Finite Temperature and Large Gauge Invariane
Ashok Das
Department ofPhysisandAstronomy,
UniversityofRohester,
Rohester,NewYork, 14627
Reeivedon1February,2001
Inthistalk,Iwillsumarizethestatusofourunderstandingofthepuzzleoflargegaugeinvariane
atnitetemperature.
I Introdution
Gauge theories are beautiful theories whih desribe
physialforesinanaturalmannerandbeauseoftheir
rihstruture,thestudyofgaugetheoriesatnite
tem-perature [1℄ is quite interesting in itself. However, to
avoidgettingintotehnialities,wewillnotdisussthe
intriaies of suh theories either at zerotemperature
oratnitetemperature. Rather,wewouldstudya
par-tiularpuzzlethatariseswhenafermioninteratswith
anexternalgaugeeld inoddspae-timedimensions.
Tomotivate,letusnotethatgaugeinvarianeis
re-alized asaninternalsymmetryinquantummehanial
systems. Consequently,wedonotexpetamarosopi
external surrounding, suh asa heat bath, to modify
gaugeinvariane. Thisismoreorlesswhatisalsofound
byexpliitomputationsatnitetemperature,namely,
that gaugeinvariane andWard identities ontinueto
hold even at nite temperature [2℄. This is ertainly
the asewhen oneistalking about smallgauge
trans-formationsforwhihtheparametersoftransformation
vanishat innity.
However,there is aseond lass ofgauge
transfor-mations, ommonlyknownaslarge gauge
transforma-tions, where the parametersdo not vanish at innity.
In odd spae-time dimensions, invariane under large
gauge transformations leads to some interesting
fea-tures in physial theories. Let us note that, in odd
spae-timedimensions,onean,inadditiontotheusual
Maxwell (Yang-Mils) term, have a topologial term
in the gaugeLagrangian known as the Chern-Simons
term. Forexample,in2+1dimensions,afermion
inter-atingwithanon-Abeliangaugeeldanbedesribed
byaLagrangiandensityoftheform
L = L
gauge mL
CS +L
fermion
= 1
2 trF
F
m
trA
(
A
+ 2g
A
A
)+ (
(i
gA
) M) (1)
wheremisamassparameter,A
amatrixvalued
non-Abelian gaugeeld in agivenrepresentation and\tr"
standsforthematrixtrae.Therstterm,ontheright
hand side,is the usualYang-Mills termwhile the
se-ond is known as the Chern-Simonsterm whih exists
onlyin oddspae-time dimensions. It is atopologial
term(sine itdoesnotinvolvethemetri)and, in the
presene of a Yang-Mils term, its eet is to provide
agaugeinvariantmasstermto thegaugeelds.
Con-sequently, suh a term is also known as a topologial
mass term [3℄. (Suh a term also breaks various
dis-retesymmetries,butwewill notgetintothat.)
Underagaugetransformationoftheform
! U
1
A
! U
1
A
U
i
g U
1
U (2)
itisstraightforwardtohekthat boththeYang-Mills
andthe fermiontermsareinvariant, while the
Chern-Simonstermhangesbyatotaldivergeneleadingto
S= Z
d 3
xL!S+ 4m
g 2
2iW (3)
where
W = 1
24 2
Z
d 3
x
tr
UU
1
UU
1
UU
1
(4)
is known as the winding number for the gauge
trans-formation. It isa topologialquantity whih is an
in-tegerandwhihgroupsallgaugetransformationsinto
topologially distint lasses. Basially, it ounts how
manytimesthegaugetransformationswraparoundthe
sphere. Forsmall gauge transformations, thewinding
number vanishessinethe gaugetransformations
van-ishatinnitywhereasnon-vanishingwindingnumbers
Letusnotefromeq. (3)thateventhoughtheation
isnot invariantunder alargegaugetransformation,if
m is quantized in units of g
2
4
, the hange in the
a-tionwould beamultiple of2i and, onsequently,the
path integral would be invariant under a large gauge
transformation. Thus, wehave theonstraint oming
from theonsistenyof the theorythat theoeÆient
of theChern-Simons termmust bequantizedin units
of g
2
4
. (Fromanoperatorpointof view,suha
ondi-tionarisesfromananalysisofthedimensionalityofthe
Hilbertspae.)
Wehavederived thequantizationof theCS
oeÆ-ient froman analysisof thelargegaugeinvarianeof
thetreelevelationandwehavetoworryifthe
quan-tumorretionsanhangethebehaviorofthetheory.
