Random matrix model at nonzero chemical potentials with anomaly effects

0.27 MB·English

by  H. Fujii

Random matrix model at nonzero chemical potentials with anomaly effects
H. Fujiia and T. Sanoa,b
aInstitute of Physics, The University of Tokyo,
Tokyo 153-8902, Japan
bDepartment of Physics, The University of Tokyo,
Tokyo 113-0033, Japan
(Dated: January 19, 2011)
Phase diagram of the chiral random matrix model with U(1)A breaking term is studied with
the quark chemical potentials varied independently at zero temperature, by taking the chiral and
meson condensates as the order parameters. Although, without the U(1)A breaking term, chiral
transition of each flavor can happen separately responding to its chemical potential, the U(1)A
1 breaking terms mix the chiral condensates and correlate the phase transitions. In the three flavor
1 case, we find that there are mixings between the meson and chiral condensates due to the U(1)A
0
anomaly,whichmakesthemesoncondensedphasemorestable. Increasingthehyperchargechemical
2
potential (µY) with theisospin and quarkchemical potentials (µI, µq) kept small, we observe that

n thekaoncondensedphasebecomesthegroundstateandatthelargerµY thepioncondensedphase
a appears unexpectedly, which is caused by the competition between the chiral restoration and the
J mesoncondensation. Thesimilar happenswhenµY andµI areexchanged,andthekaoncondensed

8 phasebecomes theground state at larger µI below the full chiral restoration.
1

] I. INTRODUCTION based on empirical nucleon interactions and chiral per-
h
turbationtheory[3]. Morerecently,colorsuperconductiv-
p
ity was extensively investigated and the kaon condensed
- The phase diagram of QCD at finite density has long
p phase acquired renewed interests there[4]. Possibility of
been challenged from the perspectives on the fundamen-
e inhomogeneous phases is also discussed recently1 [5].
h tal properties of dense matter realized in the core of a
Another motivation to consider the phase diagram
[ compact star and relativistic heavy ion collisions. The
withunequalchemicalpotentialsis thatonecandirectly
chiral symmetry breaking in light quark sector is one of
2 examine in lattice QCD simulations the N = 2 case
the crucialfeatures in low-energyQCD,which shouldbe f
v with finite isospin and zero quark chemical potentials,
7 reflected in any effective model. In the massless quark
µ (µ µ )/2 = 0 and µ (µ +µ )/2 = 0, where
7 limit with flavor number Nf = 2 or 3, chiral symmetry thIe≡measuu−refodrthe6 importanqce≡samuplingdisreal. Wenote
9 breakinggivesriseto continuouslydegeneratevacua and
thatitmaybe possibleto simulate the N =3 casewith
5 to emergence of the massless Nambu-Goldstone bosons f
9. asthemostimportantdegreesatlowenergies. Inreality, µI 6=0 and µq =µs =0, at least in principle.
The ground state at finite isospin chemical potential
0 nonzerou-andd-quarkmassesandheaviers-quarkmass
but at zero quark number density was studied with the
0 lift this degeneracy and explain the observed pseudo-
1 scalar meson spectrum. chiral Lagrangian, which revealed that the pion conden-
: sate appears once µI exceeds half the pion mass mπ/2
v Asystematfinitequarknumberdensitiesischaracter-
[6]. Indeed, in the chiral limit, the degenerate ground
i ized with the quark number chemicalpotentials µ (f =
X f state is completely chiral-rotated to the pion condensed
u, d, s) in the grandcanonical description. The simplest
r state at infinitesimal external field µI = 0. Adding a
a situation at finite density may be the case with equal finite quark mass m = 0, which make6s the pion mas-
q
chemical potentials µ = µ = µ = µ for all flavors, 6
u d s sive,we havea competitionbetween the two alignments;
where the s-quark density may be suppressed compared
q¯q and u¯γ d directionsform andµ ,respectively. In
5 q I
with the u- and d-quarks due to the mass difference. h i h i
thethree-flavorcase,kaoncondensationappearsoncethe
However, the quark chemical potentials µ are generally
f hyperchargechemicalpotentialµ exceedsthethreshold
Y
unequal in various physical situations. There are more
given by m with µ =0[7].
K I
protonsthanneutronsinnucleiandalsoinneutronstars
Thechiralrandommatrix(ChRM)modelisoneofthe
because of the electric charge of protons. In the core of
modelswhichsharethechiralsymmetrywithQCD.Thus
compactstarsinbetaequilibrium,the heavymassofthe
it will serve as a useful model providing qualitative fea-
s-quark may be overcome by the large electric chemical
tures of the QCD phase diagram. The aim of this paper
potential, whichmayresultin the appearanceofs-quark
is to explore the phase structure of the (ChRM) model,
degrees of freedom. In ultra-relativistic heavy-ion colli-
includingthepossibilitiesofthemesoncondensations,as
sionsthetotalstrangenessnumberisconstrainedtozero.
