Domain wall junctions in a generalized Wess-Zumino model

9 Pages·0.5 MB·English

by  D. Binosi

TPI-MINN-99/58-T
UMN-TH-1831/99
FTUV/99-81
IFIC/99-85
Domain wall junctions in a generalized Wess-Zumino model.
Daniele Binosi
Departamento de F`ısica Te´orica and IFIC, Centro Mixto, Universidad de Valencia-CSIC,
E-46100, Burjassot, Valencia, Spain
e-mail:[email protected]fic.uv.es
Tonnis ter Veldhuis
Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, 116 Church St,
Minneapolis, MN 55455, U.S.A.
0 e-mail:[email protected]
0 (December7, 1999)
0
2 We investigate domain wall junctions in a generalized Wess-Zumino model with a Z symmetry.
N
We present a method to identify the junctions that are potentially BPS saturated. We then use a
n
numerical simulation to show that those junctions indeed saturate the BPS bound for N = 4. In
a
J addition, we studythe decay of unstable non-BPS junctions.

8
1
I. INTRODUCTION. and (1/2,1/2) central charges appear in the supersym-

3 metry algebra. The junction tension is determined by a
v
In field theories with discrete, degenerate vacua, do- combination of the (1/2,1/2) and (1,0) central charges
1
8 mainwalls–fieldconfigurationsthatinterpolatebetween [6]. Although the two central charges contain (related)
0 different vacua – may occur. Supersymmetric theories, ambiguities,itisshownin[6,7]thatthe ambiguitiescan-
2 for which the vacuum energy is positive semi-definite, cel in physical quantities like the tension.
1 frequently have degenerate vacua. In many instances, Itis worthwhile to emphasizethat domainwallsin su-
9 multiplevacuaexistinwhichthevacuumenergyvanishes persymmetric models arenot necessarilyBPSsaturated.
9 and supersymmetry remains unbroken. The degenerate For example, in the model of [8] the BPS saturation of
/
h vacua may or may not be related by a spontaneously a domain wall depends on the values of the model pa-
-t broken discrete symmetry. rameters. In many models it is not possible to obtain
p Generalconsiderationsconcerning domainwalls in su- explicit analytical solutions to the first order BPS equa-
e persymmetric theories are presented in [1–3]. A lower tions. In those models the question whether solutions
h
boundonthe tension,the BPSbound, canbe calculated exist at all can still be answered. In [9] a necessary and
:
v without explicitly solving for the profile of the wall. Do- sufficient condition which indicates whether two vacua
Xi main walls break the translation symmetry in one direc- are connected by a PBS saturated domain wall was de-
tion, and supersymmetry is either completely broken, or veloped∗.
r
a 1/2 of it is preserved. In the latter case, a central exten- The issue of BPS saturationin junctions is more com-
sionappearsin the supersymmetry algebra. The tension plicated, because a similar condition does not exist. At
saturatestheBPSboundandisequaltothe(1,0)central this time, there is only one known model [10] in which
charge. a domain wall junction was found as an explicit ana-
In[4,5]itwasnotedthatmorecomplicatedfieldconfig- lytical solution to the BPS equations. In the absence
urationswithaxialgeometry,domainwalljunctions,may of analytical expression for the field configuration, al-
alsooccur. Theseconfigurations,ofthe“hubandspoke” ternative methods must be employedto study junctions.
type,areanaturalgeneralizationofdomainwalls,andin- Someinformationcanbeobtainedfromanalyticalresults
terpolate between more than two degenerate vacua. Far for specific limiting values of parameters or coordinates.
away from the center of the junction, the fields are ap- Complementary results can be obtained from numerical
proximatelyconstant–atthevacuumexpectationvalues simulation of the equations of motion.
of the various vacua – in sectors. These sectors are sep- For any domain wall junction, a BPS bound on the
aratedby “spokes”,where the field interpolates between junction tension can be calculated. In [7] the general
one vacuum to the next following approximate domain
wallprofiles. Inthecenterofthejunction,“thehub”,the
domain walls meet and the field configuration resembles
astring. Ifthese junctionsareBPSsaturated,1/4ofthe ∗It is not obvious that this method works also for theories
original supersymmetry is preserved and both the (1,0) with more than one chiral superfield.