Atzerotemperature,ananalysisofthequantum
orre-tionsshowsthatthetheoryontinuestobewelldened
with the tree level quantization of the Chern-Simons
oeÆient provided the number of fermion avors is
even. Theevennumberoffermionavorsisalso
nees-sary for aglobalanomaly of thetheory to vanish and
so,everythingiswellunderstood at zerotemperature.
(Namely, thequantumorretions, withaneven
num-beroffermionavorsshiftthetreelevelintegervalueto
another,therebymaintaininglargegaugeinvariane.)
At nite temperature, however, the situation
ap-pears to hange drastially. Namely, the fermions
indue a temperature dependent Chern-Simons term
leadingto [4℄
m!m g
2
4 MN
f
2jMj tanh
jMj
2
(5)
Here,N
f
isthenumberoffermionavorsand= 1
k T .
This shows that, at zero temperature ( ! 1), m
hanges by an integer (in units of g 2
=4) for an even
numberofavors. However,atnite temperature,the
CSoeÆientbeomesaontinuousfuntionof
temper-atureand,onsequently,itislearthatitannolonger
beanintegerforarbitraryvaluesofthetemperature,as
isrequired forlargegaugeinvariane. It seems,
there-fore,thattemperaturewouldleadtoabreakingoflarge
gaugeinvarianein suhasystem. This,on theother
hand, is ompletely ounter intuitive onsidering that
temperature should have no diret inuene on gauge
invarianeofthetheory. Asaresult,weareleftwitha
puzzle,whose resolution, as wewill see, isquite
inter-esting.
II C-S Theory in 0 + 1
Dimen-sion:
The2+1dimensionaltheorydesribedintheprevious
setion is quite ompliated to arry outhigher order
alulations. On the other hand, as we have noted,
Chern-Simons terms an exist in odd spae-time
di-mensions. Consequently, letus tryto understand this
puzzle of large gauge invariane in asimple quantum
mehanialtheory. Let us onsidera simpletheoryof
an interatingmassivefermionwithan Abeliangauge
eld in0+1dimensiondesribedby[5,6℄
L=
j (i
t
A M)
j
A (6)
Here,j =1;2;;N
f
labelsthefermionavors. There
areseveralthingsto notefrom this. First,weare
on-sideringan Abeliangaugeeld forsimpliity. Seond,
in this simplemodel, thegaugeeld has nodynamis
(in0+1dimensiontheeldstrengthiszero)and,
there-fore,wedonothavetogetintotheintriaiesofgauge
theories. ThereisnoDiramatrixin0+1dimensionas
wellmaking thefermionpartof the theoryquite
sim-ple as well. And, nally, the Chern-Simons term, in
thisase,isalineareldso thatwean,infat,think
of thegaugeeld asanauxiliaryeld. (Notethat we
havesettheouplingonstanttounityforsimpliity.)
Inspiteof the simpliity ofthis theory, it displays
arihstrutureinludingallthepropertiesofthe2+1
dimensional theorythatwehavedisussedearlier. For
example,letusnotethatunderagaugetransformation
j !e
i(t)
j
; A!A+
t
(t) (7)
thefermionpartoftheLagrangianisinvariant,butthe
Chern-Simonstermhangesbyatotalderivativegiving
S= Z
dtL!S 2N (8)
where
N= 1
2 Z
dt
t
(t) (9)
isthewindingnumberandisanintegerwhihvanishes
for small gauge transformations. Let us note that a
largegaugetransformationanhaveaparametriform
oftheform,say,
(t)= iN log
1+it
1 it
(10)
The fat that N has to be an integer an be easily
seentoarisefromtherequirementofsingle-valuedness
for the fermion eld. One again, in light of our
ear-lierdisussion, it islear from eq. (8)that thetheory
is meaningful only if , the oeÆient of the
Chern-Simonsterm,isaninteger.
Letusassume,forsimpliity,thatM>0and
om-pute the orretion to the photon one-point funtion
arisingfrom thefermionloopatzerotemperature.
iI
1
= ( i)N
f Z
dk
2
i(k+M)
k 2
M 2
+i =
iN
f
2
(11)
Thisshowsthat, asaresultofthequantumorretion,
theoeÆientoftheChern-Simonstermwouldhange
as
!