The phase structure at finite chemical potentials has
very rich theoretical possibilities[1–4]. Historically, in
dense nuclear matter, nucleon superfluidity, pion and 1 The p-wave pion condensed phase too accompanies
kaon condensations, and hyperon mixture were studied inhomogeneity[3].
2
a function of the quark chemical potentials µ (f=u, d, II. QCD WITH QUARK CHEMICAL
f
s) at zero temperature. POTENTIALS
In the previous work of the ChRM model with two
flavors done in Ref. [8] it was shown that the pion con- In this section we briefly review the symmetry break-
densed phase appears at finite µ above a critical value ing pattern of QCD with nonzero quark chemical poten-
I
µ as suggestedby [6], whereas it eventually disappears tials. The QCDpartitionfunctionwithN quarkflavors
Ic f
asµ isfurtherincreased,whichisunderstoodasthechi- at finite temperature and chemical potential is regarded
I
ralrestorationathighdensity. Itwasalsoindicatedthat as the average of the Dirac operators with non-Abelian
withsmallnonzeroµ thechiralrestorationoccursintwo gauge field action S :
I YM
steps: asµ isincreasedwith(e.g.) negativeµ <0fixed,
q I
the d¯d condensate discontinuously decreases first, and Nf
then(cid:10)the(cid:11)hu¯ui at slightly larger µq. The two- (or three-) ZQCD = DA det(D(µf)+mf)e−SYM, (1)
step transition along the µq axis with finite µI and/or Z Yf
µ is alsoobservedin the three flavorcase ofthe ChRM
Y
model[9, 10] and in the two and three flavor cases of the where D(µf)=γν(∂ν igAν+Bνµf) is the (Euclidean)
−
Nambu–Jona-Lasinio(NJL) model [11–14]. Dirac operator for a flavor f and mf is the mass. The
quark number chemical potential µ is introduced inde-
This two-step transition happens evidently because f
pendentlyforeachquarkflavorf withaconstantLorentz
these models do not include flavor mixing interactions
vector B =(0,0,0,1).
and then each quark sector independently responds to ν
In the massless limit m = 0 with zero chemical po-
each quark chemical potential µ . The U(1) anomaly f
f A
tential µ = 0, the classical QCD Lagrangian possesses
inQCDisknowntoinducetheflavormixingandtendsto f
U(N ) U(N ) chiralsymmetry,whichisbrokendown
unify the two chiral transitions into one. This is demon- f L f R
×
to U(1) SU(N ) SU(N ) at the quantum leveldue
strated in the NJL model with U(1) breaking effective B f L f R
A × ×
totheaxialanomaly. TheU(1) invarianceresultsinthe
interactions, which results in the usual phase diagram B
quark number conservation. Non-perturbative QCD dy-
with a single transition[14].
namicsisbelievedtobreakthissymmetryspontaneously
The U(1) anomaly effect is indispensable to build
A
to U(1) SU(N ) , generating a large part of the nu-
a low-energy effective model of hadrons in medium as B× f V
cleon mass and the light pions as the Nambu-Goldstone
well as in vacuum[16–18]. Although in a conventional
(NG) boson. In reality, nonzero quark mass term acts
ChRM model the U(1) anomaly effect is not treated
A as an external field to select the ordinary vacuum state
appropriately so as to affect the nature of the chiral
with nonzero q¯q condensate out of the nearly degener-
transition, recently we have succeeded in constructing h i
ate vacua.
the model in which the U(1) breaking term generates
A
The chiral symmetry of QCD is intact at finite quark
theflavor-numberdependenceofthetransitionorder[19].
number chemical potential as far as it is flavor-singlet,
Thismodelinthemasslessquarklimitshowsthesecond-
which is usually assumed in the context of finite density
orderphasetransitionforN =2,whereasthefirst-order
f QCD.Oncetheflavor-nonsingletpotentialissetnonzero,
one for N = 3. We explored the phase diagram of the
f it gives a stress on the flavor symmetric ground state in
model in the T–µ –m space with q¯ q as the order
q q h f fi addition to the quark mass term. Symmetry breaking
parameter[20].
pattern in this situation has been studied most conve-
It seems now quite intriguing to investigate the phase
niently with the chiral Lagrangian[6, 7].