1
method to obtainthis bound fromthe domainwalls sur- numerical simulations. The tension of the “basic” junc-
rounding the junction is presented. In this work, it was tion was also calculated, analytically to next to leading
also established that the junction tension is negative in orderin1/N,andnumericallyalsoforfinite valuesofN.
general, although “exotic” models in which domain wall Beside the “basic” junction, the vacua of the model
junctions have positive tension may exist. allowfor a multitude ofother types of junctions. As dis-
In this paper we will study domain walljunctions in a cussed further in the paper, we devise a general method
class of generalized Wess-Zumino (WZ) models with Z to identify all potential BPS saturated junction in this
N
symmetry. The Lagrangianis given by model. There is a very stringent (asymptotic) consis-
tency condition which allows only for a well defined set
= d2θ d2θ¯ (Φ,Φ†)+ d2θ (Φ)+h.c. , of intersection angles between the walls. The set of po-
L Z Z K (cid:26)Z W (cid:27) tential BPS junctions we identify contains the junctions
that appear in [15,16],where tilings of domains and net-
(1.1)
works of domain walls in this model are studied.
Weconsiderfourtypesofdomainwalljunctions inthe
where the K¨ahler potential has the canonical form
(Φ,Φ†)=ΦΦ†, and the superpotential is given by† model with N = 4, two of which belong to the class
K of potential BPS saturated, and two more that are not
1 BPS saturated. For each junction we perform a numer-
(Φ)=Φ ΦN+1. (1.2)
W − N +1 ical simulaton of the second order equations of motion
to obtain the junction profile. For the potential BPS
The action is invariant under the transformation junctions, we compare the energy of the junction to the
Φ(x,θ) e2πı/NΦ(x,θe−πı/N). Moreover,themodelhas BPS bound (which we evaluated from the domain walls
→
N physically equivalent vacua given by surrounding the junction by the methods in [7]) to de-
terminewhetherthey areindeedBPSsaturated. Forthe
φ=e2πık/N, k =0,1,...,N 1, (1.3) othertwojunctions,weinvestigatetheirstabilityandde-
−
cay. Althoughwe performour numericalcalculations for
where φis the scalarcomponentofthe superfieldΦ. De-
the value N =4, we expect our results to apply to other
spite its apparent simplicity, the model is of physical in-
values as well. Moreover, the methods we use are quite
terestbecauseitisrelatedtotheVeneziano-Yankielowicz
general, and can be applied to other models.
action [11], which is a low energy effective action for su-
The remainderof the paper is organizedas follows. In
persymmetric QCD. The parameter N is related to the
Sec. IA, we describe the possible domain walls of the
number of colors in the SU(N) gauge group.
model. TheninSec.IA1andIA2,wedescribeindetail
For N = 2, the model reduces to the renormalizable
the two types of domain walls which exist in the case
Wess-Zuminomodel. Itiswellknownthatthereisanan-
N =4. These wallsappearfar aawayfromthe centerof
alytical expression for the domain wall in this case (see
thejunctionsthatweconsider. InSec.IBwefirstdiscuss
[2], for example). In the large N limit, an analytical
the potentialBPSjunctions inthe models for genericN,
expression for the “basic” domain wall – the wall that
whichsatisfyaveryrestrictiveconsistencycondition. We
connects vacua for consecutive values of k – is presented
then focus on the N = 4 case and describe in detail the
in [12,13] to next to leading order in 1/N. In this ref-
two types of BPSjunctions (Sec.IB1 and IB2)and the
erence it is also noted that all vacua in the model are
non-BPS types (Sec. IB3) which occur. In Sec. IC we
connected by BPS saturated domain walls at finite N.
giveabriefoutline ofthe computationalschemeusedfor
The “basic” domain wall junction has also been stud-
obtaining the numerical representation of the junctions;
ied. Wedefinethe“basic”junctionasthefieldconfigura-
the paper ends with some comments in Sec. II.
tion that cycles through all N vacua, consecutively from
k =0 to k =N 1, counter-clockwisearoundthe center
−
ofthe junction. In[14]ananalyticalsolutiontothe BPS A. Domain walls.
