It shows that forW(f,x) har-monic, we can solve the two~second-order differential!equations of motion~5! and ~6! with two sets of two first-order differential equations, the set of Bogomol’nyi equtions ~7! and ~8!, and another one, given by Eqs.~23! and~24!.

We make good use of this result by imposing thatsecond set of first-order equations~23! and ~24! are stillBogomol’nyi equations, that appear via the presence ofother functionW̃. This new function is defined by

W̃f52Wx , ~26!

W̃x5Wf . ~27!

SinceW and W̃ are smooth functions, they are necessaharmonic. We then get the result that potentials definedharmonic functions can always be written in terms of twfunctions,W and W̃. We use this to introduce the complesuperpotential

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W5W1 iW̃. ~28!

The superpotentialW5W(w) now depends on the complefield w5f1 ix, and the harmonicity of bothW and W̃ ap-pears via the Cauchy condition onW. This is the way we getfrom the investigations@2–4# to the recent work in Refs@21–23#.

We guide ourselves toward the topological solutionsintroducing a topological current. This is the standard produre, and we can introduce for instance the topological crent

JBPSa 5«ab]bW~f,x!. ~29!

It is defined in terms ofW(f,x), and the topological chargeis directly identified with the energy of the topological soltion that satisfies the pair of first-order equations~7! and~8!.This definition identifies the topological charge with the eergy of the BPS solution. It expresses the fact that the mof a BPS state is completely determined by its topologicharge. This definition provides a single way to infer stabity of the BPS states. This is a nice way of defining ttopological current, but it is not complete, because it cansee sectors of the non-BPS type. It is only effective for Bsectors, and that is the reason for the label BPS assigneit. However, since we need to identify every topological setor, we should use another topological current. We can csider, for instance

Ja5«ab]bS f

xD . ~30!

It obeys]aJa50, and it is also a vector in the (f,x) plane.The topological current~30! is the current that we shall usin this work. We use this definition to obtain, for static cofigurations

Jat Ja5r tr52 g~x!, ~31!

where g(x) is the gradient energy density of the solutioThis property can be used to infer stability of junctions, fconfigurations that present energy density evenly distribuin their gradient and potential energy densities. This isbecause in this caser tr52g(x)5g(x)1p(x) gives the fullenergy density, and may be used to show that the juncprocess occurs exothermically. This result was already induced in Ref.@24#, and offers a way of showing how thprocess of junctions of defects may occur exothermicallymodels defined outside the context of supersymmetry—Sec. V B.

Up to here the calculations were done in (1,1) space-tdimensions. However, we can easily immerse the sysinto (2,1) or into (3,1) space-time dimensions. In the ficase, ind52 the solutions can be seen as domain ribboand the energy is now multiplied by the length of the ribons. In the second case, ind53 we get domain walls, andthe energy is to be multiplied by the area of the wall. Tplanar case is necessary for introducing junctions of domribbons @or of domain walls if one immerses the system

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the (3,1) space-time# and their corresponding planar neworks. There are several distinct possibilities, and in thequel we specify some cases. We recall that we singletopological sectors using pairs of vacuum states. For Bsectors the energy is obtained via the superpotential, andshall also use the notationt i j 5uW(f i ,x i)2W(f j ,x j )u,identifying the BPS sectors with tensions or energies oftopological defects.

III. SOME SPECIFIC MODELS

Our aim here is to introduce models of coupled scafields, to illustrate how one can find topological solutionsthe BPS and non-BPS type. We investigate several modecoupled scalar fields, defined with smooth functioW(f,x). We organize the subject in the four subsectiothat follow, where we examine four different models.

A. The first model

We consider the superpotential

W(1)~f,x!52lf11

3lf31mfx2, ~32!

where l and m are real parameters. This model was fiinvestigated in Ref.@2#, but here we bring new results. Thequations of motion for static field configurations are givby

d2f

dx252l2f~f221!12m2S 21

l

m Dfx2, ~33!

d2x

dx252lm~f221!x14m2f2x12m2x3. ~34!

These equations are solved by configurations that also sthe pair of first-order differential equations

df

dx5l~f221!1mx2, ~35!

dx

dx52mfx. ~36!

Solutions that solve these first-order equations are BPS stions.

This model has four absolute minima whenl/m.0, andtwo minima whenl/m,0. For l/m.0 the minima are(61,0) and (0,6Al/m). Topological solutions are solutionthat connect adjacent minima. Thus, the asymptotic behaof the fields is governed by the set of minima of the potetial. Forl/m.0 the system supports several topological stors. Their energies or tensions are given by, for solutithat connect the minima (21,0) and (1,0)

tBPS1 5

4

3ulu. ~37!

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For solutions that connect the minima (61,0) and (0,6Al/m) we get

t̄ BPS1 5

2

3ulu. ~38!

For the non-BPS solution that connects the minim(0,Al/m) and (0,2Al/m) we obtain

tnBPS1 5

4

3uluAl

m. ~39!

There are BPS solutions in the first sector, which connethe two minima (61,0). They are

f~x!52tanh~lx!, ~40!

x~x!50 ~41!

and

f~x!52tanh~2mx!, ~42!

x~x!56Al

m22 sech~2mx!. ~43!

The first pair is of the one-component type, and is importfor investigating the presence of defects inside defects—for instance Refs.@11,12,14,15#. The second pair is of thetwo-component type, and may play a role in applicationscondensed matter, as for instance in the case investigateRefs.@4,37#.

The one-component and the two-component type of sotions also appear in magnetic materials@35#. There they areknown as Ising and Bloch walls, respectively. The twcomponent Bloch wall has a peculiar behavior, that canused to understand specific features of interfaces betwmagnetic domains. The issue here is that if one looks at thsolutions as vectors in the (f,x) plane, forx spanning theinterval (2`,`) they describe a straight line and an ellipcal arc connection between the two minima (61,0), respec-tively, behaving very much like linearly and elliptically polarized light. For this reason, the Block walls can be seenchiral interfaces between domains with different magnetition, and this can be used to describe the wall motionmagnetic materials@38,39#. Furthermore, these walls can bcharged and we may introduce charge by making thex fieldcomplex, or by coupling fermions to the system.

