Chiral phase transition in a random matrix model with three flavors

0.32 MB·English

by  Hirotsugu Fujii

Chiral phase transition in a random matrix model
0 with three flavors
1
0
2

n
a
J

0 HirotsuguFujii
2
InstituteofPhysics,UniversityofTokyo,Tokyo153-8902,Japan

]
t
a MunehisaOhtani
l
-
p PhysicsDepartment,SchoolofMedicine,KyorinUniversity,Tokyo181-8611,Japan
e
h TakashiSano
∗
[ DepartmentofPhysics,UniversityofTokyo,Tokyo113-0033,Japan

1 InstituteofPhysics,UniversityofTokyo,Tokyo153-8902,Japan
v E-mail:[e-mail]
0
4
6
The chiralphase transition in the conventionalrandommatrix modelis the second order in the
3
. chirallimit,irrespectiveofthenumberofflavorsNf,becauseitlackstheUA(1)-breakingdeter-
1
0 minantinteractionterm. Furthermore,itpredictsanunphysicalvalueofzeroforthetopological
0
susceptibility at finite temperatures. We propose a new chiral random matrix model which re-
1
: solvesthesedifficultiesbyincorporatingthedeterminantinteractiontermwithintheinstantongas
v
i picture. Themodelproducesasecond-ordertransitionforNf =2andafirst-ordertransitionfor
X
N =3,andrecoversaphysicaltemperaturedependenceofthetopologicalsusceptibility.
f
r
a
TheXXVIIInternationalSymposiumonLatticeFieldTheory-LAT2009
July26-312009
PekingUniversity,Beijing,China
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Chiralphasetransitioninarandommatrixmodelwiththreeflavors TakashiSano
1. Introduction
One of the prominent features of non-perturbative QCD is spontaneous breaking of chiral
symmetryinthelightquarksector. AccordingtotheBanks-Casherrelation[1],theorderparameter
q¯q isrelated tothe Diracspectral density atzero eigenvalue inthethermodynamic limit. Within
h i
the so-called e regime characterized by the system size L4 such that mp 1/L 1 GeV, the
≪ ≪
QCD partition function is dominated by the constant pion field configurations. In this regime,
universal properties of the Dirac eigenvalue distribution near zero are legitimately analyzed in a
chiralrandommatrix(ChRM)theory[2],wherethekinetictermoftheDiracoperatorisdiscarded
and the complexity of the gauge field dynamics is represented by treating the Dirac operator as
a random matrix of constant modes. The matrix size of the constant modes 2N is considered to
be proportional to the system volume. By taking a large volume limit N ¥ , away from the e
→
regime, one can study thermodynamics of the ChRM theory as a schematic mean-field model for
QCD. One finds the ground state of the model in the chirally broken phase and can explore the
modelphasediagram byintroducing thetemperaturet [3]andthequarkchemicalpotential m [4].
There are two drawbacks, however, in the ChRM model, concerning the U (1) symmetry.
A
Firstly,theU (1)-breaking determinant termismissingintheeffectiveactionofthemodel,which
A
consequently predicts a second-order phase transition at finite temperature for any number of the
quark flavors N . Secondly, the topological susceptibility is suppressed unphysically to vanish at
f
finitetemperatures.
In an earlier work [5], the appropriate form of the ChRM model with a U (1)-breaking term
A
is speculated from aquark model withthe determinant interaction in0+1 dimensions. Inaddition
totheconstant modesintheconventional ChRMmodel, newconstant modesareintroduced soas
to reproduce the determinant interaction. These new modes are considered to be associated with
instantons andcalledtopological zeromodes. Howevertheeffectivepotentialofthestartingquark
modelisunbounded frombelowforN =3,andtherefore nophysical groundstateexists.
f
Inthis paper wepropose a new ChRMmodel [6]changing the distribution ofthe topological
zeromodes,whichresultsinthedeterminanttermappearing underalogarithm. Thentheeffective