equation for the “basic” junction in the N limit
→ ∞
was presented. In [7] it was confirmed that the “basic”
BPSsaturateddomainwallsaresolutionstothe equa-
junction is also BPS saturated for finite values of N by
tion
∂ φ=eıδ(1 φ¯N), (1.4)
x
−
† More generally we could consider the superpotential with the boundary condition that φ approaches vacuum
W(Φ) = AΦ−BΦN+1/(N +1), but for our purposes this values at x and x . Such solutions exist for
→−∞ →∞
is equivalent to Eq. (1.2) by rescaling the field and the coor- particularvaluesofthephaseδ,whicharedeterminedby
dinatesas φ→(A/B)1/Nφ and x →A(1−N)/NB−1/Nx . In the equation
µ µ
the literature, the large N limit of the model is studied with
A=N and B =1/NN−1. Im e−ıδ (φ(x )) =Im e−ıδ (φ(x + )) .
W →−∞ W → ∞
(cid:8) (cid:9) (cid:8) (cid:9)
(1.5)
2
For the model with N = 4, there are two types of TABLEI.TypeIwallswiththeircorrespondingvaluesofδ.
domain walls. The first type connects adjacent vacua,
for which k differs by one. The second type connects
oppositevacua,forwhichkdiffersbytwo. Wewilldiscuss 2. Type II domain walls.
each of these types in turn.
For definiteness, we focus on the wall connecting the
vacualabeledbyk =2andk =0. Forthiswall,thevalue
1. Type I domain walls. of δ is either 0 or π. For δ =0 (δ =π) the wall connects
the vacuum with k = 2 (k = 0) at x to the
→ −∞
Although it is not possible to find an analytical so- vacuumwithk=0(k =2)atx . Thewallispurely
→∞
lution for the domain wall profile, the characteristic fea- real. Again, we fix the integration constant so that the
turescanbereadilyuncovered. Fordefiniteness,wefocus wall is centered at the origin; with this choice, the wall
onthewallthatconnectsthevacualabeledbyk =0and is invariant under the transformation φ(x)= φ( x).
− −
k = 1. From Eq. (1.5) it follows that for this wall the The field vanishes at the origin, its derivative being
value of δ is either π/4 or 3π/4. For δ = π/4 (3π/4) (for δ =0)
− −
the wall connects the vacuum with k = 1 (k = 0) at
x=−∞ to the vacuum with k =0 (k =1) at x=∞. ∂xφ|x=0 =1, (1.9)
A general solution to the BPS equation contains one
while the wall’s tension in this case is
integration constant, indicating the freedom to shift the
center of the wall. We fix this constant by choosing the T(I) =2 (φ(x )) (φ(x )) = 16.
center of the wall to be at the origin: The wall is then 1 |W →∞ −W →−∞ | 5
invariant under the transformation φ(x) = ıφ¯( x). To
− (1.10)
studythewallclosetothecenter,itisusefultowritethe
fieldintermsofpolarvariables,φ=ρeıα. Thesymmetry Table II lists all domain walls of this type with their
of the wall dictates that α = π/4 and ∂xρ = 0 at the corresponding value of δ.
center. It then follows from the BPS equation that, for
δ = π/4, δ φ( ) φ( )
− −∞ ∞
1+ρ4 0 1 ı
∂ α = 0, (1.6) −
x |x=0 − ρ π/2 ı 1
0
π 1 ı
where ρ0 = ρ(0). The value of ρ0 can be calculated by 3π/2 −ı 1
evaluating the constant of motion along the wall, i.e. − −
Im e−ıδ (φ) at x = and at x = 0. As a conse-
TABLE II. Type II walls with their corresponding values
W ∞
que(cid:8)nce, ρ0 is t(cid:9)he sole real root of the equation of δ.
ρ5 2
ρ + 0 = √2. (1.7)
0
5 5
B. Junctions.
Numerically, ρ = 0.55514. The tension of the wall can
0
be calculated exactly, and depends only on the value of
BPS junctions are solutions to the equation
φ at the two ends of the wall; it is given by
8 eıδ
T1(I) =2|W(φ(x→∞))−W(φ(x→−∞))|= 5√2. ∂zφ= 2 (1−φ¯N). (1.11)
(1.8) Here z = x + ıy and ∂ = 1/2(∂ ı∂ ). Junctions
z x y
−
arestaticconfigurationsthathavethegeometryofahub
InTableIwepresentalldomainwallsoftypeIwiththeir
and spokes system. In the sectors between the spokes,
corresponding values for the phase δ.
the field φ approachesthe vacuum values; far away from
the hub, φ follows domain wall profiles from one sector
δ φ( ) φ( ) to the next on trajectories perpendicular to the spokes.