N
Asin2+1dimensions,itislearthattheoeÆientof
theChern-Simonstermwouldontinuetobequantized
and large gauge invariane would hold if the number
of fermion avors is even. At zero temperature, we
analsoalulatethehigherpointfuntionsduetothe
fermions in thetheoryandtheyallvanish. Thishasa
simple explanation following from the small gauge
in-variane of the theory [7℄. Namely, suppose we had
a nonzero two point funtion, then, it would imply a
quadratitermintheeetiveationoftheform
2 =
1
2 Z
dt
1 dt
2 A(t
1 )F(t
1 t
2 )A(t
2
) (12)
Furthermore,invarianeunder asmall gauge
transfor-mationwould imply
Æ
2 =
Z
dt
1 dt
2 (t
1 )
t1 F(t
1 t
2 )A(t
2
)=0 (13)
The solution to this equation is that F = 0 so that
thereannotbeaquadratitermintheeetiveation
whih wouldbeloaland yet beinvariantunder small
gaugetransformations. Asimilar analysis would show
that smallgaugeinvarianedoesnotallowanyhigher
pointfuntiontoexist atzerotemperature.
Letusalsonotethateq. (13)hasanothersolution,
namely,
F(t
1 t
2
)=onstant
Insuh aase,however,thequadratiationbeomes
non-extensive, namely, it is the square of an ation.
Wedonotexpetsuh termstoariseat zero
tempera-tureandhenetheonstanthastovanishforvanishing
temperature. As we will see next, the onstant does
not have to vanish at nite temperature and we an
have non-vanishing higher point funtions implying a
non-extensivestrutureoftheeetiveation.
The fermion propagator at nite temperature (in
therealtimeformalism) hastheform[1℄
S(p) = (p+M)
i
p 2
M 2
+i 2n
F (jpj)Æ(p
2
M 2
)
=
i
p M+i 2n
F
(M)Æ(p M) (14)
and the struture of the eetive ation an be studied in the momentum spae in a straightforward manner.
However, in this simple model, it is muh easier to analyze theamplitudes in the oordinate spae. Letus note
that theoordinatespaestrutureofthefermionpropagatorisquitesimple,namely,
S(t)= Z
dp
2 e
ipt
i
p M+i 2n
F
(M)Æ(p M)
=((t) n
F (M))e
iMt
(15)
d
Infat,thealulationoftheonepointfuntionis
triv-ialnow
iI
1
= ( i)N
f S(0)=
iN
f
2 tanh
M
2
(16)
This shows that the behavior of this theory is
om-pletelyparalleltothe2+1dimensionaltheoryinthat,
itwouldsuggest
!
N
f
2 tanh
M
2
anditwouldappearthat largegaugeinvarianewould
notholdat nitetemperature.
temperature.
iI
2
= ( i) 2
N
f
2! S(t
1 t
2 )S(t
2 t
1 )
= N
f
2 n
F
(M)(1 n
F (M))
= N
f
8 seh
2 M
2 =
1
2 1
2! i
(iI
1 )
M
(17)
Thisshowsthatthetwopointfuntionisaonstantas
wehadnotedearlierimplying thatthequadratiterm
intheeetiveationwouldbenon-extensive.
iI
3 =
iN
f
24 tanh
M
2 seh
2 M
2 =
1
2 1
3!
i
2
2
(iI
1 )
M 2
(18)
In fat, all the higher point funtions an be worked out in a systemati manner. But, let us observe a simple
method of omputation forthese. Wenote that beauseof thegaugeinvariane(Wardidentity), theamplitudes
annot depend on the external time oordinates as is lear from the alulations of the lower point funtions.
Therefore,weanalwayssimplifythealulationbyhoosing apartiulartimeorderingonvenientto us. Seond,
sineweareevaluatingaloopdiagram(afermionloop)theinitialandthenaltimeoordinatesarethesameand,
onsequently, thephasefators in the propagator(15)drop out. Therefore, letus dene a simplied propagator
withoutthephasefatoras
e
S(t)=(t) n
F
(M) (19)
sothat wehave
e
S(t>0)=1 n
F
(M); S(t<0)= n
F
(M) (20)
Then,itislearthatwiththehoie ofthetimeordering,t
1 >t
2
, weanwrite
e
S(t
1 t
2 )
M
=
e
S(t
1 t
3 )
e
S(t
3 t
2
) t
1 >t
2 >t
3
e
S(t
2 t
1 )
M
=
e
S(t
2 t
3 )
e
S(t
3 t
1
) t
1 >t
2 >t
3
(21)
d
Inotherwords,this showsthat dierentiation of a
fermioni propagator with respet to the mass of the
fermion is equivalent to introduing an external
pho-tonvertex(and,therefore,anotherfermion propagator
as well) up to onstants. This is the analogueof the
Ward identityin QED in four dimensionsexept that
it ismuh simpler. Fromthis relation, it islear that
ifwetakean-pointfuntionanddierentiatethiswith
respettothefermionmass,then,thatisequivalentto
adding another external photon vertex in all possible
positions. Namely, it should giveus the (n+1)-point
funtion upto onstants. Working outthedetails,we
have,
I
n
M
= i(n+1)I
n+1
(22)
Therefore, the(n+1)-point funtion is relatedto the
n-point funtion reursivelyand, onsequently, allthe
amplitudesarerelatedtotheonepointfuntionwhih
we have already alulated. (Inidentally, this is
al-readyreetedineqs. (17,18)).
