Nonommutative Supersymmetri Field Theories
Vitor O.Rivelles
Instituto deFsia,UniversidadedeS~aoPaulo
CaixaPostal66318, 05315-970,S~aoPaulo-SP, Brazil
E-mail: rivellesfma.if.usp.br
Reeivedon19February,2001
Wedisusssomepropertiesofnonommutativesupersymmetrieldtheorieswhihdonotinvolve
gaugeelds. Weonentrateontherenormalizabilityissueofthesetheories.
I Introdution
Although stringtheory isquitewellunderstoodin the
perturbativeregimeitsformulationinabakground
in-dependent way is almost unknown. There are many
reasons for that. String theory has toomany degrees
of freedom. It is quite diÆult to handle all of them
together. It alsoinludes thegravitationaleld whih
may havequantum utuations. And there are many
souresfornonloalitywhihisalsotroublesomeinany
theory. Oneway outof thesediÆulties is toonsider
limitsofstringtheorywhihhavesomeofthetroubles
raisedabovebutnotallofthem. Thismayallowusto
understand bettersomeaspetsof stringtheory
with-outtheompliationsofthefull theory.
Onesuh alimitis the zeroslopelimitof the
D3-braneinthepreseneofaonstantNS-NSeld[1℄. The
lowenergyeetivetheoryisaquantumeldtheory
de-formed intermsoftheMoyalprodutoverspae-time.
In nonommutativeeld theoriestheusual produtof
elds isreplaed bytheMoyalprodutof eldsgiving
risetononloaleldtheories[2℄. Usuallynonloaleld
theoriesturn outto benotwelldened butthe
nonlo-alityinduedbytheMoyalprodutisstilltratable. It
wasfoundthatthemainharateristiof
nonommuta-tiveeldtheoriesisthemixingofultraviolet(UV)and
infrared (IR)divergenes due toits nonloalstruture
[3℄. As aonsequene it is notlear that the
proper-ties of the usual ommutative eld theories are kept,
without modiations,in theirnonommutative
oun-terparts. Thisgaverisetoanintensiveresearhof
non-ommutative eld theories in Eulidean orMinkowski
spae-time.
Oneof themanifestationsof theUV/IR mixing in
the 4
theoryisasaninfraredquadratisingularityin
the propagator at one loop [3℄. Although
renormaliz-ableupto twoloops[4℄itbeomesnon-renormalizable
at higher loop orders. Models involving a omplex
salareldmaybenon-renormalizableevenatoneloop
harateristi of ommutativeeld theories, i.e., their
renormalizability.
Inwhat followswewill disuss theinlusion of
su-persymmetry in suh models and how it restores the
renormalizability. We will onentrate on the
Wess-Zumino model in 3+1dimensions [6℄ and the
super-symmetrinon-linearsigmamodelin2+1dimensions
[7℄. Inthislast asewewillseethat the
nonommuta-tivityalsodestroysthemehanismfordynamialmass
generation of the fermioni setor, and we will show
howsupersymmetryhelpstox it.
II Nonommutative Spaes
Inquantummehaniswehavetheusualommutation
relations
^ q i
;p^ j
= i~g ij
; (1)
^ q i
;q^ j
=
^ p i
;p^ j
=0: (2)
It is natural to onsider nonommutative oordinates
withommutationrelations
^ q i
;q^ j
=i ij
; (3)
where
ij
is aonstantof dimensionL 2
whih denes
anonommutativity sale. This breaks rotational (or
Lorentz)symmetrybutinthelimit!0thesymmetry
isreovered. This isanexampleof anonommutative
spae. It an be extended to spae-time but we will
onsider nonommutativity only in thespatial
oordi-natessineotherwisethereareproblemswithunitarity
[8℄.
We an understand heuristially how the UV and
IR physis gets mixed. From Eq.(1) it follows that
^q i
p^ j
ig ij
. In a similar way, from Eq.(3) it
fol-lowsthatq^ i
^q j
i ij
so weexpet thatq^^p.
Thismeansthathighenergymodeshavedrastieets
itselfalreadyatonelooplevelinthepropagatorofthe
elds.