For l/m,0 we have only two vacuum states. They a(61,0) and now we have a single topological sector, wthe same energy and topological charge obtained beforethe corresponding sector. We notice that the first pair oflutions ~40! and~41! solves the first-order equations even fl/m,0. This one-component solution is still of the BPtype for l/m,0, but now no topological defect can bnested inside it. The reason is that forl/m,0 the modelpresents no spontaneous symmetry breaking in the indedentx direction, and so it cannot support topological defein this x direction anymore.

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For l/m.0, there are non-BPS solutions connecti(0,Al/m) and (0,2Al/m) by a straight line. To see thiexplicitly we considerf50 in the specific system to get thequation of motion for the static fieldx5x(x)

d2x

dx252m2S x22

l

m Dx. ~44!

The solutions are

x~x!56Al

mtanh~Almx!. ~45!

The corresponding non-BPS energy is (4/3)uluAl/m, as al-ready presented in Eq.~39!.

This model decouples forl5m, as we have alreadyshown in Ref. @12#. In this case the symmetryZ23Z2changes to theZ4 one, but here a rotation byp/4 changes themodel into two single-field models. We also note that forl522m, the model decouples into two single-field systemshowing that the limitm→2l/2 is also uninteresting, aleast from the point of view of models of coupled scafield.

We expose other features of the model usingW(1). Wewrite

Wff(1)52lf, Wxx

(1)52mf, Wfx(1)52mx. ~46!

We see thatW(1) is harmonic whenm→2l. This meansthat the model admits another pair of first-order equatioThis pair follows from Eqs.~23! and ~24!

df

dx52mfx, ~47!

dx

dx5l2lf22mx2. ~48!

These equations also furnish solutions to the equationmotion, for m→2l. As we have already seen, form→2lthe model presents a single topological sector, already intigated. But we notice thatW(1) is also harmonic in the limitf→0, independently of the parametersl andm. We use thisinto the above Eqs.~47! and ~48! to getf50 and

dx

dx52mS x22

l

m D . ~49!

We considerlm.0. In this case the above equation is eassolved to give

x~x!5Al

mtanh~Almx!. ~50!

The pairf50 andx in ~50! is the pair of non-BPS solutionthat we have already found. Here, however, we foundnon-BPS solution as an explicit solution of the new firorder differential equations in systems of coupled real scfields. Although this is a non-BPS solution, we have alrea

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shown in Sec. II that its energy is composed of equal ptions of gradient~g! and potential~p! contributions. In termsof densities we have, explicitly

g1~x!5p1~x!51

2l2 sech4~Almx!. ~51!

We can now understand this feature by realizing thatthough this pair of solutions is non-BPS, in the reduced stem wheref→0, thex field solution is indeed a BPS solution.

B. The second model

We consider another model, defined by the superpoten

W(2)~f,x!5l~f221!x. ~52!

This model was also considered in Ref.@28#. The equationsof motion are

]2f

]t22

]2f

]x212l2~f221!f14l2fx250 ~53!

and

]2x

]t22

]2x

]x214l2f2x50. ~54!

For static fields the energy is minimized for configuratiothat obey the first-order equations

df

dx52lfx, ~55!

dx

dx5l~f221!. ~56!

The model presents two vacuum states, given by (61,0). Incontraposition with the former model, however, this nemodel presents no BPS sector, because the function~52!obeysW(2)(f251,x)50, and then givesuDW(2)u50. Thismeans that the first-order equations~55! and~56! present notopological solutions connecting the two vacuum states.the other hand, the vacuum states (61,0) can be connectedby topological but non-BPS solutions. The explicit formthe solutions is given by

f~x!56 tanh~lx!, ~57!

x~x!50. ~58!

The energy or tension of the solution is

tnBPS2 5~4/3!ulu. ~59!

This result reproduces the value presented by the simone-component BPS solution~40! and ~41! of the firstmodel.

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We compare this model with the first one to see thaprovides a nice way to understand how the BPS and nBPS solutions behave. We notice that

Wff(2)52lx, Wxx

(2)50, Wfx(2)52lf. ~60!

There is only one way to makeW(2) harmonic, which im-plies settingx→0. The new first-order equations correspoto the equations~23! and ~24! for W(2), in the limit x→0.They give

df

dx5l~f221!. ~61!

This is solved by

f~x!52tanh~lx!. ~62!

The pair of solutions formed byx50 andf as in Eq.~62! isthe pair of non-BPS solutions that we have already foundthis model. This is another example where one gets non-Bsolutions as solutions to first-order differential equatioThis is perhaps the most natural way of understanding wthis non-BPS solution has energy evenly distributed intogradient and potential portions. In terms of energy densiwe have

g2~x!5p2~x!51

2l2 sech4~lx!. ~63!

C. The third model

Let us now consider another model, defined by the suppotential

W(3)~f,x!52lf11

3mf32mfx2. ~64!

This model has the first-order equations

df

dx52l1mf22mx2, ~65!

dx

dx522mfx. ~66!

The model has only two minima, and forl/m.0 they are(6Al/m,0). They give

W(3)~6Al/m,0!572

3lAl

m~67!

and define a BPS sector. The explicit BPS solution is givby

f~x!52Al

mtanh~Almx!, ~68!

together withx50. We deal with the first-order equationsget

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xS f22l

m2

1

3x2D50. ~69!

This restriction constrains the orbits in the (f,x) plane insuch a way that no finite orbit can connect the two minimunless in the casex50, in which one reproduces the topological solution just obtained. This model has a single toplogical sector, and presents no family of topological sotions, like the family ~42! and ~43! obtained in the firstmodel.