diagram allowing the pion and kaon condensates in the
Let us consider the N = 2 case with the small u-
ChRM model with N =2 and 3 as a function of the f
f and d-quark masses m = m . At zero chemical po-
chemical potentials µ , with emphasis on the role of u d
f tential there appear nearly degenerate vacua connected
the U(1) anomaly. Similar studies were previously per-
A with each other by the SU(2) SU(2) O(4) transforma-
formed employing the NJL model with U(1)A anomaly tion, and the true groundstat×e aligns≃to the direction of
effects[14, 15, 21]. In Ref. [14] the phase structure
the quark mass term with SU(2) invariance. When the
for N = 2 is studied only in small µ region below V
f I isospin chemical potential µ = (µ µ )/2 is applied,
the threshold for the pion condensation, and the study I u − d
this invariance explicitly breaks down to U(1) SU(2)
is extended by including the pion condensed phase in ∈ V
aroundthe 3rdisospinaxis. The mass m and the poten-
Ref. [15]. Ref. [21] deals with the N =3 case restricted
f tial µ compete to fix the ground state. For µ >m /2,
at µ = 0 and µ = 0, where the possibility of kaon I I π
I 6 Y anewclassofthedegeneratevacuaappearswithnonzero
condensation is ignored.
pionfieldandthegroundstatespontaneouslybreaksthe
This paper is organized as follows. After a review of
U(1) symmetry, which is accompanied by the appear-
the QCD phase structure with finite chemical potentials
ance of an NG boson. This symmetry breaking pattern
in Sec. 2, we introduce the ChRM model with anomaly
is summarized as follows:
effects in Sec. 3. The model phase structure including
thepossibilityofthemesoniccondensationisinvestigated U(1) SU(2) µI6=0 U(1) U(1)
analytically and numerically in Sec. 4 and 5 for Nf = 2 B× V −π−c−c−o→nd B× (2)
U(1) .
and 3, respectively. Sec. 6 is devoted to a summary. B
−−−−−−→
3
ForN =3,we considerthe 2+1flavorcasewherethe configuration is ν = N N . We take the number of
f + −
−
quarkmassesm =m =m andthechemicalpotentials the near-zero modes 2N and the mean number of the
u d s
6
µ = µ =µ in general. In this case, all flavor symme- topological zero modes N to be proportional to the
u d s ±
6 6 h i
tryisbrokenexplicitlyleavingonlyU(1) U(1) U(1)in- four volume V of the system.
× ×
variancewhichrepresentstheconservationofeachflavor. ConcerningthematrixC whichrepresentsthemedium
Once(e.g.,)d¯γ5smesoncondensateisformedinaddition effects, we introduce the effective temperature T and
tothenonzerochiralcondensates,thesymmetryisspon- chemicalpotentialµ foraflavorf inthenear-zeromode
f
taneously broken down to U(1) U(1)ds, where U(1)ds sector, while they are set to zero in the topological zero
leavesthe condensate d¯γ5s inva×riant. We alsofind one mode sector[19, 23]:
NG mode corresponding to the spontaneous breaking of
U(1) 2. The symmetry(cid:10)brea(cid:11)king pattern is (µ +iT)1 0 0
f N/2
C = 0 (µ iT)1 0 . (5)
f f N/2
U(1) SU(3) µf6=0 U(1) U(1) U(1)  0 − 0 0
B V
× −−−−−→ × ×
K cond  
U(1) U(1) . (3)
ds Inthenear-zeromodesector,µ iT maybeinterpreted
−−−−−→ × f
±
as the contributions from the two lowest Matsubara fre-
quencies. The Dirac operator D with nonzero chemical
III. RANDOM MATRIX MODEL
potential µ =0 is no longer anti-hermitian, though the
f
6
partition function is invariant under µ µ . We
f f
Thechiralrandommatrixmodel[22,23]isconstructed stressherethefactthattheabsenceofthem↔edi−umeffects
based on the idea that the spontaneous breaking of the in the topological zero mode sector resolves the unphys-
chiral symmetry is dominated by the low-lying Dirac ical suppression of the topological susceptibility of the
eigenmodes, as is manifest in the Banks-Casher relation original ChRM model (see discussion in [25, 26]). This
between the chiral condensate and the spectral density formofC maybeunderstoodphysicallyasthefactthat
f
nearzero[24]. TheDiracoperatorD isthentruncatedto a topological zero mode localized near an (anti-) instan-
be a matrix within the restricted space spanned by the ton is rather insensitive to the medium effects.