−∞ ∞
Each spoke is therefore associated with a domain wall.
3π/4 1 ı
− − The hub is the center of the junction where the domain
ı 1
− walls meet.
π/4 ı 1
− A few considerations are important to consider what
1 ı
− − junctions are possible, and which of those junctions are
π/4 1 ı
− BPS saturated.
ı 1
− First, the sumof the forcesexertedonthe junction by
3π/4 ı 1
− − thedomainwallsthatareattachedtoithastovanishfor
1 ı
a static configuration.
3
Second, the value of the phase δ in the BPS equation
for the junction merely determines the orientationofthe
junction. The effective value of the phase for the BPS
domain wall equation for a wall associated with a spoke a) b)
that makes an angle α with the yˆaxis is equal to δ α.
−
Once the direction of one spoke is fixed, for example to FIG. 1. Potential BPS saturated junctions for the case
point in the yˆ direction, the value of δ is fixed. For the N =4.
remainderofthespokes,onlyspecificvaluesofαarethen
possible. Thisrestrictiononthepossiblevaluesofαturns
out to be a very stringent consistency condition. In the
case of the model with N = 4, inspection of Tables I
and II shows that this condition allows only two types
of junctions as potential BPS candidates. We call these
twotypes A andB respectively,andwe investigatethem
below.
Once it has been established that a static junction ex-
ists, another interesting issue to determine is whether or
notitisstable. IfajunctionisBPSsaturated,itisguar-
anteed to be stable (also see the discussion in [7]). For
non-BPSjunctions,theissueisnotsoclear;ifthebound-
ary conditions also allow a BPS saturated configuration,
then the non-BPS junction is at most meta-stable.
One can actually devise a generalmethod for generat-
ingallthe potentialBPSsaturatedjunctionsinourclass
of models. In fact, given the presence of the N degener-
ate vacua (1.3), one has that for N even (odd) there are
N/2 ((N 1)/2) types of domain walls. These different
−
types of walls can be labeled by the integer l = k k
1 2
| − |
or l = N k k , whichever is smaller (here k and
1 2 1
−| − |
k indicate the vacua that are connected by the wall);
2
moreover the tension of a wall, labeled by l, is equal to
FIG. 2. Potential BPS saturated junctions for the case
N =6.
2N 2πl
T(l) = 2 2cos . (1.12)
N +1r − N
1. Type A junction.
Now, we can represent each type of wall by a vector
with length equal to its tension; thus a junction will be
Here we study the junction of the type shown in
represented by the vectors of the walls that it contains
Fig.1a). Farawayfromthecenterofthejunction,there
farawayfromtheorigin,withthedirectionofthevectors
are four sectors in which the field consecutively takes on
chosenso that they point in the direction where the cor-
the four possible vacuum expectation values. These sec-
responding wall is located. In a sense, the vectors play
tors are separatedby BPS domainwalls of type I, which
the role of the spokes in the hub and spoke system, but
were discussed before.
theyhaveadditionalinformationbecauseoftheirlength.
Near the center of the junction, the field can be ex-
Crossinga spokecounter clockwisemeans that the value
panded in a power series in z and z¯. Accordingly one
of k increases by l from one domain to the next along
has
the correspondingdomainwall: Thus, two junctions will
be equivalent if they can be mapped onto each other by ∞ ∞
inversions and rotations. φ(z,z¯)= a znz¯m, (1.13)
mn
All of the potential BPS saturated junctions can then nX=0mX=0
be obtained as follows. We start with the basic N-
junction, which consists of N vectors representing walls with amn some complex coefficients‡.
withl =1(typeIwalls)atanglesof2π/N. Thenweadd
any two neighboring vectorsin the diagramrepresenting
thejunction,repeatingthisprocessuntilallunequivalent
diagramshavebeengenerated. ForN =4thisprocedure ‡One can in principle allow also for negative n and m with
gives the type A and B junctions of above (Fig. 1). In the condition m > |n| (n > |m|) if n < 0 (m < 0). These
Fig. 2 all the potential BPS junctions are shown for the negativevaluesarehoweverexcludedwhenonesubstitutethe
case N =6. expansion (1.13) in thedifferential equation (1.11).