We useW(3) to obtain

Wff(3)52mf, Wxx

(3)522mf, Wfx(3)522mx. ~70!

We see thatW(3) is harmonic. This means that we havanother pair of first-order equations that come from Eqs.~23!and ~24!. It is

df

dx52mfx, ~71!

dx

dx52l2mx21mf2. ~72!

We think of this pair as appearing from another functioW̃(3), as defined in the Eqs.~26! and ~27!. We get

W̃(3)52lx21

3mx31mf2x. ~73!

We notice thatW̃(3) vanishes along the linex50 where theformer sector was explicitly constructed.

We can useW̃(3) to write W̃ff(3)52mx, W̃xx

(3)522mx,

andW̃fx(3)52mf, showing thatW̃(3) is also harmonic, as ex

pected. As we have suggested in Sec. II, we use the comsuperpotentialW (3)5W(3)1 iW̃(3) and the complex fieldw5f1 ix. We change notation and write

W2~w!5lw21

3mw3 ~74!

as the superpotential of the third model. The new notatuses the subscriptN in WN to identify the complex superpotential that presents theZN symmetry—see Sec. V B for further details. We useW2 to get, in the vacuum statesw̄k , k51,2

W2~ w̄1!522

3lAl

m, ~75!

W2~ w̄2!52

3lAl

m, ~76!

or better

W2~ w̄k!522

3lAl

me2p i [(k21)/2], k51,2. ~77!

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We use this result to obtain the tensions

t jk(2)5uW2~ w̄ j !2W2~ w̄k!u. ~78!

In the present case there is a single topological sector,tension

t1(2)5

4

3uluAl

m. ~79!

Here we usetn(N) to identify tensions in models with comple

superpotentials having theZN symmetry—see Sec. V B fomore details.

D. The fourth model

We consider another model, defined by the superpoten

W(4)~f,x!5l̄f1m̄x21

4lf42

1

4lx41

3

2lf2x2.

~80!

This model is similar to models already investigated in R@4#. The equations of motion are

d2f

dx256m̄lfx23l̄l~f22x2!26l2f~x223f2!x2

13l2f~f223x2!~f22x2!, ~81!

d2x

dx256l̄lfx23m̄l~x22f2!26l2f2~f223x2!x

13l2~x223f2!~x22f2!x. ~82!

These equations are solved by field configurations that othe first-order equations

df

dx5l̄2lf~f223x2!, ~83!

dx

dx5m̄2lx~x223f2!. ~84!

The minima of the potential obey

f~f223x2!5l̄

l, ~85!

x~x223f2!5m̄

l. ~86!

Interesting cases are obtained withm̄50, and with l̄50.They give similar models, and we consider the casem̄50.We usef̄35l̄/l, and the minima now obey

f~f223x2!5f̄3, ~87!

x~x223f2!50. ~88!

10501

th

ial

f.

ey

The minima are given by (f i ,x i), i 51,2,3, where

f15f̄, x150, ~89!

f25f3521

2f̄, x252x35

A3

2f̄. ~90!

They form a triangle in the (f,x) plane. The ratiol̄/lspecifies the vacuum states. For instance, forl̄5l we getthe minima at

~1,0!,S 21

2,6

1

2A3D . ~91!

The minima are equidistant from each other. The trianthey form is equilateral, and the symmetry is theZ3 symme-try. The distance between the minima isd5A3uf̄u, and de-pend on the ratiol̄/l. The functionW(4) is changed to, form̄50,

W(5)5l̄f21

4lf42

1

4lx41

3

2lf2x2. ~92!

The equations of motion are solved by the first-order diffential equations

df

dx5l@f̄32f~f223x2!#, ~93!

dx

dx52lx~x223f2!. ~94!

We use Eq.~92! to get

W(5)~f1 ,x1!53

4lf̄4, ~95!

W(5)~f2 ,x2!5W(5)~f3 ,x3!523

8lf̄4. ~96!

We notice thatW(4) ~and alsoW(5)) is harmonic. This meansthat the potential of this model can be obtained by anotsuperpotential, given by

W̃(4)~f,x!52m̄f1l̄x2lf3x1lfx3. ~97!

We setm̄50 to get

W̃(5)~f,x!5l̄x2lf3x1lfx3. ~98!

The corresponding first-order equations are

df

dx5lx~x223f2!, ~99!

dx

dx5l@f̄32f~f223x2!#. ~100!

9-7

ef-

arnt

owwica

lity

the

cit

thenolveera-


Evidently, the minima are (f i ,x i), i 51,2,3, the sameminima already obtained in Eqs.~89! and ~90!. However,W̃(5) behaves differently

W̃(5)~f1 ,x1!50, ~101!

W̃(5)~f2 ,x2!52W̃(5)~f3 ,x3!53

8A3lf̄4. ~102!

This model leads to the model recently considered in R@21,22#. We proceed like in the form model to get the complex superpotential

W3~w!5l̄w21

4lw4. ~103!

The fieldw is complex,w5f1 ix. In Ref. @21# one consid-ers the casel̄5l51. In this case the potential is a particulcase of the potential introduced above. The superpotecan be seen asW35W(5)1 iW̃(5), and this allows obtaining

W3~ w̄1!53

4lf̄4, ~104!

W3~ w̄2!53

4lf̄4S 2

1

21 i

1

2A3D , ~105!

W3~ w̄3!53

4lf̄4S 2

1

22 i

1

2A3D , ~106!

or better

W3~ w̄k!53

4lf̄4e2p i [(k21)/3], k51,2,3. ~107!

We use this result to write

t jk(3)5uW3~ w̄ j !2W3~ w̄k!u. ~108!

The three tensions degenerate to the single value

t1(3)5

3

4A3uluf̄4. ~109!

