The Spetrum of D/ in QCD via Replias
D. Dalmazi
UNESP,Campusde Guaratingueta, DFQ
Av.Dr. AribertoPereiradaCunha, 333
CEP12500-000, Guaratingueta, SP,Brazil
Reeivedon15January,2002
ThisisashortreviewofthederivationofthespetrumoftheDiraoperatorinQCD4 inanite
volumeV bymeansofthe repliatrik. Thederivationis nonperturbativeintheQCD oupling
anditis inagreementwithlattie resultsas wellas hiralrandom matrixtheory(ChRMT).Our
resultsholdintheenergysalewherehiralsymmetryis brokenandthePionwavelengthismuh
biggerthanthesizeofthesystem( >>>V 1=4
).
I Introdution and motivation
Inthissetionwewillreallsomebasipropertiesofthe
spetrumof the Diraoperator in QCD
4
and explain
whyisitinterestingtostudysuhoperator,forareview
papersee[1℄. First,letusstartfromthepartition
fun-tionofQCD
4
in Eulideanspaewhihorrespondsto
integrateoverthegluonsandN
f
avorsofquarks.
Z
Nf =
Z
[dA℄e S
YM N
f
Y
f=1 D
f D
f e
R
dV
fD f+m/
f f
= Z
[dA℄e SY
M
[det(D/ +m)℄ N
f
d
where S
YM
represents the Yang-Mills ation for the
gluonsand
/ D = (
+{A
)
; A
=A
a
T
a
;a=1;;N 2
1
Theformulasinthissetionandinthenextoneholdfor
arbitraryinteger N
f
and N
. In Eulidean spaeall
gammamatriesanbehosenHermiteanandtherefore
theDira operator is anti-Hermiteanand its
eigenval-uesarepureimaginarynumberswhihshowupinpairs
{
k
and {
k
dueto
5 /
D= D/
5
. That is,
/ D y
= D/
/ D
k
= {
k k
k 2<
/ D(
5 k
) = {
k (
5 k )
Thepairingarosstheorigin
k
=0guaranteesthe
allowus touse theVaa-Wittentheorem[2℄ toderive
the hiralsymmetry breaking pattern. Positivenessof
thepartitionfuntion anbeeasilyseenasfollows,
Z
N
f =
Z
[dA℄e SY
M
[det(D/+m)℄ Nf
= Z
[dA℄e SY
M N
Y
k =1 ( {
k +m)
N
f
( {
k +m)
N
f
= Z
[dA℄e SY
M N
Y
k =1
2
k +m
2
N
f
InD =3wedonothavesuhapairingofeigenvalues
sinethereisno
5
,onsequentlythepartitionfuntion
forQCD
3
doesnothaveadenitesigningeneral[3,4℄.
ForevenN
f
oneanhoosehalfofthemassesasparity
main motivation to study the spetrum of the Dira
operator.Morepreisely,oneanndadiretrelation
betweenspontaneoushiralsymmetrybreakingandthe
densityofzeroeigenvalues. Beforewegettherewerst
reall how spontaneous symmetry breaking anour
in the simplerexampleof the2D Isingmodelin a
ex-ternal magnetieldsinethesituationisverysimilar
to QCD.
The partition funtion of the Ising model is given
byasumoverallspinongurationsoneahsiteofthe
lattie,i.e.,
Z
Ising =
X
fig e
J P
<ij> ij+H
P
N
i i
(1)
Where J > 0 and H are onstants and we have only
two possibilities for the spin on eah site
i
= 1.
Without magneti eld H = 0wehave a Z
2
symme-try
i
!
