Remarks on Topologial Models and
Frational Statistis
C.A.S. Almeida
UniversidadeFederaldoCeara-Departamentode Fsia
C.P.6030,60470-455, Fortaleza,CE,Brazil
Reeivedon22February,2001
OneofthemostintriguingaspetsofChern-Simons-typetopologialmodelsisthefrational
statis-tisofpointpartileswhihhasbeenshownessentialforourunderstandingofthefrational
quan-tumHalleets. Furthermorethese ideasare appliedtothe studyofhighT superondutivity.
WepresenthereanreentlyproposedmodelforfrationalspinwiththePauliterm. Ontheother
hand,inD=4spae-time,aShwarz-typetopologialgaugetheorywithantisymmetritensorgauge
eld, namelyB^F model, isreviewed. Antisymmetritensoreldsare onjetured asmediator
ofstringinteration. Adimensionalredutionofthepreviousmodelprovidesa(2+1)dimensional
topologialtheory,whihinvolvesa2-formBanda0-form. Somereentresultsonthismodelare
reported. Reently,therehavebeenthoughtsofgeneralizingunusualstatististoextendedobjets
inothersspae-timedimensions,andinpartiulartotheaseofstringsinfourdimensions. Inthis
ontext,disussionsaboutfrationalspinandantisymmetritensoreldarepresented.
I B ^F Models
Shwarz-type theories are purely topologial in the
sense that theirpartition funtionsare independentof
themetriandthattheonlyobservablesinthese
theo-riesare topologialinvariantsof theunderlying
spae-time manifoldM. Other observablesdesribe linking
andintersetionnumberofmanifoldsofanydimension.
Commonly alled BF systems, they are
harater-ized by a BRST-gauge xed quantum ation whih
dier from the lassial ation only by a
BRST-ommutator whih ontains the whole metri
depen-deneofthequantumation. Ontheotherhand,sine
the vaumexpetation value ofa BRST-ommutator
vanishes,theseeldtheoriesmaybeobtainedfromthe
lassialations[1℄. Furthemore,ifwedenoteasQthe
BRST-operatorwhihisnilpotent,inthesetheoriesthe
energy-momentum tensorisQtrivial,i.e.,
T
=fQ;
g (1)
where
representseldsandthemetri.
Conneted to BF systems, it is worth mentioning
that antisymmetri tensor elds theories have been
studied during the past years. They play an
impor-tantrole in the realization of thevarious strong-weak
oupling dualitiesamong stringtheories. An
antisym-metritensorofrankp 1ouplesnaturallytoan
ele-mentaryextendedobjetofdimensionp 2,namelya
(p 2)brane.
As an example of an abelian BF system onsider
the following metri independent ation on an
D-dimensionalmanifoldM.
S(D;p)= Z
M B
p ^dA
D p 1
; (2)
whereAandB areforms,pdenotingtheirrank,^
de-notingtheirwedgeprodutanddistheexterior
deriva-tive.
InpartiulartheabelianB^F four-dimensional
a-tionis
S
BF =
Z
M
4
fB^Fg: (3)
B =B
dx
^dx
;F =dA;A=A
dx
: (4)
This ationis formulatedin termsof thetwo-form
potential B while F = dA is the eld-strength of a
one-formgaugepotentialA.
Appliations:
Field theories desribing the low-energy limit of
fundamental string theories typially ontain
higher-ranktensorelds.
ThetopologialontributionomingfromBF
the-ories appear even in those physial theories with non
trivial physial Hamiltonian where the BF term
ap-pearsasaninterationterm.
Coloronnementmodels.
Axioniosmistrings.
QCDstrings.
Topologiallymassivemodels.
II Gauge invariant massive B^ F
model in D = 4.
Our starting point is an abelian gauge theory whih
ontainsthevetoreldAandtheantisymmetrield
B,andinorporatedthetopologialtermB^F inthe
four-dimensionalation[2℄
S
BF =
Z
M
4
1
2 H^
H 1
2 F^
F+kB^F
: (5)
Here H = dB is the eld-strength of a two-form
gaugepotentialB,kisamassparameter,andisthe
Hodge star(duality)operator. Theationaboveis
in-variantunderthefollowingtransformations:
ÆA=d;ÆB=d; (6)
where and are zero and one-form transformation
parametersrespetively,andgivestheequationsof
mo-tion
d
H =F (7)
and
d
F=H: (8)
Applyingd
onbothsidesof eq. (8)andusingthe
eq. (7),weget
(d
d
+ 2
)F =0: (9)
Repeatingtheproedureaboveinreverseorder,we
ob-taintheequationofmotionforH
(d
d
+ 2
)H =0: (10)
Theseequationsanberewrittenas
(+ 2
)F =0 (11)
and
(+ 2
)H=0: (12)
III Abelian gauge invariant
massive models in D = 3
Dimensional redution !B^' models.