IV. STABILITY

In this Sec. IV we deal with issues concerning stabilitythe topological solutions in (1,1) dimensions. We folloRef. @15#, recalling that the linear stability also shows hothe bosonic fields behave in the background of the classsolutions. We examine linear stability for two coupled rescalar fields using f(x,t)5f̄(x)1h(x,t) and x(x,t)5x̄(x)1j(x,t). Here

h~x,t !5(n

hn~x!cos~wnt !, ~110!

10501

s.

ial

f

all

j~x,t !5(n

jn~x!cos~wnt ! ~111!

are the fluctuations around the static solutionsf̄ and x̄. Weuse the equations of motion to get the equation for stabi

S 21̂d2

dx21U D S hn

jnD 5wn

2S hn

jnD . ~112!

For wn2>0 (wn

2,0) we have stable~unstable! solutions.

Here 1̂ is the 232 identity matrix and

U5S V̄ff V̄fx

V̄xf V̄xxD . ~113!

The bar over the potential means that after obtainingderivatives, we should substitute the fieldsf andx by theirclassical static valuesf̄(x) and x̄(x).

We consider the first model first. In this case the explicalculation gives to the matrixU (1) the elementsUi j

(1)

52m2mi j , where

m1152S l

m D 2

13S l

m D 2

f̄21S l

m12D x̄2, ~114!

m125m2152S l

m12D f̄x̄, ~115!

m2252l

m1S l

m12D f̄213x̄2. ~116!

We use the first-order equations~35! and ~36! to introducethe first-order operators

S51̂d

dx12mS l

mf̄ x̄

x̄ f̄D ~117!

and

S†521̂d

dx12mS l

mf̄ x̄

x̄ f̄D . ~118!

These operators allow writingH15S†S, where H1 is theHamiltonian corresponding to the first model

H1521̂d2

dx21U (1). ~119!

This factorizationH15S†S shows thatH1 is positive semi-definite, and so can have no negative eigenvalue. Weconclude that the BPS solutions are stable, since they sthe first-order equations we need to get the first-order optors ~117! and ~118!.

9-8

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n

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a

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ieshe

en

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a-

o

andsti-

oftieson

or-en-


In this first model, the one-component solutions cansolved explicitly. For the BPS pair~40! and ~41! we get thefollowing set of discrete eigenvalues: in the first directiothe f direction

w0(1,1)50, w1

(1,1)5A3l2 ~120!

and in the second direction, thex direction

w0(1,2)50. ~121!

For the pair of non-BPS solutions given byf50 andx inEq. ~45! we get

w̄n(1,1)5AlmH 42F S n1

1

2D2A2S 21l

m D11

4G2J 1/2

~122!

and

w̄0(1,2)50, w̄1

(1,2)5A3lm. ~123!

Sincel/m.0, we find the restriction 0,l/m<1 on theratio l/m in order to make the non-BPS solution stable.

We notice that form5l the eigenvaluesw̄n(1,1) change to

0 andA3l2, and degenerate to the valuesw̄n(1,2) , n50,1,

given above. This means that the scalar matter splits intoequal portions that present equal BPS states.

Let us now consider the second model. For the pairnon-BPS solutions given by Eqs.~57! and ~58! we get thefollowing set of eigenvalues for the bound states, in the fidirection:

w0(2,1)50, w1

(2,1)5A3l2. ~124!

In the second direction we have

w0(2,2)5A@~A1721!/2#l2, ~125!

w1(2,2)5A@~3A1725!/2#l2. ~126!

Let us now consider the third model. We use the pobtained with Eq.~68! and x50 to obtain for the boundstates the set of eigenvalues, in the first direction

w0(3,1)50, w1

(3,1)5A3lm ~127!

and, in the second direction

w0(3,2)5A2lm. ~128!

We cannot present explicit investigations concerning sbility of solutions of the fourth model, because we do nknow any explicit solution in this case. However, we cosider stability of the solutions of another model, introducin Ref. @24#. This model is defined by the superpotential

10501

e

,

o

f

t

ir

-t-d

W~f,x!5lA 4

27S x1A1

3D 3

2lS x1A1

3D 2

1lfS x1A1

3D 2

23lf1lf3. ~129!

The system presents the three minima (0,6A4/3) and (61,2A1/3). They give three BPS sectors, having energulu, 3ulu, and 4ulu. The highest energy is associated to tsector connecting the minima (61,2A1/3). This is the onlysector where we can find explicit solutions. They are givby

f~x!52tanh~3lx!, ~130!

x~x!52A1/3. ~131!

This pair of solutions represents a straight line orbit in t(f,x) plane, connecting the two vacua (61,2A1/3). Weconsider stability of the above solutions. Here the fluctutions decouple and we obtain

U1159l2@426 sech2~3lx!#, ~132!

U2259l2S 8

9D F11tanh~3lx!25

4sech2~3lx!G .

~133!

The first potentialU11 is the modified Po¨schl-Teller poten-tial. It presents two bound states, with energies

w050, w153A3l2. ~134!

The second potentialU22 is not reflectionless. It supports nbound state, and the continuum starts at zero energy.

V. FURTHER CONSIDERATIONS

The models investigated above present similar BPSnon-BPS solutions. They provide concrete ways of invegating how the scalar fields behave in the backgroundBPS and non-BPS solutions. We explore some possibiliin the next Sec. V A. In Sec. V B we investigate intersectiof defects.

A. Entrapment of the other field

In the first model, we can further understand the imptance of the topological solutions by investigating the pottial V1(f,x). It is given by

V1~f,x!51

2l2~f221!21lm~f221!x2

11

2m2x412m2f2x2. ~135!