4
Thedifferentialequation(1.11)andtheboundarycon- it. In orderto determine whether those asymptotic solu-
ditions are invariant under the transformations (γ = tions can be connected consistently and thus to investi-
π/2+π/N) gatewhetherthejunctionisBPSsaturatedweperformed
−
a numerical simulation of the second order equations of
Z2 : φ(z,z¯) φ†(e−2ıγz¯,e2ıγz); (1.14) motions for the field φ, while imposing the domain walls
→
Z : φ(z,z¯) e−2ıγφ( e2ıγz, e−2ıγz¯). (1.15) astheboundaryconditionsatlargedistancefromtheori-
N
→− − − gin. In order to allow the field to relax to its minimum
ImposingtheZ symmetrytheexpansion(1.13)takes energyconfiguration,wealsointroducedadampingterm
N
the form (see Sec. IC for details). In Fig. 3 we show the modulus
of φ after the configuration has come to rest.
∞ ∞
φ(z,z¯)= b z1+Nk(zz¯)l+c z¯N(k+1)−1(zz¯)l ,
kl kl
Xk=0Xl=0h i
(1.16)
with b and c again complex coefficients. Imposing
kl kl
in addition the Z symmetry, we find the following con-
2 1
straints on these coefficients
(cid:26) ˝f˝ 2.5
0.5
b =( 1)Nke2ıγb† , (1.17) 1.3
kl − kl 0
ckl =(−1)N(k+1)e2ıγc†kl. (1.18) --11..33 0 y
00 -1.3
Thus, if N is even, one can introduce the real coeffi- xx 11..33
cients d and e , obtaining the expansion 22..55
kl kl
∞ ∞
φ(z,z¯)=eıγ d z1+Nk(zz¯)l+e z¯N(k+1)−1(zz¯)l .
kl kl
Xk=0Xl=0h i FIG.3. Modulus of φ for thetypeA junction.
(1.19)
InordertodeterminewhetherthejunctionisBPSsat-
urated, we will first calculate the BPS bound on the en-
If on the other hand N is odd, one has
ergy inside a square of “radius” R, with the center of
b =( 1)ke2ıγb† , (1.20) the junction located at the center of the square. Assum-
kl − kl ing the junctionto be BPSsaturatedthe energyinside a
ckl =(−1)(k+1)e2ıγc†kl, (1.21) square with radius R can be calculated entirely in terms
ofquantitiesrelatedtothedomainwallssurroundingthe
so that, accordingto the value of the index k,half ofthe
junction when the radius R is much larger than the size
coefficientsisreal,the otherhalfbeingpurelyimaginary.
ofthe junction. Explicitly,for largeR,the boundonthe
In the case N =4 one has γ = π/4, so that
energy takes the form
−
∞ ∞
E
φ(z,z¯)=e−iπ/4 d z1+4k(zz¯)l+e z¯3+4k(zz¯)l . BPS =T(A)+4T(I)R. (1.23)
kl kl L 2 1
kX=0Xl=0(cid:2) (cid:3)
Here L is a distance in the direction perpendicular to
(1.22)
the x-y plane. The second term in (1.23) is the contri-
We cannowcalculatethecoefficientsd ande using bution to the energy from the four type I domain walls
kl kl
the BPS equation; it turns out that all coefficients are connected to the junction; the first term represents in-
determinedexceptfortheoneswhichmultiplytermsthat stead the energy of the junction, which is equal to
containonlyz¯,thatis,thee . Thesecoefficientshaveto
k0 R≫1
be adjusted to obtain the correct domain walls far from T(A) = ~a d~l, (1.24)
the center of the junction§. 2 I ·
square
Closetotheorigin,theconfigurationlookslikeastring.
We have provedthen that there are sensible solutions to where
theBPSequationsbothneartheoriginandfarawayfrom
a =Im φ∂ φ¯ . (1.25)
i xi
(cid:8) (cid:9)
Now,eachwallonthecontourofthesquarecontributes
the sameamountto the tension. The BPSbound onthe
§Notethat theexpansion uptotermsof eleventhorderinz
junction tension is therefore equal to
and z¯contains only two adjustable parameters.