Fields dened on suh spaes are operator valued
objets. It turns out to be more onvenient to use
elds whih are not operator valued objets but just
funtions. Thisanbeahievedthroughtheuseofthe
Weyl-Moyalorrespondene[2℄
^
(^q)!(x): (4)
We assoiate to the operator valued eld ^
(^q) a
las-sial funtion (x)through itsFouriertransform ~
(p)
as
^
(^q)= Z
dpe ipq^
~
(p): (5)
Theoperatorvaluedeld ^
satises
^
1 (^q)
^
2 (^q)=
Z
dp
1 dp
2 e
i(p
1 +p
2 )^q
1
2 p
1 p
2
1 (p
1 )
2 (p
2
); (6)
hene
^
1 (^q)
^
2
(^q)$(
1 ?
2
)(x); (7)
where
(
1 ?
2 )(x)
h
e i
1
2
x
y
1 (x)
2 (y)
i
y=x
; (8)
is the Moyal(or star) produt. Then we an work on aommutative spae in whih the usual produt of eld
is replaed by the Moyalprodut. Notie that thederivatives in the denition Eq.(8) makes theMoyalprodut
non-loal. Also,theMoyalommutatoroftheommutativeoordinatesx
gives
[x
;x
℄
MB =x
?x
x
?x
=i
: (9)
It anbeeasilyveriedthefollowingpropertiesoftheMoyalprodut:
a) e ik x
?e iqy
=e i(k +q)x
e ik ^q
; (10)
wherek^q= 1
2 k
q
.
b) (f?g)(x)= Z
dk dq ~
f(k)~g(q)e ik ^q
e i(k +q)x
; (11)
where ~
f andg~aretheFourieromponentsof f and g,respetively.
) [(f?g)?h℄(x)=[f?(g?h)℄(x): (12)
d) Z
dx(f?g)(x)= Z
dx(g?f)(x)= Z
dxf(x)g(x): (13)
e) Z
dx(f
1 ?f
2 ?:::f
n )(x)=
Z
dx(f
n ?f
1 ?:::f
n 1
)(x): (14)
f) (f?g)
=g
?f
: (15)
III Nonommutative Salar Field Theory
Letusonsiderthemassivesalareld inD=3+1dimensions
[3℄,whoseationis
S= Z
d 4
x
1
2
?
m
2
2 ?
g 2
4!
???
: (16)
Usingpropertyd)itisseenthatthepropagatorisnotaetedbytheMoyalprodut. Thisisageneripropertyof
nonommutativeeld theories. Thevertex,however,mustbesymmetrized. Inmomentum spaewehave
g 2
2 Z
d 4
x???= g
2
6 Z
dk
1 dk
2 dk
3 dk
4 Æ(k
1 +k
2 +k
3 +k
4 )
[os( 1
2 k
1 ^k
2 )os(
1
2 k
3 ^k
4
)+os( 1
2 k
1 ^k
3 )os(
1
2 k
2 ^k
4 )+
os( 1
k
1 ^k
4 )os(
1
k
2 ^k
3 )℄(k
1 )(k
2 )(k
3 )(k
4
Then,theonelooporretionforthetwo-pointfuntionis
g 2
3(2) 4
Z
d 4
k
1+ 1
2
os(k^p)
1
k 2
+m 2
: (18)
The rst term is the usual one loop mass orretion of the ommutative theory (up to a fator 1=2) whih is
quadratiallydivergent. Theseond termisnotdivergentdue tothe osillatorynature ofos(k^p). This shows
that thenonloalityintroduedbytheMoyalprodutisnotbadandleavesuswiththesamedivergenestruture
of theommutativetheory. Totakeintoaounttheeet oftheseondtermweregularizetheintegralusingthe
Shwingerparametrization
1
k 2
+m 2
= Z
1
0 d e
(k 2
+m 2
)
e 1
2
; (19)
where autowasintrodued. Wend
(2)
= g
2
48 2
[( 2
m 2
ln(
2
m 2
)+:::)+ 1
2 (
2
eff m
2
ln(
2
eff
m 2
)+:::)℄; (20)
where
2
eff =
1
1
2
+p~ 2
; p~
=
p
: (21)
Notethatwhentheutoisremoved,!1,thenonommutativeontributionremainsniteprovidinganatural
regularization. Also 2
eff =
1
~ p 2
whih divergeseither when!0orwhenp~!0.