9-9

se

n-

d

th

a

t d

eak

ytryo

ths,aliolly

so

-w

e

y

baly

fo-ht

ions

di-ol-a-alarral

wsgh

d


This model presents the one-component BPS solution~40!and ~41!. We havef251 outside the BPS defect, andf50 at x50, at the center of the defect. We then uV1(f,x) to get

V1~f→0,x!51

2m2S x22

l

m D 2

, ~136!

V1~f2→1,x!52m2x211

2m2x4. ~137!

For l/m.0 the potential~136! shows the presence of spotaneous symmetry breaking in the independentx direction inthe (f,x) plane. In this case thex field presents squaremasses given by

mx2~ in!54lm, ~138!

mx2~out!54m2, ~139!

where in and out identify the region inside and outsidedefect. We see that the ratio of masses is

mx2~ in!

mx2~out!

5l

m. ~140!

This ratio identifies the region 0,l/m,1 in the space ofparameters where the fieldf entraps the other field. Forgiven l, we see that the parameterm controls the ratio ofsquared masses. Thus, the efficiency of the entrapmenpends onm.

This model can be used as a bag model@29#, similar to themodels in Refs.@30,31#. Here we can further enlarge thmodel by changing dimensions from (1,1) to (3,1) and ming thex field complex, going from theZ23Z2 symmetry tothe Z23U(1). However, since inside the defect formed bf(x), the x field still engenders spontaneous symmebreaking forlm.0, we can only circumvent the presenceGoldstone bosons when one gauge the relatedU(1) symme-try, making it local. The model that emerges presentssymmetryZ23Ul(1). Since we started in (3,1) dimensioninside the wall the theory is effectively (2,1) dimensionThis is an alternate mechanism for dimensional reductand inside the wall the effective theory may very naturadevelop the Callan-Harvey effect@40#, if one appropriatelycouples fermions to the second field,x. This scenario seemto appear interesting, and we shall further explore this psibility in a separate report.

The first model is simpler forlm,0. In this case, spontaneous symmetry breaking no longer appear inside theformed by thef field. However, the wall is still of the BPStype. The potentialV1(f,x) changes in a way such that thratio ~140! becomes

mx2~ in!

mx2~out!

52l

2m. ~141!

10501

e

e-

-

f

e

.n,

s-

all

This result shows that the topological defect generated bfcan still entrapx mesons for 0.l/m.22, acting like a bagfor the elementary excitations of the fieldx. If we change thesymmetryZ23Z2 to Z23U(1) by making thex field com-plex, in this case we do not need to make theU(1) symme-try local, and we can still explore the presence of a glosymmetry within the wall. Work in this direction was alreadpresented in Ref.@41#, in a model inspired in the work oRef. @6#, with a potential that differs from the one here prposed. We believe that our model may give further insiginto the issues that arise in this scenario. Further applicatcan be done, as for instance in Ref.@42#, where one exploresthe presence of diffuse domain walls at intersecting flatrections within the cosmological scenario. Also, we can flow Ref. @43# to investigate issues concerning hybrid infltion, envolving features that requires at least two real scfields, the inflaton field and another one, which is in geneused to break the symmetry.

The second model is defined with the superpotential~52!.It gives the potential

V2~f,x!51

2l2~f221!212l2f2x2. ~142!

Here we have

V2~f50,x!51

2l2, ~143!

V2~f251,x!52l2x2. ~144!

Thus the squared masses associated to thex field are

mx2~ in!50, ~145!

mx2~out!54l2. ~146!

In this case the non-BPS defect always entraps the fieldx,acting like a bag for the elementaryx excitations. Here,however, thex meson is massless inside the bag. This shothat the entrapment is more efficient in this model, althouthe basic solution, the bag, is of the non-BPS type.

We can investigate the third model similarly. It is defineby the superpotential~64! and we have

V3~f,x!51

2m2S f22

l

m D 2

2m2S f22l

m Dx2

11

2m2x412m2f2x2. ~147!

We can write, forl/m.0,

V3~0,x!51

2m2S x21

l

m D 2

, ~148!

V3~6Al/m,x!52lmx211

2m2x4. ~149!

9-10

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ldser

ex-o-tion


No spontaneous symmetry breaking appears insidef-defect anymore, and so the model cannot support deinside defect. However, the squared masses of the elemtary x mesons are

m̃x2~ in!52lm, ~150!

m̃x2~out!54lm. ~151!

We get the ratio

m̃x(in)2

m̃x(out)2

51

2. ~152!

This ratio does not depend on the parametersl and m, butthe topological defect may also entrapx mesons. This isanother model, in which the topological state may simulatbag to entrap the elementary excitations of the other fieldthis case, however, the efficiency of the entrapment canlonger be controlled by the parameters that definesmodel.

The second and third models may also be immersed in(3,1) dimensional space-time. This immersion provide alnative ways to investigate the behavior of the current genated by making the second field complex, as commenabove. They give similar scenarios for charged walls, amay present distinct physical contents.

Other applications include the diverse issues recentlyvestigated in Refs.@12–15,27,44#. We can, for instance, enlarge the models in order to break the symmetry and/orsupersymmetry. This possibility may give rise to Fermi ba@27#, Fermi disks@14#, and other charged objects@44#, whichdepend on the particular space-time dimension one swith, and on the parameters that define the terms addeallow the symmetry breaking. We can also follow Re@12,13,15# to extend the scenario of nested defects to the cof nested network of defects. This picture follows as a dirgeneralization of the result of Ref.@15#, which shows explic-itly that defects may nest defects if the model engendersZ23Z2 symmetry in systems of two real scalar fields. As ware going to see below, to have intersection of defectsmust necessarily enlarge theZ2 symmetry to at least theZ3one. For this reason, we can think of a model that presthe Z23Z3 symmetry, composed of three real scalar fielone presenting theZ2 symmetry, and the other two controling the Z3 symmetry. In (3,1) space-time dimensions tfirst field engenders theZ2 symmetry, and we can certainlget to the case where it may give rise to a domain wwhich may nest the two other fields in its interior. Withthis scenario, if theZ3 symmetry related to the second anthird fields is still effective we may nest a planar networkdefects inside domain walls. We take advantage of the moinvestigated in Ref.@24# to present a model that seemswork correctly. This model is defined by

V~c,f,x!52

3l2S c22

9

4D 2

2l2c2~f21x2!