i
sine the nearest neighbor interation
is quadrati. Turning on the magneti eld webreak
expliitly theZ
2
symmetryhoweverweexpetthat in
the limit H ! 0we reoverit. It turns outthat this
is nottheaseand atlowtemperatures,evenwithout
2
measuretheamountofspontaneoussymmetrybreaking
byomputingthemagnetization:
m = lim
H!0 1
N
H logZ
Ising =
h N
up N
down i
H!0
N
(2)
It is assumed that we rst take the thermodynami
limit N ! 1 then H ! 0. Whenever m 6= 0 the
Z
2
symmetry is spontaneously broken. In QCD
4 the
roleof themagneti eld isplayed by themassof the
quarksm whose term breaks thehiralsymmetry
ex-pliitly. Theroleofthemagnetization isplayedbythe
hiralondensate<
f f >:
<
f f >
QCD(m!0) = lim
m!0 1
N
f V
m logZ
N
f
(3)
One again we take V ! 1 before m ! 0. If
<
f f
>6= 0 the hiral symmetry is broken
spon-taneously. Nextwederiveadiretrelationbetweenthe
hiralondensateandthespetraldensityoftheDira
operatorinQCD
4
. Asarststepwederivethe
parti-tionfuntion w.r.t. thequarkmass,
1
N
f
m Z
N
f =
Z
[dA℄e SY
M N
Y
k =1
2
k +m
2
Nf N
X
k =1 2m
2
k +m
2
d
Nowusingthefollowingrepresentationof Dirasdelta
funtion,
Æ(x) = 1
2 lim
m!0
1
{x+m +
1
{x+m
= 1
lim
m!0 m
x 2
+m 2
(4)
and thedenition ofthe average marosopispetral
density():
() =
*
N
X
k =1
Æ(+
k )
+
QCD(m)
(5)
It isasimpleexeriseto provethefollowingdiret
re-lation betweenspontaneoushiral symmetry breaking
andthespetrumof theDiraoperator[5℄:
lim
m!0 j<
f f >j =
V
(0) (6)
From(6)weseethatinordertohavespontaneous
hi-ralsymmetry breaking wemust havea non-vanishing
average spetraldensityattheoriginofthespetrum.
Heneforthwewillbeinterestedinalulatingthe
spe-traldensityoftheDiraoperatorloseto theoriginof
thespetrumin order to ompare withsimilar results
obtainedinlattie[7℄QCDandthroughChRMT[1℄℄.
II Partial quenhing
Asarststepitisonvenienttowritetheaverage
() = 1
2 "
Tr 1
/ D+z
z={+
Tr 1
/ D+z
z={ #
QCD(m)
d
Notiethattheprobemasszinthelastequationis
dierent fromthe originalquarkmasses m. Themass
z is alled valenequarkmass. In order toobtainthe
spetraldensitywejust havetoalulate (z)dened
below
(z) =
Tr 1
/ D+z
m
(7)
Nowthequestionis: Howdoweobtain(z)? Byusing
themathematialidentity:
Tr 1
/ D+z
= 1
det(D/+z)
z
det(D/+z) ; (8)
there are at least three dierent ways to proeed by
introduingextra vallenequarksasweexplain in the
nextthreesubsetions.
II.1 SupersymmetriMethod
Introduingtwoextravallenequarksofopposite
statis-tisinthepartitionfuntion:
Z
Nf+1;1 =
Z
dAe S
YM [A℄
det(D/+z+J)
det(D/+z)
[det (D/+m)℄ N
f
d
weanobtain(z)using(8),
(z) =
Tr 1
/ D+z
mf
= lim
J!0
J logZ
Nf+1;1
Eah method has its own advantages and
prob-lems and they all depend on a onvenient hoie of
parametrizationofthemanifoldoftherelevantdegrees
offreedom. Thesupersymmetrimethodbeomes
um-bersomeforhigherpointspetraldensitiesandrequires
an extension of the hiral symmetry breaking pattern
tothesupersymmetriasebutitfurnishes[6℄
nonper-turbativeresults(in thevalenequarkmassz )whih
areinfullagreementwithlattie[7℄simulationsandthe
ChRMT(see[1℄). Tolearnmoreaboutthismethodsee
thebook [8℄.
II.2 Bosonireplias
Inthisaseweintrodueanarbitrarynumberof
ex-trabosoniquarkswiththesamemassz,alledreplias
Z
N
f ;n
= Z
[dA℄e SY
M [A℄
[det (D/ +m)℄ N
f
det (D/+z) n
:
Andobtain(z)bymeansofthereplialimitn!0:
(z) =
Tr 1
/ D+z
m
f
= lim
n!0 1
n
z logZ
N
f ;n
II.3 Fermionireplias
This aseis similar to thelast one,but insteadof
bosoni replias we now add an arbitrary number of
fermioniquarkreplias,
Z
n+N
f =
Z
dAe
SYM[A℄
det (D/+z) n
/ D+z
mf
n!0
n
Bothreplia methodsanonlybeused,in general,
perturbatively in the vallene quarkmass z. It turns
outthatbosonirepliasdonotwork[11℄.