With this, we annow determine thefull eetive
ationofthetheoryat nitetemperaturetobe
= i
X
n a
n
(iI
n )
=
iN
f
2 X
n
(ia=) n
n!
M
n 1
tanh M
2
= iN
f log
os a
+itanh M
sin a
(23)
wherewehavedened
a= Z
dtA(t) (24)
There are several things to note from this result.
First of all, the higher point funtions are no longer
vanishing atnitetemperatureandgiverisetoa
non-extensivestrutureoftheeetiveation. More
impor-tantly,when weinlude allthehigherpointfuntions,
the omplete eetive ation is invariant under large
gaugetransformations,namely,under
a!a+2N (25)
theeetiveationhangesas
! +NN
f
(26)
whih leaves the path integral invariant for an even
number of fermion avors. This laries the
puz-zle of large gauge invariane at nite temperature in
this model. Namely, when weare talking about large
hanges (large gauge transformations), we annot
ig-norehigherordertermsiftheyexist. Thismayprovide
aresolutiontothelargegaugeinvarianepuzzlein the
2+1dimensional theoryaswell.
III Exat Result:
In the earlier setion, we disussed a perturbative
tem-perature whih laried the puzzle of large gauge
in-variane. However,thisquantum mehanialmodelis
simple enough that we an also evaluate the eetive
ation diretly and, therefore, it is worth asking how
the perturbative alulations ompare with the exat
result.
Theexatevaluation oftheeetiveationanbe
done easily using theimaginarytime formalism. But,
rst, let us note that the fermioni part of the
La-grangianineq. (6)hastheform
L
f
= (i
t
A M) (27)
where wehavesuppressed thefermionavorindex for
simpliity. Letusnotethat ifwemakeaeld
redeni-tionoftheform
(t)=e i
R
t
0 dt
0
A(t 0
)
~
(t) (28)
then,thefermionipartoftheLagrangianbeomesfree,
namely,
L
f =
~
(i
t M)
~
(29)
Thisisafreetheoryand,therefore,thepathintegral
anbeeasilyevaluated. However,wehavetoremember
that theeldredenitionin(28)hangesthe
periodi-ity onditionfor thefermion elds. Sine the original
fermioneldwasexpetedtosatisfyanti-periodiity
()= (0)
itfollowsnowthatthenewelds mustsatisfy
~
()= e ia
~
(0) (30)
Consequently,thepathintegralforthefreetheory(29)
hastobeevaluatedsubjettotheperiodiityondition
of(30).
Althoughtheperiodiityondition (30)appearsto
be ompliated, it is well known that this anbe
ab-sorbed by introduing a hemial potential [1℄, in the
presentase,oftheform
= ia
(31)
With the additionof thishemial potential, thepath
integral an be evaluated subjet to the usual
anti-periodiity ondition. The eetiveationan nowbe
easily determined
= ilog
det(i
t M+
ia
)
(i
t M)
!
N
f
= iN
f log
osh
2 (M
ia
)
osh M
2 !
= iN
f log
os a
2
+itanh M
2 sin
a
2
(32)
IV Large Gauge Ward Identity:
Itis learfrom theaboveanalysis that, to see iflarge
gaugeinvarianeisrestored,wehavetolookatthe
om-pleteeetiveation. Inthe0+1dimensional model,
it was tedious, but we an derivethe eetive ation
in losed form whih allows us to analyze the
ques-tionof large gaugeinvariane. On the other hand,in
the theory of interest, namely, the 2+1 dimensional
Chern-Simonstheory, we do notexpet to be able to
evaluate the eetive ation in a losed form.
Conse-quently, wemust look foran alternatewayto analyze
thequestionoflargegaugeinvarianeinamore
realis-timodel. Onesuh possiblemethodmaybeto derive
aWard identity for large gauge invariane whih will
relatedierentamplitudesmuhliketheWardidentity
forsmallgaugeinvarianedoes. Insuhaase,evenif
weannotobtaintheeetiveationin alosedform,
weanat leasthekifthelargegaugeWardidentity
holdsperturbatively.