1l2~f21x2!22l2f~f223x2!. ~153!

10501

hectn-

aInoe

er-r-dd

-

es

rtsto

.set

e

e

ts,

l,

fel

It behaves standardly in (3,1) space-time dimensions ansuch thatV(c)5V(c,0,0) obeys

V~c!52

3l2S c22

9

4D 2

. ~154!

We consider that the fieldc5c(x) depends only onx. Thus,it supports BPS defects with the explicit form

c~x!523

2tanh~A3lx!. ~155!

The corresponding energy density is 3A3l2. We also noticethat the potentialsVin(f,x)5V(0,f,x) and Vout(f,x)5V(63/2,f,x) are such that

Vin~f,x!527

8l21l2~f21x2!22l2f~f223x2!

~156!

and

Vout~f,x!5l2~f21x2!22l2f~f223x2!

29

4l2~f21x2!. ~157!

We see that theZ3 symmetry of the pair of fields (f,x) ispreserved both inside and outside the domain wall generby the first field,c. In the region inside the domain walVin(f,x) has minima at the points

S 3

4,0D , S 2

3

8,6

3

8A3D . ~158!

In this region inside the domain wall the squared mascorresponding to the two mesons are

mf2 ~ in!5

9

4l2, ~159!

mx2~ in!5

27

4l2. ~160!

In the region outside the domain wall,Vout(f,x) has theminima @24#

S 3

2,0D S 2

3

4,6

3

4A3D . ~161!

In this case the squared masses aremf2 (out)5mx

2(out)5(27/2)l2. These results show that the fieldc engenders anefficient mechanism for the entrapment of the pair of fie(f,x) in this model. We shall further explore this and othrelated issues in another work.

B. Intersection of defects

The process of intersection of defects was alreadyplored in Ref.@20#, in the case of supersymmetric field theries. There, the basic idea was to show that the intersec

9-11

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thretria

m

h

enexes

le

s

a

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te

ctn

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althisym-er-tingtheBPSra-

tedwn,andrks

ea-q.

al

tedey

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oc-


of two extended objects is energetically admissible whecan be seen as a reaction that occurs exothermically.may be translated into the expressiont ik,t i j 1t jk , wheret ik , t i j , andt jk are the energies or tensions associated tofusion of the two defects, and to the individuals defects,spectively. The crucial point here is that in supersymmetheories the topological chargeT of the defect appears ascentral charge, and for BPS states one getst5uTu. If one usest i j 5uTi j u, t jk5uTjku, andt ik5uTi j 1Tjku the triangle inequal-ity t ik<t i j 1t jk follows naturally. This triangle inequality isstrictly valid in supersymmetric systems described by coplex superpotentials, whose values at the vacuum statesfine points in the complex plane, not aligned into a straigline segment.

The fourth model is the supersymmetric model that opthe possibility for the presence of stable intersection oftended objects, giving rise to a triple junction of BPS statas recently shown in Ref.@21#—see also Refs.@22,23#. Thismodel was considered in Sec. III D, and the basic modegiven byw2(1/4)w4. Here, however, we consider the cas

WN5l̄w21

N11lw (N11) ~162!

with l and l̄ real parameters. This model presents theZNsymmetry. The vacuum states are singular points of theperpotential. They obeywN5l̄/l, and theN solutions are,using l̄5aNl with a real and positive,

w̄k5ae2p i [(k21)/N] , k51,2, . . . ,N. ~163!

The topological sectors connect pairs of adjacent vacua,in general the number of BPS sectors isN(N21)/2. Theirenergies or tensions are given in terms of the values ofsuperpotential in the singular points, that is

t i j(N)5uWN~ w̄ i !2WN~ w̄ j !u. ~164!

The tensions can be classified according to the integern, n51,2, . . . ,<@N/2#, where@N/2# stands for the greatest integer not greater thanN/2 itself. These tensions have thexplicit values

tn(N)5

Na(N11)

N11uluA2F12cosS 2p

n

ND G . ~165!

The integern identifies topological sectors described via tconnection between a given vacuum, and itsn-th neighbor:n51 for connections between first neighbors,n52 for sec-ond neighbors, and so forth. We can check that thesesions are ordered in the formtk

(N),tk11(N) , for k51, . . . ,

,@N/2#, and N>4. This order also shows that the direconnection between two vacuum states is always less egetic than any other possible connection. For instance,t2

(N)

,2t1(N) , N>4, and t3

(N),t2(N)1t1

(N),3t1(N) , N>6, and so

forth.More recently, in Ref.@24# we considered another mode

The model is defined by the potential

10501

itis

e-c

-de-t-

s-,

is

u-

nd

e

n-

er-

V~f,x!5l2f2S f229

4D1l2x2S x229

4D12l2f2x22l2f~f223x2!1

27

8l2.

~166!

It presents theZ3 symmetry, but it is described by a potentithat cannot be written in terms of some superpotential. Incase the key issue relies on finding a way out of supersmetry to show that the fusion of defects still occurs exothmically. In the supersymmetric case one can use interesproperties to make the reasoning naturally simple. Innonsupersymmetric case, however, we have found non-solutions that presents the interesting property of having gdient and potential portions of the energy evenly distributo compose the total energy of the defect. As we have shosuch a property is also shared by all the BPS solutions,so we can use it to present a new reasoning, that wowithin and beyond the context of supersymmetry. The rsoning follows from the topological current presented in E~30!. Here we have

r~x!5S f8~x!

x8~x!D . ~167!