5
∞
T(A) =4 a(I)dx, (1.26)
2 Z
−∞
where a(I) = Im φ(I)∂ φ¯(I) and φ(I) is the profile of a
x
type I wall along(cid:8)the xˆ axis(cid:9). Using a numerical repre-
sentation of the type I domain wall, we determined that 1
T(A) = 2.8614. ˝f˝ 2.5
2 − 0.5
We also calculated the energy in a square with radius 1.3
0
R as a function of R directly from our numerical repre-
0 y
sentation of the junction using the equation --11..33
00 -1.3
E R xx 11..33
= ∂xφ¯∂xφ+∂yφ¯∂yφ+V dxdy, (1.27) 22..55
L Z
square(cid:8) (cid:9)
whereV =∂φW∂φ¯W¯ isthescalarpotential. InFig.4we FIG.5. Modulus of φ for thetypeB junction.
comparethisenergywiththeBPSbound. Itisclearthat
theenergyconvergestotheboundforlargeR(compared
To analyze whether this type of junction is BPS satu-
to the size of the junction) and therefore the junction
rated we follow a procedure similar to the one used for
does indeed saturate the BPS bound.
the type A junction. We firstcalculatethe boundonthe
EHRL energy in a square of radius R for large R, assuming the
20 junctiontobeBPSsaturated. Thisboundtakestheform
E
15 BPS =T(B)+2T(I)R+√2T(II)R ∆T. (1.28)
L 2 1 1 −
10 ThefirsttermrepresentstheBPSboundonthetension
ofthejunction. Thesecondandthethirdtermrepresent
5 the contribution to the energy from the two type I walls
and the type II wall (the factor of √2 appears because
R the spoke representingthe type II wallis in the diagonal
0.5 1 1.5 2 2.5
direction). Specialcaremustbetakenbecausethe spoke
representing the type II wall intersects the sides of the
FIG. 4. Energy inside a square of radius R as a function
squarediagonally,so that the energy contributedby two
of R (dots) compared to the BPS bound (solid line) for the
triangularareasneedstobesubtracted. Thisistheorigin
typeA junction.
of the fourth term, ∆T, which takes the form
∞
2. Type B junction. ∆T =4 ∂xφ(II)∂xφ¯(II)xdx, (1.29)
Z
0
The second type of junction that is potentially BPS where φ(II) is the profile of a type II wall along the xˆ
saturatedhastwospokescorrespondingtotypeIwallsat axis.
an angle of π/2. A third spoke associated with a type II The field φ on the type II wall connecting the k = 0
domain wall intersects each of the other spokes at an sector to the k =2 sector of this junction is purely real,
angleof3π/4. TheconfigurationissketchedinFig.1b). so that it does not contribute to the BPS bound on the
Thistypeofjunctionmaybeobtainedfromajunction tension of the junction. On the other hand the tension
of type A by pinching two neighboring spokes together receivesequalcontributionsfromeachofthetypeIwalls.
in a symmetric fashion, thereby eliminating one domain. The BPS bound on the tension of type B junctions is
Weperformedanumericalsimulationofthesecondorder therefore half of the tension of a type A junction
equations of motion, with the appropriate domain wall
1
profilesasboundaryconditions,andincludingadamping T(B) = T(A). (1.30)
2 2 2
term(seeagainSec.ICfordetails). InFig.5weshowthe
modulus of the field φ after the configuration has come We canfinally evaluate∆T usingthe numericalrepre-
to rest. sentation of the type II wall, finding in this way
1
∆T = T(A). (1.31)
−2 2
This isa curiousequality,as∆T is aquantity calculated
from a type II wall, and T(A) can be calculated from a
2
6
type I wall profile. We have no further comments on the
natureoftheequalityhere,butasaconsequencethetotal
energy in a square with large radius is equal for type A
and type B junctions.
We also calculated the energy in a square with radius
R as a function of R directly from our numerical rep- 1
resentation of the type B junction using Eq. (1.27). In ˝f˝ 2.5
0.5
Fig. 6 we compare this energy with the BPS bound. It 1.3
is clear that for this junction too, the energy in a square 0
0 y
convergestotheboundforlargeR(comparedtothe size --11..33
of the junction). The type B junctions therefore also 00 -1.3
saturate the BPS bound. xx 11..33
22..55
EHRL
20
FIG.8. Modulus of φ for thenon-BPS four-junction.