ItturnsoutthatthelargegaugeWardidentitiesare
highlynonlinear[8℄,aswewouldexpet. Hene,
look-ingforthemwithintheontextoftheeetiveationis
extremelyhard(althoughitanbedone). Rather,itis
muhsimplertolookatthelargegaugeWardidentities
intermsoftheexponentialoftheeetiveation. Let
usdene
(a)= ilogW(a) (33)
Namely,weareinterestedinlookingattheexponential
oftheeetiveation(i.e. uptoafatorofi,W isthe
basideterminantthatwouldarisefromintegratingout
thefermioneld). Wewillrestritourselvestoasingle
avor of massive fermions. The advantage of
study-ingW(a) asopposedto theeetive ationliesin the
fat that, in order for (a) to havethe right
transfor-mationpropertiesunder alargegaugetransformation,
W(a) simply has to be quasi-periodi. Consequently,
from the study of harmoniosillator (as well as
Flo-quettheory),weseethat W(a) hasto satisfyasimple
equationoftheform
2
W(a)
a 2
+ 2
W(a)=g (34)
where and g are parameters to be determined from
thetheory. Inpartiular,letusnotethattheonstantg
andependonparametersofthetheorysuhas
temper-aturewhereasweexpettheparameter,alsoknownas
theharateristiexponent,to beindependentof
tem-peratureandequaltoanoddhalfintegerforafermioni
mode. However,all these properties should
automati-allyresultfromthestrutureofthetheory. Letusalso
noteherethattherelation (34)is simplytheequation
W(a)= g
2
+Aos(a+Æ)= g
2
+
1
osa+
2
sina (35)
The onstants
1 and
2
appearingin thesolution anagain bedetermined from thetheory. Namely, from the
relationbetweenW(a)and (a),wereognizethatweanidentify
2 1 = 2 W a 2 a=0 = a 2 i 2 a 2 ! a=0 2 = W a a=0 =i a a=0 (36)
Fromthegeneralpropertiesofthefermiontheorieswehavedisussed,weintuitivelyexpetg=0. However,these
shouldreally followfromthestrutureofthetheoryand theydo,aswewill showshortly.
The identity(34)is alinearrelationasopposedto theWard identityin termsof theeetiveation. Infat,
rewritingthisin termsoftheeetiveation(usingeq. (33)),wehave
2 (a) a 2 =i 2 (a) a 2 ! ige i (a) (37)
So, let us investigate this a little bit more in detail. We know that the fermion mass term breaks parity and,
onsequently,theradiativeorretionswould generateaChern-Simonsterm,namely,inthistheory,weexpet the
one-point funtion to be nonzero. Consequently, by taking derivative of eq. (37) (as well as remembering that
(a=0)=0), wedetermine(Thesupersriptrepresentsthenumberofavors.)
( (1) ) 2 = " (1) a 2 3i 2 (1) a 2 (1) a 1 3 (1) a 3 # a=0 g (1) = " 2i 2 (1) a 2 + (1) a 1 3 (1) a 3 # a=0 (38) d
This is quite interesting, for it says that the two
pa-rametersin eq. (34)or (37)anbedeterminedfrom a
perturbativealulation. Let usnote heresomeofthe
perturbativeresultsinthistheory,namely,
(1) a a=0 = 1 2 tanh M 2 2 (1) a 2 a=0 = i 4 seh 2 M 2 3 (1) a 3 a=0 = 1 4 tanh M 2 seh 2 M 2 (39)
Using these, weimmediately determine from eq. (38)
that ( (1) ) 2 = 1 4 ; g (1)
=0 (40)
sothattheequation(37)leadstothelargegaugeWard
identityforasinglefermiontheoryof theform,
2 (1) a 2 =i 1 4 (1) a 2 ! (41)
Furthermore,wedeterminenowfromeq. (36)
(1) 1 =1; (1) 2
=itanh M
2
(42)
Thetwosignsinof (1)
2
simplyorrespondstothetwo
possiblesignsof (1)
. With this then,wean solvefor
W(a) in the single avorfermiontheory and we have
(independentofthesignof (1)
)
W (1)
f
(a)=os a
2
+itanh M 2 sin a 2 (43)
whihanbeomparedwitheq. (32). ForN
f
avors,
similarly,weandeterminetheWardidentitytobe
(N f ) a 2 =iN f 1 4 1 N 2 f (N f ) a 2 2 ! (44)
wherethenonlinearityoftheWardidentityismanifest.
Similarly, we andetermine the large gauge Ward
V Bak to 2+1 dimensions:
Theanalysisofvarious0+1dimensionalmodelsshows
[9℄howthelargegaugeinvarianepuzzlegetsresolved.