Dimensional redution is usually done by
expand-ing the elds in normal modes orresponding to the
ompatiedextradimensions,andintegratingoutthe
extradimensions. Thisapproahis veryusefulindual
models and superstrings. Here,however,weonly
on-sidertheeldsinhigherdimensionstobeindependent
oftheextradimensions.
Inthisase,weassumethat oureldsare
indepen-dentoftheextraoordinatex
3
:From(3),on
perform-ingdimensionalredutionasdesribedabove,wegetin
threedimensions
S = Z
M
3
fB^d+V ^Fg; (13)
whereV andarea1-formanda0-formelds
respe-tively.
WereognizethatB^distopologialinthesense
that thereis noexpliitdependene onthespae-time
metri. Onehastostressthatthistermmaynotbe
on-fused with the two-dimensional versionof the B^F,
whih involves a salar and a one-form elds.
More-over,atermthat isequivalentto thefour-dimensional
B^Ftermispresentination(13)(theso-alledmixed
Chern-Simonsterm,V ^F).
Non-Chern-Simonsgaugeinvariantmassive
modelsin D=3:
Now, in order to show thetopologial mass
gener-ation for the vetorand tensor elds, we onsider the
modelwiththetopologialtermB^d,andwith
prop-agationforthetwo-formgaugepotentialBandforthe
zero-formeld, representedbytheation
S= Z
M3
1
2 H^
H+ 1
2 d^
d+B^d
; (14)
where the seond term is a Klein-Gordon term, is
a mass parameter and H = dB is a three-form
eld-strengthofB.
The ation above is invariant under the following
where and are zero and one-form transformation
parametersrespetively.
We follow herethe samesteps that has beenused
by Allen et al. [2℄ in order to show the
topologi-al mass generation in the ontext of B ^F model.
Thus, we nd the equations of motion for salar and
tensor elds, whih are respetively d
H = d and
d
d = H: Consequently, we obtain the equations
(d d + 2
)d=0and(d
d
+ 2
)H=0:
Theseequationsanberewrittenas
(+ 2
)
=0 (16)
and
(+ 2
)H =0: (17)
Therefore,theutuationsofandH aremassive.
Obviously,thesetwopossibilitiesannotours
simul-taneously. Indeed, in the most interesting ase, the
degreeoffreedom ofthemasslesseld is"eatenup"
bythegaugeeldB tobeomemassiveandtheeld
ompletelydeouples fromthetheory[3℄.
IV N = 1 D = 4 massive B ^ F
! N = 2 D = 3 massive B^'
models
N=1 D=4massiveB^F model.
Let us begin by introduing the N = 1 D = 4
supersymmetri BFextendedmodel. For extendedwe
meanthatweinludemasstermsfortheKalb-Ramond
eld. Thismasstermwill beintrodued herefor later
omparison to the tridimensional ase. Atually, this
onstrution an be seen as a superspaeand abelian
versionof the soalled BF-Yang-Mills models. These
modelsaredesribedbytheation
S
BF YM =
Z
M4 Tr
kB^F + g 2 4 B^ B : (18)
Note that, on-shell, (18)is equivalent to thestandard
YM ation. This formalismwasused by Fuito et al.
[4℄in ordertostudyquarkonnement.