15
However, this junction has a negative mode. It is un-
10
stable againstlocal perturbations that break its symme-
try,anditdecaysintoaconfigurationwiththreedomains
5
separatedby two domain walls, as shownin Fig. 9. Two
domains with the same vacuum expectation value con-
R
0.5 1 1.5 2 2.5 nect, and the resulting two domain walls are pushed out
of the center.
FIG. 6. Energy inside a square of radius R as a function
of R (dots) compared to the BPS bound (solid line) for the
typeB junction.
3. Non-BPS junctions.
1
˝f˝ 2.5
HerewediscussstaticjunctionsthatarenotBPSsatu- 0.5
1.3
rated. For such junctions the values of δ associatedwith
0
the various spokes are not consistent. However, the to- 0 y
--11..33
tal force on the junction vanishes. Moreover, they are
00 -1.3
invariantunderanextendedsymmetrygroup,andthere-
xx 11..33
fore extremize the energy.
22..55
The first junction of this type that we discuss is dis-
played in Fig. 7 a). It consists out of four sectors sepa-
rated by spokes which intersect at π/2 angles. However,
FIG.9. Modulus of φ after the non-BPS four-junction has
in contrast to the BPS four-junction, the spokes are as- decayedintoaconfigurationwiththreedomainsseparatedby
sociated with type II walls. two walls.
The second type of non-BPS walls we discuss is the
eight-junction shown schematically in Fig. 7 b). Com-
paredtotheBPSfour-junction,thisjunctionhaswinding
a) b) number two. We againfound by numericalsimulationof
the second order equations of motion that a static con-
figuration of this type exists. We show the modulus of
FIG.7. Non-BPS saturated junctions in thecase N =4.
the field φ for this configuration in Fig. 10.
Wefoundbyanumericalsimulationofthesecondorder
equations of motion that a static configuration of this
type does indeed exist. In Fig. 8 we show the modulus
of φ for the final configuration at rest.
7
figuration. Atthesametime,this dampingstabilizesthe
forward predicting scheme.
Inordertogenerateunstablejunctions,westartedour
simulation with an initial configuration that was invari-
ant under the same symmetry transformations as the
1 boundary condition. As these symmetry are not broken
(cid:26) ˝f˝ 10 by the equations of motion, the final configurationis the
0.5
5.2 lowest energy configuration consistent with the symme-
0 tries. Suchaconfigurationmaybeunstableagainstlocal
0 y
--55..22 perturbations which break the symmetry.
00 -5.2
xx 55..22
1100 II. SUMMARY
In this paper we have studied in detail domain wall
FIG.10. Modulusofφforthenon-BPSjunctionwithwind-
junctions in a generalizedWess-Zumino model. We have
ing numbertwo.
presented a method to identify all potentially BPS sat-
urated junctions, and we have described a procedure to
Wealsoobservedthatthisjunctioncandecayintotwo
determinewhetherthesejunctionsindeedsatisfytheBPS
four-junctions with winding number one when a sym-
bound. We showed that in the case N = 4 (the lowest
metry breaking perturbation is introduced. The two re-
valueofN forwhichthereismorethanonepotentialBPS
sulting junctions repel each other, as shown in Fig. 11
junctions), these junctions are in fact BPS saturated.
We noticedthatthe eight-junctiononlydecayswhenthe
On the basis of our results, in conjunction with the
perturbationissufficientlylarge. Fornow,weleaveopen
results for the basic junction for generic value of N [7]
the question whether this indicates that the junction is
and the large N results for the basic junction [6], we
meta-stable, or that this behavior is an artifact of the
speculatethateverypotentialBPSsaturatedjunctionfor
finite size of the lattice we used.
any N in this model indeed saturates the BPS bound.
ACKNOWLEDGMENTS
Useful discussion with M. Shifman and V. Vento are
1 gratefully acknowledged. One of us (D.B.) would like
(cid:26) ˝f˝ 10 to acknowledgewarmhospitality extended to him at the
0.5
Theoretical Physics Institute of the University of Min-
5.2
0 nesota, where part of this work was realized. This work
0 y was supported in part by the Department of Energy un-
--55..22
00 -5.2 der Grant No. de-FG-94ER40823, and by Ministerio de
xx 55..22 Educaci´on y Cultura under Grant No. DGICYT-PB97-
1100 1227.
FIG. 11. Modulus of φ after the non-BPS eight-junction
has decayed into a configuration with two typeA junctions.
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9