A ruial feature in this was the existene of
non-extensivehigherordertermsintheeetiveation. To
further understand this feature, we have also studied
the eetive ation for a fermion interating with an
external gauge eld in 1+1dimensions [10℄. In that
ase, the eetive ation is non-loal, as it should be
at nitetemperature,but extensive. Furthermore,the
eetiveationshowsnon-analytiity,asonewould
ex-petatnitetemperature,unlikethe0+1dimensional
model,whereonedoesnotexpetanynon-analytiity.
Followingtheresultsofthe0+1dimensionalmodel,
it was shown [11℄ that, for the speial hoie of the
gauge eld bakgrounds where A
0 = A
0 (t) and
~
A =
~
A(~x ),theparityviolatingpartoftheeetiveationof
afermioninteratingwithanAbeliangaugeeldtakes
theform
PV
= ie
2 Z
d 2
xartan
tanh M
2 tan(
ea
2 )
B(~x )
(45)
However,beausethehoie ofthebakgroundis very
speial, itwould seemthat thismaynotrepresentthe
omplete eetive ationin ageneral bakground. In
fat, in higherdimensions,suh as 2+1,onealso has
totaklewiththequestionofthenon-analytiityofthe
thermal amplitudeswhih leadstoanonuniquenessof
theeetiveation[1,12℄.
Withtheseissuesinmind,wehavestudiedthe
par-ity violating part of the four point funtion in 2+1
dimensionsatnite temperature. Thealulationsare
learly extremely diÆult and we have evaluated the
amplitudes at nite temperaturebyusing themethod
of forward sattering amplitudes [13℄. Without going
into details, let me summarize the results here [14℄.
First, the parity violating partof the box diagram is
nontrivial at zero temperature and omes from an
ef-fetiveationoftheform
4
T=0 =
e 4
64M 6
Z
d 3
x
F
(
F
)F
F
(46)
This is Lorentz invariantand is invariant under both
small andlargegaugetransformationsandis
ompati-ble withtheColeman-Hilltheorem[15℄.
Atnitetemperature,however,theamplitudeisnot
manifestlyLorentzinvariant(beauseoftheheatbath)
and is non-analyti. We have investigated the
ampli-tudeintwointerestinglimits. Namely,inthelongwave
limit(allspatialmomentavanishing),theleadingterm
of theamplitude, at hightemperature,anbeseento
ome fromaneetiveationoftheform
4
LW =
e 4
512MT Z
d 3
x
0ij E
i (
1
t E
j )(
1
t E
k )(
1
t E
k )
(47)
where ~
Erepresentstheeletrield. Thereareseveral
thingsto note here. First,this is an extensive ation,
be it non-loal. Seond, it is manifestly large gauge
invariantandnally,theleadingbehaviorathigh
tem-peraturegoesas 1
T .
In ontrast, we an evaluate the amplitude in the
stati limit (allenergies vanishing) where we nd the
presene of both extensive as well as non-extensive
terms at high temperature. However, the extensive
termsare suppressed by powers of T and the leading
termseemstoomefromaneetiveationoftheform
4
S =
e 4
4T 2
tanh M
2
tanh 3
M
2 Z
d 3
xa 3
B
(48)
Thisoinideswiththeamplitudethat willomefrom
eq. (45) and has the leading behavior of 1
T 3
at high
temperature. Suhatermisnotinvariantunderalarge
gaugetransformation. However, we an now derivea
large gaugeWard identity for the leading part of the
statiationandthesolutionoftheWardidentity
o-inideswiththeformgivenineq. (45). This,therefore,
lariesthemeaningoftheeetiveationinthe
spe-ialbakground,namely,itrepresentstheleadingterm
intheeetiveationin thestatilimit.
VI Higher order orretions:
Sinewehavealulatedtheboxdiagramatnite
tem-perature, we an also ask about possible higher loop
orretionsto the CS oeÆient. Let us note that, at
zerotemperature,thereisaresultduetoColemanand
Hill [15℄, whih says that in an Abelian theory, there
annotbeanyorretionto the CSoeÆientbeyond
one loop. Their result basially uses two simple
as-sumptions,i)smallgaugeinvarianeandii)analytiity
oftheamplitudesinthemomentumspae. Smallgauge
invarianeis,ofourse,knowntobetrueatnite
tem-perature. However,aswehavepointedoutearlier,
am-plitudes beomenon-analyti in the momentum spae
at nite temperature. Therefore, the seond
assump-tionofColeman-Hillbreaksdownatnitetemperature
and onemay expet higher loop orretionto the CS
oeÆientat nitetemperature.