Theoneloopeetiveationisthen
Z
d 4
p 1
2 (p
2
+M 2
+
g 2
96 2
(~p 2
+1= 2
) g
2
M 2
96 2
ln
1
M 2
(~p 2
+1= 2
)
+:::)(p)( p); (22)
whereM istherenormalizedmass. Letustakethelimits!1andp~!0. Ifwetakerstp~!0thenp~ 2
<< 1
2
and
eff
= showingthat wereovertheeetiveommutativetheory
Z
d 4
p 1
2 p
2
+M 02
(p)( p): (23)
If,however,wetake!1thenp~ 2
>> 1
2
and 2
eff =
1
~ p 2
andweget
Z
d 4
p 1
2
p 2
+M 2
+ g
2
96 2
~ p 2
g 2
M 2
96 2
ln
1
M 2
~ p 2
+:::
(p)( p); (24)
d
whih is singular when p~ ! 0. This shows that the
limit !1 doesnotommutewith thelow
momen-tum limitp~! 0so that there is amixing of UV and
IR limits.
Thetheoryisrenormalizableatonelooporderifwe
do nottake p~! 0. What about higher loop orders?
Suppose we have insertions of one loop mass
orre-tions. Eventuallywe will haveto integrate over small
values of p~whih diverges when ! 1. Then we
nd an IR divergene in a massivetheory. This
om-bination of UV and IR divergenes makesthe theory
non-renormalizable.
Therearealsoexamplesofnon-renormalizable
the-oriesalreadyatonelooporder[5℄. Foraomplexsalar
theoryisone-loopnon-renormalizablewhile
??
?
givesaonelooprenormalizablemodel.
Then the questionis whether it would be possible
tondatheorywhih isrenormalizableto allloop
or-ders. Sinethe UV/IR mixing appearsat the level of
quadrati divergenes a andidate theory would be a
supersymmetri theory beause it does nothave suh
divergenes[9,10℄. Asweshallseethisindeedhappens.
IV Nonommutative W
ess-Zumino model
L
0 =
1
2
A
A+
1
2
B
B+
1
2
i6 ; (25)
L
m =
1
2 F
2
+ 1
2 G
2
+mFA+mGB 1
2
m ; (26)
L
g
= g(F?A?A F?B?B+G?A?B+G?B?A
? ?A ?i
5
?B); (27)
d
where A and B are bosoni elds, F and G are
aux-iliary elds and is a Majorana spinor. The ation
is invariant under the usual supersymmetry
transfor-mations. TheyarenotmodiedbytheMoyalprodut
sine they are linearin the elds. The elimination of
the auxiliary elds through their equations of motion
produesquartiinterations. Intermsoftheomplex
eld =A+iB we get
?
?? whih is
non-renormalizableinthenonommutativease. Thisasts
doubtsabouttherenormalizabilityofthemodelbutas
weshallseesupersymmetrysavestheday.
Asusual,thepropagatorsarenotmodiedby
non-ommutativityduetothepropertyd). Theyaregiven
by
AA
(p) = (p) i
p 2
m 2
+i
; (28)
FF
(p) = p 2
(p); (29)
AF
(p) =
FA
(p)= m(p); (30)
S(p) = i
6p m
: (31)
Takingintoaountthesymmetriesthevertiesare
FA 2
vextex: igos(p
1 ^p
2
); (32)
FB 2
vextex: igos(p
1 ^p
2
); (33)
GAB vertex: 2igos(p
1 ^p
2
); (34)
A vertex: igos(p
1 ^p
2
); (35)
B vertex: ig
5 os(p
1 ^p
2
): (36)
The degree of superial divergene for ageneri 1PI
graphisthen
d()=4 I
AF I
BF N
A N
B 2N
F 2N
G 3
2
N ; (37)
d
whereN
O
denotesthenumberofexternallines
assoi-ated totheeld O andI
AF and I
BF
are thenumbers
of internal lines assoiated to the mixed propagators
AF and BF, respetively. Inallaseswewill
regular-ize the divergentFeynman integralsby assuming that
asupersymmetri regularizationshemedoesexist.