Asourbasisupereldationwetake[5℄
S SS BF = 1 8 Z d 4 xf i[ Z d 2 B W Z d 2 B _ W _ ℄ + g 2 2 [ Z d 2 B B + Z d 2 B _ B _
℄g: (19)
where W
is aspinor supereld-strenght,B
is a
hi-ral spinor supereld,
D B =0, and g are massive
parameters. Theirorresponding-expansionsare:
W
(x;;
) = 4i
(x) [4Æ
D(x)+2i( ) F (x)℄ +4 2 _ _ (20) B
(x;;)=e i [i (x)+ T (x)+ (x)℄, (21) where T =T () +T [℄ = 4i( ) B +2"
(M+iN):
(22)
Our onventions for supersymmetri ovariant
derivativesare D +i _ _ D _ _ i _ . (23)
We all attention for the eletromagneti
eld-strenghtand the antisymmetri gaugeeld whih are
ontainedin W
andB
,respetively. Intermsofthe
omponentselds,theation(19)anbereadas
S= Z d 4 xf[ i 2 + 2 B e F DN℄ + 2 _ _ + _ ( ) _ +g 2 [ 1 8 + + 1 2 B B 1 2 M 2 +N 2 ℄g = Z d 4 x[( i 2 5 + 2 + 2 B e F DN) +g 2 ( 1 8 + 1 2 B B 1 2 M 2 +N 2
)℄:(24)
Inthelastequalityabove,thefermionieldshavebeen
organizedasfour-omponent Majoranaspinors as
fol-lows = _ ; = _ ; = _ ; (25)
and we denote the dual eld-strenght dening e F 1 2 " F
. Furthermore,weusethefollowing
identi-ties = + 5 = = + . (26)
Wehavenotonsideredouplingwithmattereldsand
apropagationterm forthegaugeelds. Onthe other
hand,oursupespaeBFtermwasonstrutedinavery
simple way. A quite similar onstrution was
Theo-diagonal massterm(or
5
) hasbeen
shown by Brooks and Gates, Jr. [7℄ in the ontext of
super-Yang-Mills theory. Notethat theidentity
5 = i 2 " (27)
revealsaonnetionbetweenthetopologialbehaviour
denoted by the Levi-Civita tensor "
; and the
pseudo-esalar
5 :
So, it is worthwhile to mentionthat this term has
topologial origin and it an be seen as a fermioni
ounterpart of the BF term. In our opinion, this
fermionimasstermdeservesmoreattention.
The N=2 D=3massiveB^'model
Wewill now arry outa dimensional redution in
thebosonisetorof(24). Hene,afterdimensional
re-dution, the bosoni setorof (24) an be written as
[5℄ S bos: = Z d 3 xf[" V F +" B
' DN℄
+g 2 [ 1 2 B B V V 1 2 M 2 +N 2
℄g;(28)
where V
is a vetorial eld and ' represents a real
salareld. Notiethat thersttermin r.h.s. of (28)
an be transformed in the Chern-Simons term if we
identify V
A
. The seond one is the so alled
B^'term.
Nowletusproeedto thedimensional redutionof
thefermionisetorofthemodel. First,note thatthe
Lorentz group in three dimensions is SL(2;R ) rather
than SL(2;C) in D = 4. Therefore, Weyl spinors
withfourdegreesoffreedomwillbemappedintoDira
spinors. Sotheorretassoiationskeepingthedegrees
offreedomareskethedas
= _ ! = i = _ ! = i = _ ! = i : (29)
From(29),wendthat
! 1 2 ( + + + ) ! 1 2 ( + b b + b b + ) 5 ! 1 ( + +
+ ): (30)
where hatted index means three-dimensional
spae-time.
Thus,thedimensionallyreduedfermionisetorof
(24)maybewritten
S ferm: = Z d 3 xf i 4 ( + + + )+ 4 ( + b b + b b + )+ g 2 16 ( + + +
)g.(31)
TheationS=S
bos: +S
ferm:
isinvariantunderthe
following supersymmetry transformations (from now
on,greekindiesmeanthree-dimensionalspae-time):
Æ = iD ( ) F Æ = iD ( ) F ÆF = i ( ) i ( ) ÆD = ( + ) (32) Æ( i
) = Æ
=i e T e T ; Æ e T = + ; Æ( i
) = Æ
= i ( ) T ( ) T , (33)
where and are supersymmetri parameters, whih
indiates that we have two supersymmetries in the
aforementionedation.
V Frational statistis - anyons
Thefrationalstatistis[8℄withitstheoretialand
ap-pliableonsequenesplaysaninterestinginterplayrole
between quantum eld theory and ondensed matter
physis. Previous speulations [9℄ that the frational
quantum Hall eet ould be explained by
quasipar-tiles (anyons) obeying frational statistis were
on-rmedandthebehaviouroftwo-dimensionalmaterials
suh asvortiesin superuid heliumlms may be
ex-plained by frational statistis. As it is known, the
presene ofChern-Simons termsin (2+1) dimensional
gaugetheoriesinduefrationalstatistis. Insuh
the-ories, it has been known that there exist exitations,
alled anyons, whih ontinously interpolate between
bosons and fermions. In the well-known physial
re-alization, anyons are omposite quasi-partiles where
magnetiux-tubesareattahedtohargedpartiles.