(quasi-)zeromodes,andthematrixelementsaretreated
Using the Dirac operator (4), we define the partition
as random variables. Explicitly, in the representation
function of the ChRM model with fixed N and N as
+ −
where γ =diag(1, 1), we have
5
−
Nf
D = iR†+0 CT iR+0 C , (4) ZN+,N− = dRe−NΣ2trRR† det(D(µf)+mf) (6)
(cid:18) (cid:19) Z fY=1
whereRisarectangularcomplexrandommatrixandCis
with a Gaussian weight for the matrix R. The typical
a non-random matrix responsible for the effects of tem-
scale of the chiral symmetry breaking is fixed by the pa-
perature and chemical potentials. Note that the block
rameter Σ. The complete partition function is obtained
structure follows from the chiral symmetry γ ,D =0.
{ 5 } by summing over the numbers of topological zero modes
When the matrix R is not square but rectangular,the
N and N with a distribution function P(N ) as
Dirac operator D has ν exact zero eigenvalues with ν + − ±
| | | |
being the difference between the numbers of the rows
ZRM = P(N )P(N )Z . (7)
and the columns of R. This is the realization of the in- + − N+,N−
dex theorem for the topological charge and the number NX+,N−
of exact zero modes in ChRM model. Hence, in order
ThePoissondistributionwouldbeappropriateforP(N )
to include the anomaly effects, we need to deal with the ±
in a dilute instanton system[17]. Adopting the Poisson
non-square matrix properly. To this end, we categorize
distribution, however, one finds that the effective po-
thezeromodesintotwospecies,thenear-zeromodesand
tential becomes unbounded from below[27]. Instead, we
the topological zero modes[19]. The former are assumed
choose P(N ) to be a binomial distribution[19],
tobeN left-andN right-handedlow-lyingmodesgener- ±
ated in gluon dynamics, while the latter are interpreted
ainssttahnetomnosdoesr eNa−chaonftiw-ihniscthanistolnoscailnizeadgnaeuagreofineeldofcNon+- P(N±)=(cid:18)γNN±(cid:19) pN±(1−p)γN−N±, (8)
figuration. Then the size of the matrix R is taken as
(N +N ) (N +N ) and the topologicalcharge of the withparametersγandp. Thebinomialdistributionmod-
+ −
× elsthesituationthat,ifwedividethefour-volumeV into
γN cells,aninstantonappearsinoneofthecellswiththe
probabilityp,barringdouble occupancy. Inotherwords,
2 It is known that the numbers of the broken generators and the we introduced a repulsive interaction among instantons.
NG bosons can differ in some cases, e.g., feromagnetism, kaon Thismodificationresultsinaboundedeffectivepotential
condensedphasewithµI =0,etc.[28,29]. with a stable ground state[19].
4
After the standardbosonizationmanipulation, we can first order. Then the T–µ phase diagram becomes simi-
express the partition function at T =0 as larto the N =2 casewith smallquarkmasses. For more
f
detailed discussions, see Refs. [19, 20].
ZRM( , µˆ)= dS e−2NΩ (9) IV. PHASE DIAGRAM: Nf =2 CASE
M
Z
A. Effective potential
with the effective potential
We study the situation where µ = µ with the de-
Σ2 1 S+ µˆ u 6 d
Ω(S; M, µˆ)= 2 trS†S− 2logdet µˆM S†+ † generated quark masses mu =md ≡m. It is convenient
(cid:20) M (cid:21) todefine the (averaged)quarkchemicalpotential µq and
γ
log αdet(S+ )+12 , (10) the isospin chemical potential µI as
− 2 | M |
1
where the bosonic field S CNf×Nf is the order pa- µq = 2(µu+µd), (12)
∈
rameter matrix corresponding to the bi-fermion field 1
Sfg ∼ ψ¯RfψLg, and M = diag(mu,md,...,mNf) and µI = 2(µu−µd). (13)
µˆ=diag(µ ,µ ,...,µ ) arethe mass andthe chemical
potential muatrdices, resNpfectively. The term involving the TheorderparametermatrixcanbeparametrizedasS =
parameters γ and α = p/(1 p) breaks the U(1)A sym- λa(φa+iρa) with the U(2) generators λa (a=0,1,2,3).