We have expliitly omputed the twoloop
orre-tion to the CS oeÆient at nite temperature [16℄.
Parameterizingtheself-energyinaovariantgaugeas
(p;u)=
1
(p;u)+i
p
2
(p;u) (49)
where u
represents theveloity of theheat bath, we
ndthat,in thestatilimit,thetwolooporretionto
theCSoeÆientathigh temperaturestakestheform
(2)
2
(0)=(2m 3M) e
4
2 2 ln
T
This result expliitlyshows that the Coleman-Hill
result breaks down at nite temperatures. F
urther-more, theform ofthe orretion is interestingin that
it diverges as m ! 0. This is the usual
manifesta-tion of infrared divergene and shows that, although
thezerotemperaturetheoryiswelldenedinthelimit
ofvanishing m,there areinfrareddivergenesat nite
temperature. Inaddition,sinethereisatwoloop
or-retiontotheCSoeÆient,onemayaskthestruture
oftheeetiveationwhenhigherorderorretionsare
taken into aount. This an bedetermined from the
largegaugeWardidentitythatwehavederivedandit
determinestheformoftheparityviolatingeetive
a-tion,satisfyinglargegaugeinvariane,at anyorder to
be
PV
=tan 1
2 0
(0)tan ea
2
Z
d 2
xB (51)
where 0
(0) representstheorretionto theCS
oeÆ-ienttothat order.
VII Non-Abelian theories:
Our main interestwasin the studyof the questionof
largegauge invarianeat nite temperaturein a
non-Abelian theory. As we have shown, the question of
largegaugeinvarianeiswellunderstoodinanAbelian
theory. However, not muh is known about the
non-Abelian theory yet. To that extent, let us note that
eventheColeman-HillresultwasderivedforanAbelian
theory. Inthissetion, letme desribebrieyhowthe
Coleman-Hillresultan begeneralizedtonon-Abelian
theories(atzerotemperature).
Thereareseveralqualitativedierenesbetweenthe
Abelianandthenon-Abeliantheories. First,whilethe
Abeliantheoryiswelldenedevenwhen thetreelevel
CSoeÆientvanishes,theinfrared divergenesinthe
non-AbeliantheoryaresevereifthereisnotreelevelCS
term. Seond,whiletheAbeliantheoryanbedened
in any gauge, the non-Abelian theory is well dened
only in a selet lass of infrared safe gauges suh as
theLandau gauge,theaxialgaugeet. Finally, inthe
Abeliantheory, the CS oeÆientis agauge
indepen-dentquantitywhilein thenon-Abeliantheory,theCS
oeÆient is gauge dependent. As a result, it is not
lear how to generalizethe Coleman-Hill resultin the
non-Abelianase.
On the other hand, it is known from various
ar-gumentsthat, in a non-Abelian theory, it is theratio
4m
g 2
whihhasaphysialmeaningand,therefore,must
begaugeindependent. In fat, aoneloop alulation
veriesthatinallinfraredsafegaugestheoneloop
or-retiontothisratioisgivenby
4m
g 2
! 4m
g 2
+N (52)
It is,therefore, meaningfultogeneralize the
Coleman-that needs to be quantized for large gauge invariane
tohold.)
Thisanindeedbedoneasfollows[17℄. First,letus
note that, although theCS oeÆientis gauge
depen-dent in general, ittakeson aphysial meaningin the
axialgauge. This isbeauseintheaxialgauge,
4m
g 2
! 4m
g 2
(1+
2
(0)) (53)
Sine this ratio is physial, in this gauge
2
(0) does
arryaphysialmeaning. UsingtheWardidentitiesin
theaxialgauge,itisstraightforwardtoshowthat the
CS oeÆientdoesnotreeiveanyorretionsbeyond
oneloop provided i)small agugeinvariane holdsand
ii)amplitudes areanalytiin themomentumspae. A
onsequeneofthisisthattheratio 4m
g 2
doesnothave
any orretion beyond one loop in this gauge.
How-ever, this is a gauge independent quantity and hene
it holds in any gauge that this ratio doesnot reeive
anyorretionbeyondone loop. Thus, weunderstand
somefeaturesofthezerotemperaturenon-Abelian
the-ory whih areparallelto theAbelian theoryandwhat
remains isto understand systematiallyifthe issueof
large gauge invariane also gets resolved in a parallel
manner.