Reently, there have been thoughts of
stringsin four dimensions[10℄. AbelianBF models in
four dimensionshas also been exploited in dual
mod-els of osmi strings, and axioni blak hole theories
wheretheaxionhargeisphysiallydetetableonlyby
externalosmistringsinafourdimensional
Aharanov-Bohmtypeproess[11℄.
Anezirisetal. [12℄showedthatmoregeneral
statis-tis an exist in (3+1) dimensions. Statistial phases
of BF theoryan beseen toarise from ertainosmi
string and superstring phenomena, as well as in the
Nambu-Gotostringtheorymodiedwiththeinlusion
of theKalb-Ramondeld(Beld) [13℄.
VI Linking number -
interse-tion number
Inareentinterestingwork,AshtekarandCorihi[14℄
showed that there is a preise in whih the
Heisen-berg unertainty between uxes of eletri and
mag-netieldsthroughnitesurfaesisgivenbytheGauss
linking numberoftheloopsthat bound thesesurfaes.
Topologialeldtheoriespresentsobservablesother
thanthepartitionfuntion. Wittenhasarguedthatin
thesetheoriesWilsonloopsareappropriatemetri
inde-pendentandgaugeinvariantobjets. Polyakovhas
re-latedthevauumexpetationvaluesofWilsonloopsin
theabelianChern-SimonstheorytothelassialGauss
linking numberoftwoloops.
IntheaseofBFsystems,weanreinterpretingthe
linking numberastheintersetion numberofoneloop
withadisboundedbytheotherloop. So,this
observ-able hasanatural generalizationto other dimensions.
Considering theation (2), the elds B
p and A
D p 1
allowus toform thefollowingmetriindependentand
gaugeinvariantexpressions("Wilsonsurfaes"):
W[L℄=exp Z
L A
;W[℄=exp Z
B
(34)
where and L are disjointompat and orientedp
and (D p 1)-dimensional boundaries of two
ori-entedsubmanifold of anD-dimensional oriented
man-ifold M: This formalism was presented by Blau and
Thompson[15℄,whoprovedthattheexpetationvalue
W(;L)=hW
B ()W
A
(L)iisequaltothelinking
num-berofthe"surfaes".
VII Frational statistis in D=3
from B ^' term?
S =S
0 +
Z
d 3
x
2 "
B
'+
g
2 J
B
+hj'
;
(35)
where g;h are oupling onstants, J
and j are
ur-rentsandsoures. S
0
dependsonlyoneldsthat
orig-inate urrents and soures. Integrating out the elds
B
and',wearriveat
S
eff =S
0 hg
4 Z Z
d 3
xd 3
yJ
(x)h B
(x)'(y)ij(y):
(36)
From(35) andusing theLandau gauge, is easy to
seethat
hB
(x)'(x)i="
x
G(x y); (37)
where
G(x y)= 1
4 1
jx yj
; (38)
Therefore
hB
(x)'(x)i= "
4
(x y)
jx yj 3
; (39)
The orrelation funtion hB
(x)'(y)i is
tanta-mounttotheorrelationfuntion hA
(x)A
(y)iofthe
pureChern-SimonstheoryintheLandaugauge
(trans-verse propagator). The eetive ation (36) an be
rewrittenas
S
eff =S
0 hg
4 1
4 "
Z Z
d 3
xd 3
yJ
(x) (x y)
jx yj 3
j(y)
(40)
and
S
eff =S
0 hg
4
(linking number ) (41)
On the other hand, Blau and Thompson [15℄
sug-gest appliation of their formalism to the ase where
B isazero-formandAisaone-form,involvinga
link-ing number of a point P and a irle , through the
expression
W
B (P)W
A
(d)=exp(B(P)+ I
A) (42)
whereadisdisboundedby.
These results support our speulation that wean
VIII Pauli's term and frational
statistis in D=3.
As it is known, the presene of Chern-Simons terms
in (2+1) dimensional gauge theories indue frational
statistis[16, 17℄. Stern [18℄ was the rst, as far as
weknow,tosuggestanonminimaltermintheontext
oftheMaxwell-Chern-Simonseletrodynamiswiththe
intention of mimiking ananyonibehaviorwithout a
pureChern-Simonslimit. Thistermanbeinterpreted
asatreelevelPauli-typeoupling,i. e.,ananomalous
magnetimoment. It isaspei featureof (2+1)
di-mensions that the Pauli oupling exists, not only for
spinningpartiles,butalsoforsalarones[19℄.