metry S Seiθ even in the− =0 limit[16, 17]. In the Theusualgroundstatebreaksthechiralsymmetryspon-
thermody→namiclimit, N M,thegroundstateisfound taneously, having nonzero chiral condensate φ0. At fi-
as a solutionof the saddle→po∞intequation, ∂Ω/∂Sfg =0. nite µI the pion condensedphase where ρ1,2 are nonzero
Using the solution S¯, the chiral condensate can be com- may be favored. Therefore, we adopt here the following
puted for a flavor f as Ansatz:
φ iρ +ρ
ψ¯ ψ = ∂ Ω(S¯; ,µˆ)= Σ2S¯ . (11) S = iρ uρ 1φ 2 (14)
f f ff 1 2 d
∂m M − (cid:18) − (cid:19)
f
(cid:10) (cid:11) with real order parameters φ , φ and ρ . Notice that
u d 1,2
By generalizing the mass matrix to the source matrix
nonzeroµ explicitlybreakstheSU(2)isospininvariance
=(sa+ipa)λa/√2withthegeneratorsλa normalized I
Mtr(λaλb) = 2δab, other scalar (chiral) and pseudoscalar downtoU(1)invariancegeneratedbyλ3,andthenφu 6=
φ in general. Substituting this form into the effective
d
(meson) condensates are evaluated by differentiating in
potential (10), we obtain
sa and pa, respectively.
In the preceding works[19, 20], we have studied the Σ2
ground state of this model for Nf = 2 and 3 at finite Ω= 2 (φ2u+φ2d+2|ρ|2)
temperature T and equal chemical potential µ. Regard- 1
ingtheanomalyparametersαandγ,wehavefoundthat, log (σu+µu)(σd µd)+ ρ2
− 2 − | |
for the large anomaly parameters, αγ > Σ2, this model
doesnotshowchiralrestoration,andth∼ereforeweshould 1log(cid:2)(σu µu)(σd+µd)+ ρ2(cid:3)
− 2 − | |
uthsee cthhieraalnpohmaasleytrpaanrsaimtioetne.rsIninthtehechriergailolnimwithitchheaclhloirwasl γ log(cid:2)α(σuσd+ ρ2)+1 2 , (cid:3) (15)
− 2 | |
phase transition at finite T and zero µ is found to be
(cid:2) (cid:3)
second (first) order for N =2 (3). This flavor-number where σ = φ + m. This model reduces to the one
f f f
dependence comes from the U(1) anomaly term. Ex- studied in Ref. [8] when the anomaly term is neglected,
A
tending to the µ = 0 case, we find that the transition γα=0,asitshould. Notethatthe potentialdependson
on the T–µ plane6changes from the second order to the ρ andρ onlythrough ρ2 =ρ2+ρ2 due tothe residual
1 2 | | 1 2
first order at a tri-critical point as µ is increased in the U(1) symmetry. Hereafter, we shall arbitrarily choose
N =2 case. For N =3 the first-order phase boundary the meson condensate such that ρ = 0 and ρ = ρ,
f f 1 2
separates the T–µ plane into two regions. With increas- whichbreakstheU(1)invariance. Nonzeroρ=( u¯γ d
5
ing the quark mass, the thermal transition gets weak- d¯γ u )/2 signals the pion condensation. h i−
5
ened and eventually turns to a smooth crossover, while The saddle pointequationswith respectto φ , φ and
u d
(cid:10) (cid:11)
the transitionatlargerµ thanacriticalpoint remainsof ρ respectively yield
5
∂Ω 1 σ µ σ +µ γασ
=Σ2φ d− d + d d d =0, (16)
∂φ u− 2 (σ +µ )(σ µ )+ρ2 (σ µ )(σ +µ )+ρ2 − α(σ σ +ρ2)+1
u (cid:20) u u d− d u− u d d (cid:21) u d
∂Ω 1 σ +µ σ µ γασ
=Σ2φ u u + u− u u =0, (17)
∂φ d− 2 (σ +µ )(σ µ )+ρ2 (σ µ )(σ +µ )+ρ2 − α(σ σ +ρ2)+1
d (cid:20) u u d− d u− u d d (cid:21) u d
∂Ω 1 2ρ 2ρ 2γαρ
=2Σ2ρ + =0. (18)
∂ρ − 2 (σ +µ )(σ µ )+ρ2 (σ µ )(σ +µ )+ρ2 − α(σ σ +ρ2)+1
(cid:20) u u d− d u− u d d (cid:21) u d
Let us first classify the solutions with ρ=0. Remember For αγ =0 this recoversthe solution of [8]
that when αγ = 0, the chiral condensates φ and φ
u d
decouplefromeachother. Thus,intheidealcaseofm= ρ2 =µ µ + 1 =µ2 µ2+ 1 . (22)
0,wehavefoursolutionsbymakingcombinationsofφ = 0 u d Σ2 q − I Σ2
f
0 and φ =0, depending on the values of µ . If αγ =0,
f 6 f 6 Inthelargeµ2 region,ρ2 >0solutiondisappearsandthe
however, the condensate φd cannot vanish exactly once I
U(1) symmetry is restored. The second-order transition
φ = 0 because of the flavor mixing, while the trivial
u
6 lineisgivenbyahyperbolaontheµ –µ plane,obtained
solution φ = φ = 0 is still intact. Thus, in the chiral q I
u d
by setting ρ=0 in Eq.(21). This pioncondensedregion
limit with αγ = 0, we expect four types of solutions, (i)
6 isenlargedbyafactor(1 αγ/Σ2)−1 ascomparedtothe
ordinary phase with broken chiral symmetry, (ii) phase
−
case without anomaly.