Thusr tr52g(x), and for static solutions that presents equgradient and potential portions we getr tr5g(x)1p(x)5«(x), or better

t i j 5E2`

`

dxr i jt r i j . ~168!

We consider fusion of two defects in the model investigain Ref. @24#. The topological defects there obtained obg(x)5p(x). Thus, we can use

~r i j 1r jk! t~r i j 1r jk!5r i jt r i j 1r jk

t r jk1r i jt r jk1r jk

t r i j .~169!

Also, we take advantage of the symmetry of the model tothat if the pair (f,x) represents a solution in a given sectoall the other solutions can be obtained by rotating the sotion according to theZ3 symmetry. This means that we cawrite

S f i j8 ~x!

x i j8 ~x!D 5S cos~a i j ! sin~a i j !

2sin~a i j ! cos~a i j !D S f8~x!

x8~x!D . ~170!

Herea i j is the angle between the given solution (f,x) andthe solution (f i j ,x i j ). In the Z3 model under considerationa i j can only be 2p/3 and 4p/3. We now use Eqs.~168!,~169!, and ~170! to obtain t ik,t i j 1t jk . This result showsthat the three-junction in the model under considerationcurs exothermically.

9-12

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VI. NETWORKS OF DEFECTS

The recent Refs.@21–23# have introduced investigationconcerning defect junctions and networks in supersymmesystems, and now we turn our attention to this possibilWe consider junctions in the plane, which may give riseplanar networks of defects. To implement this possibilitywork in the (2,1) space-time, in the plane (x,y). We identifythe axes (x,y) with the axes (f,x) of the space of configurations. We illustrate this situation by considering the modefined by the potential in Eq.~166!. The topological solu-tions in the planar case are@24#

f (2,3)6 52

3

4, ~171!

x (2,3)6 56

3

4A3tanhSA27

8lyD , ~172!

and

f (1,2)6 5

3

8 F173 tanhS 1

2A27

8l~y1A3x! D G , ~173!

x (1,2)6 5

3

8A3F16 tanhS 1

2A27

8l~y1A3x! D G ,

~174!

and

f (1,3)6 5

3

8 F163 tanhS 1

2A27

8l~y2A3x! D G , ~175!

x (1,3)6 52

3

8A3F17 tanhS 1

2A27

8l~y2A3x! D G .

~176!

They may be used to represent the three-junction in thewall approximation.

Let us consider the first model, in the casel/m.0. Theminima are atv15(1,0), v25(0,Al/m), v35(21,0), andv45(0,2Al/m). The energies or tensions of the BPS anon-BPS sectors aretBPS

1 52 t̄ BPS1 5A(m/l)tnBPS

1 5(4/3)ulu.For a givenl, the value ofm allows obtaining three distinc

FIG. 1. A possible array of defects, as suggested by themodel of Sec. III A in the case ofl/m,1. The vacuum states arrepresented byv151, v252, v353, andv454, and the thinnerand thicker lines stand for BPS and non-BPS defects, respecti

10501

ic.

o

l

in

situations. The first concerns the casem.l. In this case theminima give rise to an array of defects of the form shownFig. 1, in the thin wall approximation.

The second case concernsm5l, and the third onem,l. In the thin wall approximation they are shown in Fig.and in Fig. 3, respectively. The second case seems to givintersection of four half-defects, but this is not true sincem5l the two fields decouple. Thus, the apparent interstion of four half-defects represents in fact the uninterestcase of two non-intersecting defects.

The first and third cases seem to be dual to each otsince they are linked by the interchangel↔m. This dualityis very much like thes↔t duality that the diagrams depictein Fig. 1 and in Fig. 3 remember. This scenario excludesself-dual case, wherem5l.

The basic diagrams of Fig. 1 and Fig. 3 can be usedgenerate planar networks of defects. We can generate awork in the two following ways. The s-network, which appears forl/m,1, and the t-network, forl/m.1. They aredepicted in Fig. 4 and in Fig. 5, in the case of regular heagonal networks in the thin wall approximation. In the fircase we impose that the triangle with verticesv1 ,v2 ,v4 isequilateral. This givesl/m51/3. In the second case we impose that the triangle with verticesv1 ,v2 ,v3 is equilateral. Itgivesl/m53, the result we should expect from duality. Wnotice that the first~second! imposition subdivides the distance between the horizontal~vertical! vacua into three equapieces, which is the condition for the formation of a netwoof regular hexagons.

st

ly.

FIG. 2. Another array of defects, suggested by the first modeSec. III A in the case ofl/m51.

FIG. 3. Another possible array of defects, as suggested byfirst model of Sec. III A in the case ofl/m.1.

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For m53l, we have two sides of the hexagon formednon-BPS states, with tensionA(1/3)tBPS

1 . The other foursides are formed by BPS states, with tension (1/2)tBPS

1 . Form5(1/3)l, two sides of the hexagon are formed by BPstates, with tensiontBPS

1 , and the other four sides are alsformed by BPS states, with tension (1/2)tBPS

1 .We think of a planar network of defects. We notice tha

t-network meansl/m.1, and this allows hexagonal neworks with two distinct BPS states, having two different tesions, tBPS

1 and t̄ BPS1 5(1/2)tBPS

1 , showing that each threejunction of this t-network is at the threshold of stability. Fl/m,1, in the case of s-networks, however, the situationdifferent since now we must have two non-BPS statesbuild the hexagonal cell. Since the ratiol/m,1 controls theenergy of the non-BPS state, every three-junction may othe inequalityt i j ,t jk1tki .

We can consider the case wheretnBPS1 5 t̄ BPS

1 . This givesm54l and the tensions degenerate to the single va(2/3)ulu. The hexagonal cell is no longer regular. The an

FIG. 4. The s-network, depicted as a regular hexagonal netwof defects in the case ofl/m51/3. The thicker lines represent nonBPS states.