Mathemat-ially,thereasonforthefailure isthelakof areplia
symmetry breaking saddle point to expand around.
Fromnowonwewillonentrateontheuseoffermioni
replias. From (9) we see that all weneed is the
ex-tended partition funtion Z
n+Nf
(z;m). We need to
knowthepartition funtionat lowenergieswhere
hi-ralsymmetryisbroken. Thisisthesubjetofthenext
setion.
III Low-energy QCD
When a ontinuum symmetry G is broken down to a
subgroup H spontaneously, then Goldstone theorem
Ontheotherhand,itisbelievedthat hiralsymmetry
isbrokenspontaneouslyin QCD atlow-energy. Sine
at low-energy (large distanes) we expet the lightest
partiles to play a major role, it is natural from the
abovetoexpetthat theGoldstonebosonsarethe
rel-evantdegreesoffreedominlow-energyQCD. In
pra-tie however we should take into aount that hiral
symmetry is only an approximate symmetry of QCD
beausethequarksare massivesothelow-energy
the-ory should also depend on the quark masses and in
away that it reprodues the ovariane properties of
theQCD partitionfuntionunderhangesofthemass
matrix. AssumingalsoLorentzovarianeandthe
ex-isteneofthethetavauumoneanarriveatthehiral
eetiveLagrangianbelow(see[12,13℄)atleading
or-derin aderivativeexpansion,
L
hiral =
F 2
4 Tr(
~
U
~
U y
)
2 Tr(e
{
N
f
M y
~
U+e {
N
f
M ~
U y
)
d
Where ~
U 2G=H,thequantityF isthepiondeay
on-stantand
is the innite volume limit of thehiral
ondensate. The quarkmass matrix M is in
prini-ple, without replias, a N
f N
f
matrix. The oset
G=H depends on the fermion representation and on
the number of olours, For instane, for fermions in
the fundamental representation and N
3, i.e., for
the Dyson index = 2, the QCD ation in the
hi-rallimit m!0is invariantunder independentglobal
rotationsofleftandright-handedfermionswhihleads
to a U
L (N
f )U
R (N
f
), however when all the angles
of the rotation are equal we have a U(1)
transfor-mation whih is known to be broken by the hiral
anomaly, thus reduing the quantum axial symmetry
to SU
L (N
f )SU
R (N
f
). If the hiral ondensate is
nonvanishing the hiral symmetry is broken down to
the subgroup of rotationswhih are equal to left and
right-handedfermions,i.e., vetorSU(N
f
)sinethose
annotbebrokenspontaneouslyaordingtothe
Vafa-Witten theorem. Thereforeweend upwith the oset
(SU
L (N
f )SU
R (N
f
))=SU(N
f
) whihisequivalentto
G=H = SU(N
f
). The other ases = 1;4 an be
similarlyderived(see[13℄foralearderivation ).
Theeetive Lagrangian L
hiral
is agood
approx-imationwhen distanesarelargeenough(low-energy),
i.e.,V 1=4
>>1=
QCD
andthequarkmassesaresmall.
Theregime where weare interestedin orresponds to
furtherassumethatthetypialpionwavelengthismuh
biggerthanthesizeofthesystem,i.e.,1=m
=
>>
V 1=4
Inthis spei domain thepion wavefuntion is
likeaonstantinsidetheboxandthezeromomentum
mode U
0
dominates,suh that weannegletthe
ki-netitermin L
hiral
. Weanwrite ~
U =
UU
0
andthe
in-tegraloverthenonzeromodes
U willbeome,inthe
in-nitevolumelimit,anoverallfatortotheintegralover
theonstantmodeU
0
heneforthalledsimplyU. For
=2weshouldhaveanintegraloverSU(N
f
),however
in the setorof xed topologial harge (number of
instantonsminusthenumberofanti-instantons)wean
enlargeit toan integraloverU(N
f
) asfollows. First,
onean obtainthe partition funtion in the setor of
xedtopologialhargeZ (N
f )
fromthepartition
Z (N
f )
=
Z
2
0 de
{
Z (N
f )
()
= Z
2
0 de
{ Z
SU(Nf) dUe
1
2 Tr(e
{
N
f
MU y
+e {
N
f
M y
U)
Z (N
f )
=
Z
U(N
f )
dU ( detU)
e 1
2 Tr(MU
y
+M y
U)
; (=2) (10)
d
Notie that we have ombined ddU(SU(N
f )) !
dU(U(N
f
))and M=mV
.