with restored chiral symmetry, (iii) & (iv) phases with
nearly vanishing chiral condensate for only one of the Beforeanalyzingthegroundstatesolutionnumerically,
flavors,uandd. Wenumericallyfindthatthetransitions we remark here the symmetry of the effective poten-
between these four phases are of first order, and hence tial (10). First the effective potential is invariant under
these four phases remain at small m=0. the respective charge conjugations, µu µu and/or
6 µ µ , when ρ = 0. In addition, t→he e−ffective po-
Wenextconsiderinwhatconditionstheρ=0solution d d
→ −
6 tential (10) is invariant under u d. Reflecting these
appears. To this end, we expand the thermodynamic
↔
invariantoperations, the phase diagramhas an eightfold
potential with respect to ρ at the ordinaryground state,
structure on the µ –µ or µ –µ plane. Nonzero con-
∂Ω/∂φ =0 and ρ=0: u d q I
u,d
densation ρ of the charged pion, however, breaks the re-
Ω(ρ;m,µ )=Ω (m,µ )+Ω (m,µ )ρ2+ . (19) spective charge conjugation symmetry leaving only the
f 0 f 2 f
··· simultaneous one (µ ,µ ) ( µ , µ ). Together with
u d d u
→ − −
Then one finds the coefficient Ω up to (m,µ2) as the flavorsymmetry, u d, afourfoldstructureremains
2 O I in the phase diagram. O↔n the µ –µ plane, therefore, it
q I
−1 is sufficient to investigate the phase diagram in the first
2µ2 µ2
Ω =M2 I 1+ q , (20) quadrant, µq >0 and µI >0.
2 π − (φ20−µ2q)2 (φ20−µ2q)2Σ2!
where φ0 is the scalar condensate at m = µI = 0. We B. Numerical result
seethatthefinitepionmassM2 =mΣ2/φ disfavorsthe
π 0
pion condensation,while the finite chemicalpotential µ
I In the top panel of Fig. 1, we show the phase diagram
favors it. From the condition Ω =0, the critical isospin
2 ontheµ –µ planewithparameters,α=0.5,γ =1,Σ=
chemical potential is found to be µ2 M2 within this q I
leading approximation3. Ic ∝ π 1 and m = 0. The dotted lines denotes the first-order
phase transitions with respect to the scalar condensates
When m = 0 i.e. m2 = 0, infinitesimal µ selects out
π I φu and φd when the possibility of the pion condensation
the ground state solution of φ = φ = 0 and ρ = 0
u d is neglected. In this restricted case we find around the
6
amongthedegeneratevacua. WefindthatfromEq.(18)
origintheordinaryphasewhereφ andφ =0. Allowing
u d
the non-trivial solution satisfies the relation 6
the pion condensation, we find that it completely covers
the region of the ordinary chirally broken phase and it
−1
αγ
ρ2 =µuµd+ Σ2− αρ2+1 . (21) elixnteenindsFitgo.t1h.eLlaetrguesrl|oµoIk| raetgtihone,samsaslhlochwenmwiciathlptohteesnotliiadl
(cid:18) (cid:19)
region first. The ordinary phase with nonzero φ = φ
u d
and ρ = 0 and the pion condensed phases with ρ = 0
are coexisting along the line µ = 1(µ µ ) = 06 in
I 2 u − d
3 Note that we use a different notation Mπ for the pion mass in the chiral limit. With an infinitesimal µI, however, the
this model from the physical one mπ because the physical unit ordinarygroundstate is totallyrotatedawaytothe pion
intheChRMmodelisnotfixed. condensedphase. Thisbehaviorwasalreadyfoundinthe
6
chiralsigmamodel[6]. Thisistobeexpectedbecausethe
ChRMmodelissimilartothepotentialtermofthechiral
sigma model at small chemical potential µ and small 2
f
symmetry breaking m[8]. Next, in the large-µ region
f
in Fig. 1, we see that the chiral symmetry restoration
occurs. Increasingµ withµ keptsmall,wefindasecond 1.5
I q
order phase transition from the pion condensed phase
to the chiral restored phase. On the other hand, if we ρ
µI
increase µ along with the fixed µ line, we find a first- 1
q I
order phase transitionfrom the pion condensed phase to φ
d
the phase where one or both of the chiral condensates
φu and φd melt away, depending on the size of µI. It is 0.5
remarkable that the phase diagram with ρ = 0 reflects
the symmetry of (µ ,µ ) (µ ,µ ) with (φ ,φ )
q I I q u d
↔ ↔
(φd,φu), while in the real ground state with ρ = 0, this 0
6
symmetry no longer exists. As we mentioned, the phase
0 0.5 1 1.5 2
diagram has the symmetry of (µ ,µ ) ( µ , µ ).