The one loop analysis an be done in a
straight-forwardway. As in theommutativeaseall tadpoles
ontributionsadduptozero. Wehaveveriedthis
ex-pliitly. Theself-energyofAanbeomputedandthe
divergentpartisontainedin theintegral
16g 2
Z
d 4
k
(2) 4
(1+ 1
2
os(k^p))
(pk) 2
(k 2
m 2
) 3
: (38)
The rst term is logarithmially divergent. It diers
thisdivergeneiseliminatedbyawavefuntion
renor-malization. TheseondtermisUV onvergentandfor
smallpitbehavesasp 2
ln(p 2
=m 2
)andatuallyvanishes
forp=0. ThenthereisnoIRpole. Thesameanalysis
anbearriedoutfortheothers elds. ForF wend
that thedivergentpartis
4g 2
Z
d 4
k
(2) 4
(1+ 1
2
os(k^p)) 1
(k 2
m 2
) 2
: (39)
Thersttermislogarithmiallydivergentandanalso
beeliminatedbyawavefuntionrenormalization. The
seond term diverges as ln (p 2
=m 2
) as p goes to zero.
However its multiple insertions is harmless. For the
Thetermontainingos(k^p)behavesas6pln(p 2
=m 2
)
and vanishes aspgoes to zero. Therefore, there is no
UV/IR mixingintheself-energyasexpeted.
Toshowthat themodel isrenormalizablewemust
also look into the interations verties. The A 3
ver-tex hasnodivergentpartsasintheommutativease.
Thesamehappensfortheotherthree pointfuntions.
For the four point verties no divergene is found as
in the ommutativease. Hene,thenonommutative
Wess-Zuminomodelisrenormalizableatoneloopwith
awave-funtionrenormalizationandnoUV/IRmixing.
To go to higher loop orders we proeed as in the
ommutativease[11℄. Wederivedthesupersymmetry
Ward identities for the n-point vertex funtion. Then
weshowedthat thereisarenormalizationpresription
whihisonsistentwiththeWardidentities. Theyare
the same as in the ommutativease. Andnally we
xed theprimitivelydivergentvertex funtions. Then
we found that there is only a ommon wave funtion
renormalizationasintheommutativease. Ingeneral
weexpet
'
R =Z
1=2
'; m
R
=Zm+Æm; g
R =Z
3=2
Z 0
g:
(40)
Atoneloopwefound Æm=0andZ 0
=1. Weshowed
thatthisalsoholdstoallordersandnomass
renormal-izationisneeded.
Being the only onsistent nonommutative
quan-tum eld theory in 3+1 dimensions known so far
it is natural to study it in more detail. As a rst
stepin this diretion weonsidered thenonrelativisti
limitof thenonommutativeWess-Zumino model[12℄.
We foundthelowenergyeetivepotentialmediating
thefermion-fermionandboson-bosonelastisattering
in the nonrelativistiregime. Sine nonommutativity
breaks Lorentz invarianeweformulatedthetheoryin
theenterofmassframeofreferenewherethe
dynam-is simplies onsiderably. Forthe fermions we found
thatthepotentialissigniativelyhangedbythe
non-ommutativitywhilenomodiationwasfoundforthe
bosonisetor. Themodiationsfoundgiverisetoan
anisotropidierentialrosssetion.
V Nonommutative
Gross-Neveu and Nonlinear Sigma
Models
Anothermodelwherenonrenormalizabilityisspoiledby
thenonommutativityistheO(N)Gross-Neveumodel.
Thismodelisperturbativelyrenormalizablein1+1
di-mensions and 1=N renormalizablein 1+1and 2+1
dimensions. In both asesit presentsdynamial mass
generation. ItisdesribedbytheLagrangian
L= i
2 i
6
i +
g
4N (
i i
)(
j j
); (41)
where
i
;i = 1;:::N, are two-omponent Majorana
spinors. Sine it is renormalizable in the 1=N
expan-sionin1+1and2+1dimensionswewillonsiderboth
ases. Asusual, weintrodueanauxiliaryeld and
theLagrangianturnsinto
L= i
2 i
6
i
2 (
i i
) N
4g
2
: (42)
Replaing by +M where M is the VEV of the
originalwegetthegapequation(inEulideanspae)
M
2g Z
d D
k
(2) D
M
k 2
E +M
2
=0: (43)
ToeliminatetheUVdivergeneweneedtorenormalize
theouplingonstantby
1
g =
1
g
R +2
Z
d D
k
(2) D
1
k 2
E +
2
: (44)
In2+1dimensionswend
1
g
R =
jMj
2
; (45)
and thereforeonly for 1
gR +
2
> 0it is possible to
haveM 6=0,otherwiseM isneessarilyzero. Nosuh
arestrition exists in 1+1 dimensions. In any ase,
wewillfousonlyinthemassivephase. The
propaga-torfor isproportionaltotheinverseofthefollowing
expression
iN
2g iN
Z
d D
k
(2) D
k(k+p)+M 2
(k 2
M 2
)[(k+p) 2
M 2
℄
; (46)
whihisdivergent. Takingintoaountthegapequationtheaboveexpressionreduesto
(p 2
4M 2
)N
2
Z
d D
k
(2) D
1
(k 2
M 2
)[(k+p) 2
M 2
℄
; (47)
Thenonommutativemodelisdenedby
S
GN =
Z
d D
x
i
2
6
M
2
1
2
?( ? ) N
4g
2 N
2g M
: (48)
d
Elimination of the auxiliary eld results in a
four-fermioninterationofthetype
i ?