We onsider here an Abelian Chern-Simons-Higgs
theorywheretheomplexsalareldsouplesdiretly
to the eletromagneti eld strength (Pauli-type
ou-pling). The Lagrangianof the model under
investiga-tionis
L=jr
j
2
+
2 "
A
A
A
b+
2 b
2
(43)
where r
(
ieA
i
g
4 "
F
). Note that
this ovariantderivativeinludesboththeusual
mini-malouplingandtheontributionduetoPauli'sterm.
Here A
is the gauge eld and the Levi-Civita
sym-bol"
isxedby"
012
=1andg
=diag(1; 1; 1).
Themultipliereldbhasbeenintroduedtoimplement
theovariantgauge-xingondition.
Beforequantizingthetheory,weanalyzetheabove
LagrangianintermsofHamiltonianmethods. Herewe
followtheapproahusedbyShinetal. [20℄. Wearry
out the onstraint analysis of this model, in order to
obtainaonsistentformulationofthetheory.
The anonial momenta of the Lagrangian (43),
whih an be easily seen by onsidering its temporal
andspatialomponentsseparately,aregivenby
0
=0; (44)
b = A
0
; (45)
j
=
2 "
ij
A
i i
g
2 "
ij
[
(D
i
) (D
i )
℄
g 2
2
j
A
0 jj
2
+ g
2
4 (
0 A
0 )jj
2
; (46)
=(
0
)+ieA
0
+i g
4
" ij
F
ij
; (47)
=(
0
) ieA
0
i
g
" ij
F
ij
; (48)
where
0 ,
j
;
b
; and
aretheanonialmomenta
onjugatetoA
0 ; A
j
,b;and
respetively. Alsowe
haveused"
ij ="
0ij ,D
i =
i ieA
i
andi;j=1;2.
The anonial momenta (44) and (45) do not
in-volveexpliittime dependene and heneareprimary
onstraints. Performing the Legendre transformation,
theanonialHamiltoniananbewritten as
H
=
+jDj 2
+ieA
0
(
)+" ij
A
0
i A
j
+A
i
i
b
2 b
2
i g
2 "
ij
j A
0 [
(D
i
) (D
i )
℄
g 2
4
i A
0
i
A
0 jj
2 g
4 "
ij
F
ij [
(D
0
) (D
0 )
℄
g 2
8 F
ij
F
ij jj
2
:(49)
Now,inordertoimplementtheprimaryonstraints
inthetheory,weonstruttheprimaryHamiltonianas
H
p =H
+
0 +
1 (
b +A
0
); (50)
where
0 and
1
are Lagrange multiplier elds.
Con-serving in timetheprimary onstraintsyieldsthe
se-ondaryonstraints
1 =
0
0; (51)
2 =
b +A
0
0; (52)
whih are also onserved in time and where the
sym-bolindiatesweakequality,i. e.,theonstraintsan
be identially set equal to zero only after omputing
the relevant Poisson brakets. Thus there is no more
onstraintandtheaboveequationsaretheset offully
seond-lass onstraints. On the other hand, there is
norst-lassonditionsandso,nogaugeonditionsto
bedetermined intheory. Thisisaneetofthegauge
xing ondition imposed previously. As it is known,
thelakofphysialsignianeallowsthatthe
seond-lass onstraintsanbeeliminated by meansof Dira
brakets(DB's).
Following the standard Dira brakets formalism
andquantizingthesystem,weobtainthefollowingset
ofnon-vanishingequal-timeommutators:
[A
0
(x);b(y)℄=iÆ 2
(x y) (53)
[A
i (x);
j
(y)℄=iÆ
ij Æ
2
(x y) (54)
[(x);(y)℄=[
(x);
(y)℄=iÆ 2
Afterahievingthequantizationweproeedto
on-strut the angular momentum operator and ompute
theangularmomentum ofthemattereld .