VIII Conlusion:
In this talk, we have disussed the question of large
gauge invariane at nite temperature. We have
dis-ussedthe resolutionof theproblemin asimple0+1
dimensionalmodel. WehavederivedtheWardidentity
for largegaugeinvarianein this model. Wehave
an-alyzed the boxdiagram in 2+1 dimensionsand have
obtained theform of the eetive ation at zero
tem-perature. Wehavealsoobtainedtheamplitudeaswell
asthequartieetiveationsinthelongwaveaswell
as stati limits, at nite temperature. The LW limit
hasonlyextensivetermsintheationwhihgoesas 1
T
athightemperatureandisinvariantunderlargegauge
transformations. Theleadingterminthestatiation,
however, isnon-extensive, goesas 1
T 3
at high
temper-ature and oinides withthe eetiveation proposed
earlierforarestritivegaugebakground. Thisationis
notinvariantunderlargegaugetransformations.
How-ever,using alargegaugeWardidentity, wean
deter-mine the full leading order ation in the stati limit
whihoinides withtheeetiveationobtainedin a
restritivegauge bakground. We have shown
expli-itly that higher loop orretions to the CS oeÆient
do notvanish at nite temperature. This violationof
theColeman-Hillresultisaonsequeneofthefatthat
oneoftheirassumptions(namely,analytiityofthe
am-plitudes) breaks down at nite temperature. We have
It is a plaesure to thank the organizers for their
hospitality. Ihavelearntalotfromallofmy
ollabora-torsintheeld,namely,K.Babu,J.Barelos-Neto,F.
Brandt,A. J.daSilva,G. Dunne,J.Frenkel,P.
Pani-grahi,K.RaoandJ.C.Taylor,andIamgratefultoall
of them. This workwassupported inpartbytheU.S.
Dept. ofEnergyGrantDE-FG02-91ER40685.
Referenes
[1℄ Seeforexample,A.Das,FiniteTemperatureField
The-ory,WorldSienti,1997.
[2℄ A. Das and M. Hott, Mod. Phys. Lett. A 9, 3383
(1994).
[3℄ S. Deser, R. Jakiw and S. Templeton, Ann. Phys.
140,372(1982).
[4℄ K. S.Babu, A. Dasand P. Panigrahi, Phys.Rev.D
36,3725 (1987);I.AithisonandJ.Zuk, Ann.Phys.
242,77(1995);N.Brali,C.FosoandF.Shaposnik,
Phys.Lett.B 383, 199(1996); D. Cabra, E. Fradkin,
G.RossiniandF.Shaposnik, Phys.Lett.B383,434
(1996).
[5℄ G.Dunne,R.JakiwandC.Trugenberger, Phys.Rev.
D41,661(1990).
[6℄ G. Dunne, K. Lee and C. Lu, Phys. Rev.Lett. 78,
3434 (1997).
[7℄ A.DasandG.Dunne, Phys.Rev.D57,5023(1998).
[8℄ A.Das,G.DunneandJ.Frenkel,Phys.Lett.B472,
332(2000).
[9℄ J.Barelos-NetoandA.Das, Phys.Rev.D58,085022
(1998);J.Barelos-NetoandA.Das, Phys.Rev.D59
087701 (1999); A.Das andG. Dunne, Phys.Rev.D
60,085010(1999).
[10℄ A. Dasand A. J. daSilva, Phys. Rev.D 59,105011
(1999).
[11℄ S.Deser, L. Griguolo and D. Seminara, Phys.Rev.
Lett. 79, 1976 (1997); C. Foso, G. Rossini and F.
Shaposnik, Phys. Rev. 79, 1980 (1997); S. Deser,
L.GriguoloandD.Seminara, Phys.Rev.D57,7444
(1998);C.Foso,G.RossiniandF.Shaposnik, Phys.
Rev.D56,6547(1997).
[12℄ A.DasandM.Hott, Phys.Rev.D50,6655(1994).
[13℄ J.Frenkel and J. C.Taylor, Nu. Phys.B374, 156
(1992);F.T.BrandtandJ.Frenkel, Phys.Rev.D56,
2453(1997).
[14℄ F.T.Brandt,A.DasandJ.Frenkel, Phys.Rev.D62,
085012 (2000); F. T. Brandt,A. Das and J. Frenkel,
hep-ph/0004195.
[15℄ S.ColemanandB.Hill,Phys.Lett.B159,184(1985).
[16℄ F.T.Brandt, A.Das, J.Frenkeland K.Rao, Phys.
Lett.B492,393(2000).
[17℄ F.T.Brandt,A.DasandJ.Frenkel, Phys.Lett.B494,
339(2000);F.T.Brandt,A.DasandJ.Frenkel,