i ?
j ?
j
. However
amoregeneral four-fermioninteration may involvea
termlike
i ?
j ?
i ?
j
. This lastombination does
nothaveasimple1=N expansionand wewillnot
on-siderit. TheMoyalprodutdoesnotaetthe
propa-gatorsand thetrilinearvertexaquiresaorretionof
os(p
1 ^p
2
)withregardtotheommutativease. Hene
thegapequationisnotmodied,whilethepropagator
forthe isnowproportionaltotheinverseof
iN
2g N
Z
d D
k
(2) D
os 2
(k^p)
k(k+p)+M 2
(k 2
M 2
)[(k+p) 2
M 2
℄
: (49)
d
Now thedivergentpartis nolonger aneledand this
turnsthemodelinto anonrenormalizableone.
Onthe other side, the nonlinear sigmamodel also
presents troublesin its nonommutative version. The
nonommutativemodelisdesribedby
L= 1
2 '
i (
2
+M 2
)'
i +
1
2 ?'
i ?'
i N
2g
; (50)
where '
i
, i = 1;:::;N, arereal salarelds, is the
auxiliaryeldandM isthegeneratedmass. The
lead-ingorretiontothe'self-energyis
i Z
d 2
k
(2) 2
os 2
(k^p)
(k+p) 2
M 2
(k); (51)
where
is the propagatorfor . As for the aseof
the salar eld this an be deomposed as a sum of
a quadratially divergent part and a UV nite part.
Again there is the UV/IR mixing destroying the 1=N
expansion.
VI Nonommutative
Super-symmetri Nonlinear Sigma
Model
The Lagrangian for the ommutative supersymmetri
sigmamodelisgivenby
L= 1
2
'
i
'
i +
i
2 i
6
i +
1
2 F
i F
i +'
i F
i +
1
2 '
i '
i 1
2 i
i i
'
i N
2g
; (52)
whereF
i
,i=1;:::;N,areauxiliaryelds. Furthermore,;and aretheLagrangemultiplierswhihimplement
thesupersymmetrionstraints. Afterthehangeofvariables!+2M,F !F M'whereM=< >,and
theshifts!+M and!+
0
, where
0
=<>,wearriveatamoresymmetriformfortheLagrangian
L =
1
2 '
i (
2
+M 2
)'
i +
1
2 i
(i6 M)
i +
1
2 F
2
i +M
2
' 2
i +
1
2
0 '
2
i
+ 1
2 '
2
i +'
i F
i 1
2 i
i
i '
i N
2g
N
g
M: (53)
Nowsupersymmetryrequires
0
= 2M 2
andthegapequationis
Z
d D
k
D i
2 2
= 1
soaoupling onstantrenormalizationis required. We nowmust examinewhether thepropagatorfor depends
onthethisrenormalization. Wendthatthetwopointfuntion for isproportionaltotheinverseof
(p 2
4M 2
)N
2
Z
d D
k
(2) D
1
(k 2
M 2
)[(k+p) 2
M 2
℄
; (55)
whihisidentialtotheGross-Neveuase. Notiethat thegapequationwasnotused. Thenitenessoftheabove
expressionis aonsequeneofsupersymmetry.
Thenonommutativeversionofthesupersymmetri nonlinearsigmamodelisgivenby
L =
1
2 '
i (
2
+M 2
)'
i +
1
2 i
(i6 M)
i +
1
2 F
2
i +
2 ?'
i ?'
i
1
2 F
i
?(?'
i +'
i ?)