IV Spetral density via replias
Now that weknowthe partition funtion for an
arbi-traryquarkmassmatrixweanreturntoourfermioni
repliasformula:
Tr 1
/ D+z
QCD() = lim
n!0 1
n
z logZ
(Nf+n)
andinsteadoftheoriginalquarkmassmatrixwehave
extranmassesz:
M=diag(m
1 ;;m
Nf
;z;;z)
As afurthersimpliation werestritourselvestothe
quenhed (N
f
=0) asewherethe quarkmass matrix
onlyontainstherepliasmasses:
M=z1
nn
(11)
.
Withtheorrespondingpartitionfuntion:
Z (n)
(z) = Z
U(n)
dU (detU)
e z
2 Tr(U+U
y
)
(12)
It is very fortunate that there is an exatformula
forZ (n)
(z), atleast forintegernumberof replias n,
wehaveuptoanoverallonstant[14℄:
Z (n)
(z) = det(I
+i j
(z)) ; i;j=1;;n (13)
With I
(z) being a modied Bessel funtion of rst
kind. For example, for n = 1 and n = 2 replias we
Z (n=1)
= I
(z)
Z (n=2)
= I 2
(z) I
+1 (z)I
1 (z)
Although formula(13)isexatitis onlytruefor
inte-gernumberofreplias,but inordertotakethereplia
limit n ! 0 we need to know the partition funtion
Z (n)
for arbitrary omplex number of replias. That
is, we need to ontinuethe partition funtion to
non-integer number of replias whih is the most diÆult
tehnial problem in replia alulations. We still do
not know how to do itfor arbitrary nite massz but
one aneither do perturbations forz !0(Taylor
ex-pansions)or takez ! 1(saddlepointalulations).
It turns out that, in general, the rst approximation
an not be analytially ontinued to arbitrary
num-ber of replias due to the presene of the so alled
t'Hooft - De Wit poles. On the other hand the
sad-dle point approximation is muh more suitable. For
largez theintegral(12)isloalized aroundthe saddle
points whih are solutions of the saddle point
equa-tions Æ(U +U y
) = 0 whih imply U 2
= 1. Those
equations have innitesolutions whih andivided in
n+1 lasses labelled by an integer p = 0;1;;n.
Eahlassofsolutionisrepresentedbyitsdiagonal
el-ement U
p
= diag(1;1;;1; 1;; 1) where the
labelpountsthenumberofentries 1inthediagonal
and onsequently n p entries +1. Foragiven value
ofpwehave
e z
2
Tr(Up+U y
p )
= e (n 2p)z
(14)
Thereforeitislearthatforz!+1thesolutionp=0
(replia symmetri) dominates and all other solutions
are exponentially suppressed. However, there omes
an importantsubtlety. Namely, we areatually
inter-ested in the spetral density whih will be alulated
betweenthem is innitesimally small all saddle point
lassesshouldinpriniplebetakenintoaount. Thus,
weshould sumoverallofthem,
Z (n)
(z) = n
X
p=0 Vol
U(n)
U(n p)U(p)
Z
dV
p ( detV
p )
e z
2 Tr(V
p +V
y
p )
(15)
where wehaveusedthedeomposition
U =U
0 V
p U
y
0 ; U
0 2
U(n)
U(n p)U(p)
and fatoredoutthevolumeofredundantsolutionsin
eah of the n+1 lasses. The unitary matrix V
p is
nn and blok-diagonal. It is onvenient to write
V
p
in termsofmatriesH andhwhih areHermitean
(n p)(n p) and pp respetively, suh that
theexpansionaroundthediagonalsaddlepointsmeans
V
p =
1 iH =2
1+ih=2
1 ih=2
nn
= 0
1 H
2
2 +
1+ h
2
2 +
1
A
;
The integral over dV
p
now redues to integrals over
Hermiteanmatrieswhihanbeeasily evaluated and
ontinuedtonon-integernumberofreplias. Typially,
forinstane forp=0,wehavefor z!1,
Tr(H 4
)
= R
dH
nn e
(z=2)Tr(H 2
)
Tr(H 4
)
R
dHe
(z=2)Tr(H 2
)
= n+2n
3
z 2
(16)
Theresultonther.h.s. of (16)is ananalytifuntion
of the number of replias and so will be the integral