q I → ± q ± I µ
In the bottom panel of Fig. 1 we show the result with
q
m = 0.1 = 0. Generally, finite m acts as an external
6
alignmentfield for φ , φ =0. So there is a competition
u d
between the two alignmen6t fields m and µ2. In a small
I
µ region,the finite m wins andthe ordinaryphase with
I 2
nonzero φ and φ appears pushing the pion condensed
u d
phase aside, as shown in Fig. 1. If µ exceeds a critical
I
value proportional to m , the pion condensation phase
π 1.5
appears. It is estimated from Eq. (20) with our model
parameters as µ = φ4M2/2 = 0.276 at µ = 0 . We
Ic 0 π q
hereremarkthenumbperofthephasetransitionsinsmall µI 1 ρ
µ regionwhereρ=0. Whenµ isincreased,thesystem
I q φ
experiences just one phase transition from the ordinary d
phase with φ > 0 and φ > 0 to the symmetric phase
u d 0.5
with φ 0 and φ 0 if µ is sufficiently small. If
u d I
∼ ∼ | |
µ is large but still below the critical value, we have
| I| φ , φ
twotransitionsbetweentheordinaryphaseandthesym- u d
0
metric phase; e.g., for µ > 0, the first one is from the
I
0 0.5 1 1.5 2
phase of φ > 0, φ > 0 to φ 0, φ > 0, and the
u d u d
∼
second is from φ 0, φ > 0 to φ 0, φ 0. In µ
u d u d
∼ ∼ ∼ q
the study of the ChRM model without the anomaly ef-
fect in [8], a two-step phase transition is found. We find
that, even with ρ = 0, the anomaly term mixes the φ
u
and φd yielding a one-step transition. This mechanism FIG. 1. Phase diagram for Nf = 2 on µq–µI plane in the
chiral limit (top) and at m = 0.1 (bottom). Parameters are
is general and is found in an NJL model study[14]. We
α = 0.5, γ = 1, and Σ = 1. Nonzero condensates in the
expect that this is also the case in QCD. As a result, we
respective regions are indicated with the letters, ρ and φ .
have one triple point and two critical end points 4 in the u,d
The first (second) order phase boundary is denoted in solid
phase diagram shown in the bottom of Fig. 1. (In the
(dashed) lines. For comparison, the phase boundary in the
N = 3 case we will find more triple points and critical
f case where the meson condensation is ignored, is denoted in
end points.) Varying the anomaly parameters α and γ, dotted line showing theeightfold symmetry.
we have confirmed that the position of the triple point
movestowardthelargerµ andµ regionasthestrength
q I
of the flavor mixing is increased.
Finally we comment on the large µ region regarding
I may form a pion-like Cooper pair[6]. However, since
the possibility of BCS-like condensate. At large µ in
I theChRMmodelneglectsthespace-timedimensions,the
QCD, large Fermi seas of (e.g.) u-quark and d¯-quark
physics of the Fermi surface does not exist and the su-
are formed, and the attraction between these quarks
perconducting phase is tricky. For possible extensions of
the ChRMmodels to dealwith the diquarkcondensates,
seeRef.[30]. Inourmodelwithoutsuchanextension,we
4 Acriticalendpointisthepointwhereacriticallineistruncated findasimpleterminationofthe pioncondensedphaseat
bymeetingafirst-orderphaseboundary. large µ .