potentialbecomesstableforanyN ,anddescribesasecond-orderphasetransitionforN =2while
f f
afirst-ordertransitionforN =3. Moreover,thetopologicalsusceptibilityshowsphysicalbehavior
f
asafunction oftemperature inourmodel.
2. Chiral random matrix model atfinite temperature
In analogy with the QCD partition function, the ChRM model with N flavors of mass m is
f f
definedasanaverageofthequarkdeterminants [3]
Nf
Zn = dWe−NS 2trW†W (cid:213) det(D+mf), (2.1)
Z f=1
wheretheDiracoperatorhasbeenreplacedwithananti-Hermitematrixofconstant modes
D= iW†+it01N n /2 iW+it01N−|n |/2! (2.2)
−| |
2
Chiralphasetransitioninarandommatrixmodelwiththreeflavors TakashiSano
with an (N+n /2) (N n /2) complex matrix W. The Gaussian random distribution of W is
× −
a simple realization of the complexity of the gluon dynamics. Chiral symmetry is retained as a
fact D,g5 =0with g5 =diag(1N+n /2, 1N n /2). The temperature effect has been introduced in
{ } − −
the Dirac operator withaconstant t, which maybe interpreted as the lowest Matsubara frequency
p T. One can easily show that this matrix D has n exact zero eigenvalues, which are interpreted
| |
as the zero modes accompanied by the topological charge n . The complete partition function is
obtained after the sum over n weighted by the quenched topological susceptibility t of the pure
gluondynamics:
Zq = (cid:229)2N e−2(2nN2)t einq Zn , (2.3)
n = 2N
−
wheretheq anglehasbeenintroduced.
After rewriting the determinant with the fermion variables and doing the Gaussian integral
of W in Zn , we find a four-fermion vertex interaction, which can be unfolded by introducing an
N N auxiliary variable S q†q . Weperform the fermion integral to obtain an expression for
f × f ∼ L R
Zn intermsofS,
n det(S+M)n (n 0)
Zn =Z dSe−NS 2tr(SS†)det (S+M)(S†+M)+t2 N−|2| ×(det(S†+M†)−n (n ≥<0) , (2.4)
(cid:0) (cid:1)
whereM =diag(m ,...,m ).
1 Nf
Inthethermodynamiclimitwesetn =0andevaluateEq.(2.4)withthesaddlepointequation.
ForS(cid:181) 1 andM =0, theN dependence isfactored outinthe saddle point equation, implying
Nf f
a second order phase transition for any number of Nf. Furthermore, Zn is non-analytic in n as
theintegrand contains atermwith n ,whichcauses theunphysical suppression ofthetopological
| |
susceptibility atfinitetemperatures [7]. Notethat n disappears whent =0.
| |
3. Chiral random matrix model withdeterminant interaction
Near-zeromodesandtopological zeromodes–Letusfirstrecalltheinstantongaspicture. An
isolated instanton isa localized object accompanying aright-handed exact fermion zero mode. In
a dilute system of N instantons and N anti-instantons, we expect N right-handed and N left-
+ +
− −
handed zero modes even at finite temperatures. In an effective theory at long distances, effects of
the instantons should be integrated out, which willresult inU (1)-breaking effective interactions.
A
Thefundamentalassumption inourmodelistheclassification oftheconstantmodesintothenear-
zeromodesandthetopologicalzeromodes[5,6]. Wedealwiththe2N near-zeromodesappearing
intheconventional modelsandadditionally theN +N topological zeromodeswhichweregard
+
−
asthemodesaccompaniedbytheinstantons. Inourmodel,N fluctuateaccordingtotheinstanton
distribution, andthenumberoftheexactzeromodesisgiven±byn =N N . Eventuallywesum
+
− −
overN andN assumingacertaindistribution withthemeanvalueofO(N).
+
−
WewritedownaGaussianChRMmodelfordefinitenumbersofzeromodesas[5,6]
Nf
ZN = dAdBdXdYe NS 2tr(AA†+BB†+XX†+YY†)(cid:213) det(D+m ) (3.1)
N+,N − f
− Z f=1
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Chiralphasetransitioninarandommatrixmodelwiththreeflavors TakashiSano
with
0 iA+it1 0 iX
N
iA†+it1 0 iY 0
D= N  , (3.2)
0 iY† 0 iB
 