over dV
p
in (15). Still we have to ontinue the sum
and the volume fatorin (15) to arbitrary number of
replias. Fortunately,the samevolume fatorappears
in[17℄, pluggingbak theirontinuationin ourresults
itamountsto takeinto aountonly p=0and p=1
saddlepointswhiletheothersaddlepointsdropoutin
thereplia limit. Consequently,
Tr 1
/ D+z
=
Tr 1
/ D+z
;p=0 +
Tr 1
/ D+z
;p=1
Besides the real ontribution of the p=0 saddlepointalulated in [15, 16℄ thehiralondensate(z)gets an
imaginarypartommingfromthenontrivialp=1saddlepoint[11℄:
Tr 1
/ D+z
= 1
i( 1)
e 2z
2z
+ (4
2
1)(1 i( 1)
e 2z
)
8z 2
+
From the disontinuity equation we nally obtain the mirosopi spetral density of the Dira operator in the
quenhedase(N
f !0):
s ()=
1
2 "
Tr 1
/ D+z
z={+ +
Tr 1
/ D+z
z= {+ #
;Nf=0
s () =
1
h
1
os(2 )
2
+
1 4
2
8 2
(1 sin(2 )
+ (4 2
1)(4 2
9)
os(2 )
64 3
+
+ (4 2
1)(4 2
9)
( 6+(19 4 2
)sin(2 ))
2 7
3! 4
Ourresultsholdforarbitrarytopologialharge and
theosillatingtermsallamefromthenontrivialsaddle
point p =1. Using asymptoti expressions for Bessel
funtionswefoundfullagreementwiththe
orrespond-ingChRMTanalytialresult:
ChRMT
s
() = jj
2
J
()
2
J
+1 ()J
1 ()
untiltheorder1= 4
wherewestoppedouralulation.
It is remarkable that for half-integer values of the
largemassexpansionterminatesandthereplia result
beomesexat .
V Conlusions and perspetives
Wehavebeenabletoalulatethemirosopispetral
density
S
() oftheQCDDiraoperatorlosetothe
origin of the spetrum by means of fermioni replias
of quarks. Our results are perturbative and hold for
large arguments ! 1. In this region theyoverlap
previousresultsobtainedviaChRMT,lattie
simula-tions and the supersymmetri method. Contributions
ofallsaddlepointsisruialfortheorretresult. We
haveassumedtheontinuationtoarbitrarynumberof
repliasofthework[17℄whihhasnotyetbeenproven
to be unique. For instane, asommentedin [18℄,we
ould add to our partition funtion, dened for
inte-gernumberofreplias,anarbitraryextrafatorwhih
vanishesforintegernumberofrepliasbutgivesa
non-trivialontributioninthereplialimitlike,e.g.,
lim
n!0 1
n
Z (n)
+asinn
= lim
n!0 1
n Z
(n)
+a
sinetheextratermisarbitrarysoitisthereplia
re-sult. Clearly we might try to nd relations between
the partition funtion for dierent number of avors
and topologial harges to avoidthe addition of suh
arbitrarytermshoweverwestilldonothaveaproofof
uniquenessofourrepliaresult.
Theoppositelimitz!0annotbeontinuedinn
in general,thoughwefound forthespeialase =0
that (z)=z=2+ . This isaverynontrivialresult
[11℄ whih isalso in agreementwith other non-replia
alulations(see[1,6℄).
Conerning future alulations we should mention
that we have assumed N
3 and fermions in the
fundamental representation ( = 2) but there are
other ases N
= 2( = 1) and adjoint fermions
( =4) whihmightbeexploredaswellinD=4and
D =3. Besides, inlusion ofhemialpotential6=0
is phenomenologiallyand mathematially interesting
(see[1, 19℄). The use of Virasoro onstraints around
ofinterestifwewanttounderstandloalorrelatorsas
xedbytheVirasoroonstraints[20℄.
Aknowledgments
ThisworkisaresultofaollaborationwithJ.J.M.
Verbaarshotduringmysabbatialleave(sponsoredby
FAPESP)atthePhysisDepartmentofSUNYatStony
Brook. Theauthor ispartiallysupportedbyCNPq.
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