FIG. 5. The t-network, depicted as a regular hexagonal netwof defects in the case ofl/m53.

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between the two BPS states changes froma5120 to a52u, whereu is slightly lesser than 63.5 degrees. We notthat we cannot make the tensions and angles degeneramultaneously, and this indicates instability of the suggesnetworks.

Effects that contribute to stabilize the network may appwhen we go beyond the classical level and introducequantum corrections, since the quantum contributions mmodify the classical scenario. For instance, in the casel/m.1, in a t-network all the defects are BPS defects, and thenergies or tensions are protected against receiving quancorrections. On the other hand, forl/m,1, in the case of as-network there are non-BPS states whose tensions areprotected and may receive quantum corrections, changthe classical valuetnBPS

1 , contributing to stabilize the network.

The first model does not allow the appearence of a nwork of planar squares, although squares can also fillplane. In our model this would require theZ4 symmetry,which is not effective in that model.

The fourth model leads to the model considered in Re@21,22#. In this case the three-junction is formed by BPdefects, and we can form an hexagonal array. Here,three-junction is stable, because each leg carries the stension. But it does not generate a BPS network, sincetwo adjacent three-junctions must have different windingaccomodate the three vacuum states in the network, andwould imply that each side of the hexagonal cell mustseen as BPS and anti-BPS, simultaneously, frustratingdestabilizing the network—see Refs.@21–23#.

We consider another model, investigated in Ref.@24#.This model is defined by no superpotential, and the argumabove is no longer valid. TheZ3 symmetry seems to allowthe presence of a network of defects, also in the formregular hexagonal array. In the thin wall approximation, thexagonal network that appears in this case is interesbecause the solutions are explicitly known, form straight-lsegments joining the vacuum states, and have the sameergy or tension, (9/4)A27/8ulu. The Z3 symmetry that ap-pears in the vacuum sectors is also effective in the sectorthe topological defects. The inequalityt ik<t i j 1t jk is strictlyvalid in this case, ensuring stability of the hexagonal nwork, as depicted in Fig. 6 in the thin wall approximation

VII. COMMENTS AND CONCLUSIONS

In the present work we have examined several modelcoupled scalar fields. We have found some explicit solutioof the BPS and non-BPS type. We have also investigalinear stability, checking the behavior of the bosonic maton the background of the BPS and non-BPS defects.BPS states are solutions that obey first-order differenequations, and the non-BPS states are not. The BPS sengender the property of having energy evenly distributedthe gradient and potential portions. However, in the systeof two coupled fields that we have introduced, the non-Bstates also engender the same feature, as we have expshown in this work. The point here is that the non-BPS staof the systems of coupled fields are BPS states of sim

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systems, which are projections of systems of coupled fieonto a single, specific field direction.

The first model is the model investigated in Sec. III A.presents several properties. In particular, it has a rich Bsector, the sector that connect the minima (1,0) and~-1,0! byone-component and two-component BPS states. Thecomponent state describe a family of solutions, parametrby the ratio between the coupling constants that specifysystem. These one-component and two-component statelow diverse applications, like the applications as bag modin Refs.@27#, as defects inside defects in Refs.@11,12,14,15#,and in applications to condensed matter, to polimeric chahaving two relevant degrees of freedom to representchain—see Refs.@4,37#.

In applications in the form of bag models, in additionthe models already introduced in Refs.@30,31# we have otherpossibilities, given by the first, second and third modelstroduced in Sec. III. These models differ from the modconsidered in Ref.@41#, and we believe that this new direction may shed further light on the issues explored in twork. In particular, it seems that our suggestions are easieexamine, and may offer an analytical understanding ofissues put forward in Ref.@41# under numerical investigations. We recall that the model investigated in@41# is defined

FIG. 6. The regular hexagonal network of non-BPS defegiven by theZ3 model defined by the potential of Eq.~166!.

t.

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by a potential that cannot be reduced to any of the modconsidered in the present work.

In application to defect inside defect, much work walready done in Refs.@11–15#. The natural extension is nowrelated to issues where one allows the presence of local smetries, as in the work on superconducting strings@6#. In thefirst model of Sec. III theZ23Z2 symmetry can be enlargeby enlarging each one of the two independent symmetrAs we have already commented on in Sec. III, we caninstance make thex field complex, in this case changing thZ23Z2 symmetry toZ23Ul(1), if one also includes theAbelian gauge field. This opens new possibilities, providifor instance a natural scenario to the Callan-Harvey eff@40#. Another possibility includes models that presentsZ23Z3 symmetry, which may provide scenarios for domawalls that nest planar networks of topological defects. Thissues are presently under consideration, and we hope tport on them in the near future.

We have also commented on junctions of defects, andsubsequent formation of networks of defects. In this caseneed to enlarge the spatial dimension fromd51 to at leastthe d52 case. In the planar case, we have exemplified hto get planar solutions from the unidimensional kinks of tmodel. Here we have considered three models, the firstthe fourth model of Sec. III, and another one, first considein Ref. @24#. These systems are all different: the first modadmits a supersymmetric extension of the typeN51, andseems to admit no stable network of defects, at least clacally; the fourth model presents junctions that break 1/4persymmetry, but it also seems to admit no stable networBPS defects. The other model does not present a supermetric extension, but the defect solutions get even contritions from the gradient and potential portions of the enerand so they behave very much like BPS states. This is crufor showing that the fusion of defects is a reaction thatcurs exothermically, and so the three-junction generatestable network of defects.

ACKNOWLEDGMENTS

We thank A. L. L. Guedes, A. S. Ina´cio, J. R. S. Nasci-mento, R. F. Ribeiro, and J. Zanelli for discussions. We athank CAPES, CNPq, and PRONEX for partial support.

s

s.

y

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Documents

Bags, junctions, and networks of BPS and non-BPS defects