 iX† 0 iB† 0 
 
 
wherethematrixAcorrespondstothenear-zeromodesoftheconventionalmodelandtheN N
+
× −
matrixBrepresentsthetopological zeromodes. ThematricesX andY inducemixingamongthese
modes. Notethatthetemperaturetermt isintroduced onlyforthenear-zeromodes. Followingthe
same steps as in Sec. 2, we find the sigma model representation for this ChRM model, which is
analytic inN aswellasN unlikeEq.(2.4):
±
ZNN+,N = dSe−NS 2trS†S det (S+M)(S†+M†)+t2 N det(S+M)N+ det(S†+M†)N−. (3.3)
− Z
(cid:2) (cid:3)
Distribution ofthetopological zeromodes–Thecomplete partition function isobtained after
summing ZN over the instanton numbers N and N . Here we simply assume independent
distributionsN+P,N(N− )forN andN ,i.e., + −
+
± −
Zq =N+(cid:229) ,N einq P(N+)P(N−)ZNN+,N− =Z dSe−2NW (S;t,m,q ). (3.4)
−
LetusfirstconsiderP(N )inadiluteinstantongaspicture. Foraone-instanton configuration, one
may assign a weight k c±ompared with a no-instanton configuration, and multiply a factor N (cid:181) V
taking into account the integration over the instanton location. For a configuration with N
+( )
−
(anti-)instantons, wethenhaveaPoissondistribution
1
P (N )= (k N)N (3.5)
Po ± N ! ±
±
where the factorial N ! appears as the symmetry factor. The summation with P (N ) in
+( ) Po
− ±
Eq.(3.4)resultsintheexponentiation ofthedeterminant term[5]:
W =1S 2trSS† 1lndet (S+M)(S†+M†)+t2 1k [eiq det(S+M)+e iq det(S†+M†)].
Po 2 −2 −2 −
(3.6)
(cid:2) (cid:3)
This determinant term is commonly incorporated in effective models as the U (1) anomaly term.
A
HoweverthispotentialisunboundforN =3intheChRMmodelbecausethetermdet(S+M)
f
∼
f 3 forlargeS=f 1 dominates overtheothertermsinW .
Nf Po
It should be noticed here that the fermion coupling distorts the N distribution itself. With
±
includingthedeterminanttermofthetopologicalzeromodesinEq.(3.3),theeffectivedistribution
forN reads
+
1
P (N )= (k Nd)N+ (3.7)
Po +
N !
+
with d = det(S+M), and similarley for N . This means that the average value of N increases
indefinitel|y with increa|sing d f Nf as N −=k Nd. However, the possibility of infin±itely many
∼ h ±i
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Chiralphasetransitioninarandommatrixmodelwiththreeflavors TakashiSano
f
f
1.4 1.4
1.2 1.2
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2 0.2 0.2
0 0.15 0 0.15
0 0.5 0.1 m 0.6 0.8 0.1 m
t 1 1.5 2 0 0.05 t 1 1.2 1.4 0 0.05
Figure1:Chiralcondensatef asafunctionoft andmforN =2(left)and3(right).
f
degrees offreedom N within afinitevolume isinadequate for aregularized low-energy effective
±
theory. WeneedacutoffforN .
±
HerewesetexplicitlyamaximumvalueofO(N)forN . Wesplitthefinitespace-timevolume
intog N cellswithg beingaconstantofO(1),andassignap±robability pforacelltobeoccupiedby
asingle(anti-)instanton and(1 p)foracellunoccupied. Thisassumption resultsinthebinomial
−
distributions forN :
±
g N
P(N )= pN (1 p)gN N . (3.8)
± N ! ± − − ±
±
Forasmall pandalarge g N,thebinomial distribution P(N )isaccurately approximated withthe
Poisson distribution withthemeang Np. Butitcannot bea±good approximation foralarge p. The
binomialdistributionprovidesastringentupperboundg N forthenumberofmodesN ,incontrast
±
tothePoissondistribution. Thecorresponding effectivepotential forSisfoundtobe
W (S;t,m,q )=1S 2trSS† 1lndet (S+M)(S†+M†)+t2
2 −2
1g ln eiq a det((cid:2)S+M)+1 +ln e iq a(cid:3)det(S†+M†)+1 (3.9)
−2 −
h (cid:16) (cid:17) (cid:16) (cid:17)i
witha = p/(1 p).
−
Thevarianceofthetopological chargen =N N forthebinomialdistribution iscomputed
+
as 2Nt =2Ng p(1 p), where t isthe quenched to−polo−gical susceptibility. In the presence of the
−
fermioncoupling, thissusceptibility willbereplaced with
a d
t˜=g p˜(1 p˜)=g . (3.10)
− (a d+1)2
4. Ground stateand fluctuations
In this section we shall study ground state properties of the system with equal mass, M =