Thesymmetrienergy-momentumtensoranbe
ob-tainedbyouplingtheeldstogravityandthenvarying
theationwithrespettog
:
T
=
2
p
g ÆS
Æg
=(r
)
(r
)+(r
)
(r
)
A
b A
b
g
(jr
j
2
A
b):(56)
The angularmomentum operator in (2+1)
dimen-sionsisgivenby
L= Z
d 2
x" ij
x
i T
0j :
Hene
L= Z
d 2
x" ij
x
i f(
j +
j
) ieA
j J
0
i g
2 "
jl F
l0
(
) A
0
j b
+A
j
0 b i
g
2 A
j "
k l
k [
(D
l
) (D
l )
℄
+i g
2
2 A
j
k (jj
2
F 0k
)g; (57)
where
J
0
=if
g
2e "
ij
i [
(D
j
) (D
j )
℄
+i g
2
2e
i (jj
2
F 0i
)g (58)
isthetemporalomponentoftheonservedmatter
ur-rent.
The key point here is that Gauss' law is no more
a onstraint, while J
0 and T
ontain derivatives of
A
. Note that, due to its topologial harater, the
Chern-Simonstermdoesnotontributetothe
energy-momentumtensor. Theseaspetsareattributedtothe
non-linearityintroduedbyPauli'sterm.
Therotationalpropertyoftheeldisobtainedby
omputing theommutator [L;(y)℄. Using equations
(53-55)and(57),itiseasytoseethat
[L;(y)℄=" ij
y
i
j [e
Z
d 2
x" ij
x
i A
j J
0 ;℄
+i g
2 "
ij
"
jk y
i F
k 0
: (59)
Thisommutatoran berewrittenbymeansofthe
Q= Z
d 2
xJ
0 (x)
andbeomes
[L;(y)℄=" ij
y
i
j
e 2
4 [Q
2
;(y)℄+i g
2 "
ij
"
jk y
i F
k 0
(60)
or,inmorefamiliarnotation
[L;(y)℄=i(yr)(y) e
2
2
Q(y)+i g
2
yE(y):
(61)
The rst term in the right hand side of eq. (61)
representsthe intrinsi spinand the seond is the
so-alled rotationalanomaly, whih is responsible forthe
frationalspin. UnliketheChern-Simonsterm(whose
ontribution is related with magneti eld), the Pauli
terminduesananomalousontributionforthespinof
the system, whih depends on eletri eld [21℄. We
stressthat,herethenonminimalouplingonstantisa
freeparameter.
Itisworthmentioningthatalltheproedureabove
anbearriedoutevenifthereisnoChern-Simonsterm
intheLagrangian(43). Inthisasetheanomalous
on-tributiontospinwouldjust omefromthePauliterm.
Now we will disuss the above result in
onne-tionwiththeoriesinthebroken-symmetryphase.
Boy-anovsky[22℄hasfoundthatthelow-lyingexitationsof
aU(1)Chern-Simonstheoryininterationwitha
om-plexsalareldinabrokensymmetrystatearemassive
bosonswithanonialstatistis. Heexplainedhisresult
asduetothesreeningoflong-rangeforesinabroken
symmetryphase. Inthis phaseloalizedharge
distri-butionsannot besupported, whih is supposed tobe
essentialforfrational spin. Onthe other hand,if we
onsideranon-minimallyoupledAbelian-Higgsmodel,
thelong-distanedampingeetbythe"photon"mass
no longer exists. This is an indiation that Pauli's
term,whihinduesananomalousspin,anberelevant
forthestudyofbrokensymmetrystates(superuid)in
theontextofeetivetheoriesinondensedmatter.
Innonrelativistilimit,CarringtonandKunstatter
[23℄haveshownthat anomalous magnetimoment
in-terations gives riseto both the Aharonov-Bohm and
Aharonov-Casher eets. They have speulated
pos-sible anomalous statistiswithout the CS term. As a
matteroffat,webelievethatthis(inarelativisti
the-ory)wasprovedhere. Ontheotherhand,theAbelian
Chern-Simons term an be generated by means of a
spontaneoussymmetry breakingof anonminimal
Pauli-termat tree-level(withthe nonminimaloupling
on-stantgasafreeparameter)anonstituteaneetive
theorywhihbringusinformationaboutphysial
mod-elsinbrokensymmetryphase.
Aknowledgments
I would like to thank my ollaborators R. R.
Landim, D. M. Medeiros, F. A. S. Nobre and M. A.
M.Gomeswhihontributedformanyresultsreported
here. This work was supported in part by Conselho
NaionaldeDesenvolvimentoCientoeTenol
ogio-CNPqand Funda~aoCearensedeAmparoa
Pesquisa-FUNCAP.
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