1
2 ?
i ?
i 1
2 (
?
i ?'
i +
?'
i ?
i )
N
2g
NM
g
: (56)
Notie that supersymmetryditates theform of thetrilinear verties. Also, the supersymmetrytransformations
arenotmodiedbynonommutativitysinetheyarelinearandnoMoyalprodutsarerequired.
Thepropagatorsare thesameasin the ommutativease. Theverties haveosine fatorsdue to theMoyal
produt
' 2
vertex:
i
2 os(p
1 ^p
2
); (57)
'F vertex: ios(p
1 ^p
2
); (58)
vertex:
i
2 os(p
1 ^p
2
); (59)
' vertex: ios(p
1 ^p
2
): (60)
We again onsider the propagatorsfor the Lagrange multiplier elds. Now the propagator is modied by the
osine fatorsandisproportionalto theinverseof
(p 2
4M 2
)N
2
Z
d D
k
(2) D
os 2
(k^p)
(k 2
M 2
)[(k+p) 2
M 2
℄
: (61)
It iswellbehavedbothin UVandIR regions. Thepropagatorsforand areproportionaltotheinverseof
N
2 Z
d D
k
(2) D
os 2
(k^p)
1
[(k+p) 2
M 2
℄[k 2
M 2
℄
; (62)
and
N
(6p+2M)
2 Z
d D
k
(2) D
os 2
(k^p)
1
[(k+p) 2
M 2
℄[k 2
M 2
℄
; (63)
respetively. Theyarealsowellbehavedin UVandIR regions.
Thedegreeofsuperialdivergeneforageneri1PIgraphis
d()=D
(D 1)
2 N
(D 2)
2 N
' D
2 N
F N
3
2 N
2N
; (64)
d
whereN
O
isthenumberofexternallinesassoiatedto
the eld O. Potentiallydangerousdiagrams are those
ontributingto theself{energiesof the 'and elds
sine, in priniple,they are quadrati and linearly
di-vergent,respetively. Forthe self-energiesof ' and
wendthat theydivergelogarithmiallyand theyan
respetive eld. The same happens for the auxiliary
eld F. The renormalizationfatorsfor them are the
same so supersymmetry is preserved in the
nonom-mutativetheory. Thisanalysisanbeextendedto the
n-point funtions. In 2+1 dimensions we nd
leading order of 1=N. However, in 1+1 dimensions
there some peuliarities. Sine the salar eld is
di-mensionless in 1+1 dimensions any graph involving
anarbitrarynumberofexternal'linesisquadratially
divergent. In the four-point funtion there is a
par-tialanellationofdivergenesbutalogarithmi
diver-genestillsurvives. Theountertermneededtoremove
itannotbewrittenintermsof R
d 2
x'
i ?'
i ?'
j ?'
j
and R
d 2
x'
i ?'
j ?'
i ?'
j
. A possible wayto remove
this divergeneis bygeneralizingthedenition of 1PI
diagram alongthe lines suggestedin [13℄ forthe
om-mutative nonlinear sigma model. However the osine
fators do not allow us to use this mehanism whih
astsdoubtabouttherenormalizabilityofthe
nonom-mutativesupersymmetriO(N)nonlinearsigmamodel
in1+1dimensions.
VII Conlusions
We have shown that it is possibleto build onsistent
quantum eld theories in nonommutative spae. It
seemsthatsupersymmetryisanessentialingredientfor
renormalizability. The models studied heredonot
in-volvegauge elds and this onsiderably simplies the
situation. All verties are deformed in the same way
bytheMoyalprodutandthiswasessentialtoanalyze
theamplitudes. Withgaugeeldsthesituationismuh
moreompliatedbeausethevertiesaredeformedin
dierent ways. However, supersymmetri gauge
theo-riesmaystillhaveabetterbehavior.
Aknowledgments
This work was done in ollaboration with H. O.
Girotti,M.Gomes andA.J.daSilva. Itwaspartially
supported by Funda~ao deAmparo aPesquisa do
Es-tado de S~ao Paulo (FAPESP), Conselho Naional de
DesenvolvimentoCientoeTenologio(CNPq),and
PRONEXunderontratCNPq66.2002/1998-99.
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