m1 ,forsimplicity. SettingS=f 1 withrealf andwithq =0,weobtain asimpleformofthe
Nf Nf
grandpotential:
W (f ;t,m)=1N S 2f 2 1N ln (f +m)2+t2 1g ln a (f +m)Nf +1 2 . (4.1)
2 f −2 f −2
(cid:2) (cid:3) (cid:12) (cid:12)
(cid:12) (cid:12)
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Chiralphasetransitioninarandommatrixmodelwiththreeflavors TakashiSano
Nf = 2 Nf = 3
2 s0 2 s0
ps0 ps0
s s
1.5 ps 1.5 ps
2M 1 2M 1
0.5 0.5
0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2
t t
Figure2:Temperaturedependenceofthemesonicmassesintheflavor-singletscalar(s0)andpseudo-scalar
(ps0)channelsandintheflavor-nonsingletscalar(s)andpseudo-scalar(ps)channelsform=0(thicklines)
andm=0(thinlines). Wesetm=0.1(m =0.0265)asnonzeroquarkmassforN =2(3).
c f
6
ThefactorN cannotbefactored outinthepotential W becauseoftheanomalytermhere.
f
Inthethermodynamiclimitwecalculatethequarkcondensate q¯q (cid:181) f forS =1,a =0.3and
h i
g =2inFig.1. Thechiralphasetransition isthesecondorderforN =2inthechirallimit,while
f
it is the first order for N =3 and for m<m =0.0265. The mesonic masses can be defined as
f c
W (S)=W +1M2s a2+1M2 p a2+ with S=f +l a(s a+ip a)/√2 parametrized with U(N)
0 2 sa 2 psa ···
generators (tr(l al b)=2d ab). Thetemperature dependence ofthemassesareshowninFig.2. For
pseudo-scalar flavor-nonsinglet masseswefindtheGell-Mann–Oakes–Renner relation
(f +m)2M2 =mS 2(f +m) m q¯q , (4.2)
ps
∼− h i
if we identify f +m as the pion decay constant fp . On the other hand, the flavor singlet-masses
haveanadditional contribution fromtheanomalytermas
M2 =M2 D M2, M2 =M2 +D M2 (4.3)
s0 s 0 ps0 ps 0
−
with D M2 N t˜/(f +m)2. The would-be Nambu-Goldstone mode becomes massive due to the
0 ≡ f
coupling to the U (1) interaction D M2, and the mass gap is related to the (replaced) quenched
A 0
susceptibility t˜,similarlytotheWitten-Veneziano formula.
The topological susceptibility is obtained as c top = ¶ 2W (S(q );q )/¶q 2 q =0 with the saddle
|
pointsolution S(q )=f +ih (q )l 0/√2forsmallq ,andwefind[7,6]
0
1 1 1
= + . (4.4)
c t˜ t
top m
Here t˜ is the modified susceptibility defined in Eq. (3.10). The fermion coupling screens c via
top
thecontribution
S 2m(f +m) M2(f +m)2
t = = ps . (4.5)
m
N N
f f
Noting that theq dependence can beabsorbed into thequark massterm asmeiq /Nf, onecan show
theaxialWardidentity
m2 m
c = c q¯q . (4.6)
top ps0
−N −N h i
f f
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Chiralphasetransitioninarandommatrixmodelwiththreeflavors TakashiSano
Nf = 2 Nf = 3
1 1
m = 0.10 m = 0.10
0.8 00..0051 0.8 0 .m01c
0) 0)
(top 0.6 (top 0.6
c(t) / top 0.4 c(t) / top 0.4
c 0.2 c 0.2
0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2
t t
Figure3: Temperaturedependenceoftopologicalsusceptibilityc (t)forN =2and3.
top f
Since thepseudo-scalar mesonintheflavor singlet channel hasnonzero mass(4.3)because ofthe
U (1)-breaking term, thepseudo-scalar singlet susceptibility remains finiteinthebroken phasein
A
the chiral limit. Thus for the small but nonzero quark mass m, the decrease of c follows the
top
chiralcondensate q¯q f withincreasingt,whichisclearlyobserved inFig.3.
h i∼
5. Summary
We have presented a chiral random matrix model where the determinant interaction is incor-
porated bysumming up the topological zero modes with thebinomial distribution [6]. Themodel
has a stable ground state unlike the model of [5], and describes the N dependence of the chiral
f
phase transition. Atthesametimeitreproduces thephysical temperature dependence ofthetopo-
logical susceptibility. This model can be applied to the 2+1 flavor case at finite temperature and
density [8].
ThisworkissupportedinpartbyGrants-in-AidofMEXT,Japan(No.19540269and19540273).
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