I
| |
7
V. PHASE DIAGRAM: Nf =3 CASE Note that µY = µq µs in our convention. When µI =
−
µ = 0, three quark chemical potentials are equal to
Y
µ . At nonzero µ and µ , we allow the possibilities for
q I Y
A. Effective potential
the pion and kaon condensates in addition to the chiral
condensates. Wethenapplythe followingAnsatzforthe
order parameter matrix,
We explore here the phase structure in the 2+1 flavor
case with two mass parameters, m = m m and m ,
varyingthree chemicalpotentialsµuu,µd adnd≡µs indepens- S = φρu ρφud −ρρsu , (26)
dently. They are recast to the quark chemical potential − ud d ds 
ρ ρ φ
µq,theisospinchemicalpotentialµI andthehypercharge su − ds s
 
chemical potential µ defined respectively as
Y where φ , φ , and φ are the chiral condensates, ρ is
u d s ud
thepioncondensateandρ andρ arethekaonconden-
su ds
sates. WiththisAnsatz,weobtaintheeffectivepotential
1
µ = (µ +µ ), (23) as an function of six order parameters,
q u d
2
1
µI = (µu µd), (24) Ω=Ω0+Ωa, (27)
2 −
1 where Ω is the potential of the conventional ChRM
µ = (µ +µ 2µ ). (25) 0
Y u d s
2 − model[9],
Σ2 1
Ω = φ2 +φ2+φ2+2ρ2 +2ρ2 +2ρ2 log (σ2 µ2)(σ2 µ2)(σ2 µ2)
0 2 u d s ds su ud − 4 u− u d − d s − s
+ (cid:0)ρ4 (σ2 µ2)+2ρ2 ρ2 (σ σ µ µ(cid:1))+2ρ2 (cid:8)(σ2 µ2)(σ σ µ µ )+(cyclic perm. of u,d,s) 2, (28)
ds u− u ds su u d− u d ds u− u d s− d s
(cid:2) (cid:3)(cid:9)
and Ω is the anomaly part5, for being the ground state. Below, we just assume that
a
only one component of the meson condensates becomes
Ω = γ log α σ σ σ +σ ρ2 +σ ρ2 +σ ρ2 +1 2. nonzerointhemesoncondensedphaseandleaveitsproof
a −2 u d s u ds d su s ud as an open issue6.
(29)
(cid:2) (cid:0) (cid:1) (cid:3)
Preceding the numerical results, let us summarize
Because Ω is a function of ρ2 , ρ2 and ρ2 , we always three key points for qualitative understanding of the
ud ds su phase structure.
have a trivialsolutionρ =0 for the saddle point equa-
fg
(I)analogy with the chiral sigma model —As is shown
tions. Onthe other hand,oncechiraland/ormesoncon-
inRef.[8],theconventionalChRMmodelforsmallchem-
densates become nonzero, they act as source terms for
icalpotentialsandquarkmassesisequivalenttothezero
the other chiral condensates owing to the anomaly term
momentum part of the chiral Lagrangian. Hence the
Ω , and therefore σ =0 is no longer a solution.
a f
phase structure of the ChRM model in the small chemi-
We confirmed numerically in the ChRM model that
cal potential region must have a similar structure as the
two or more meson condensates do not appear in the
chiralsigma model. It is knownin the sigmamodel with
ground state at the same time. If only one type of
N =3 thata second-orderphase transitionoccurs from
the meson condensate ρ is nonzero, the Ω part of f
fg 0
theordinarychirallybrokenphasetothepioncondensed
the potential becomes a sum of two contributions, Ω =
0
Ωfg(ρ ,φ ,φ )+Ωh(φ ), where Ωfg is nothing but the phase or the kaon condensed phase at certain finite µI
0 fg f g 0 h 0 and µ . The critical chemical potential is roughly esti-
potential of the two-flavor (fg) ChRM model without Y
mated as µ m /2 and µ m for the pion and
anomaly, and Ωh0 corresponds to the single-flavor (h) kaoncondenIsa∼tionsπ,respectivYely∼. ThKere is a competition
ChRMmodel. Inspiteofthisflavorseparation,thephase
between the pion and kaon condensed phases for finite
diagram of N = 3 is different from that of N = 2 be-
f f µ and µ , and the phase transition between the two is
cause the anomaly term introduces the coupling among I Y
found to be first-order. We also remark that the state
threeflavorsandwehaveacompetitionbetweenthepion
with two meson condensates having nonzero values can
(fg =ud)and the kaon(fg =ds or su) condensedphases
not be even a meta-stable state in this analysis.
5 Consequences oftheanomalymixingbetweenthechiralanddi- 6 We note that coexistence of the p-wave pion and kaon conden-
quarkcondensates arestudiedinRef.[31]. sates innuclearmatterwasstudiedpreviously[32].
8
(II)chiralrestoration—Incontrasttothechiralsigma
2
model, the ChRM model includes the chiral restoration
dynamics,whichwillresultina new kindofcompetition
of two meson condensed phases in the region at large
chemicalpotential. Toillustratethesituation,letuscon- 1.5
ρ
siderthecaseinthechirallimit,whereµY >µI >µq =0 ud
andthe K0-condensedphaseρ =0 with φ =0is cho-
us d
6 6
senasthegroundstate. Wenowfurtherincreaseµ with
Y Y 1
µ and µ fixed. This kaon condensed phase will simply µ
q I ρ
remainasthegroundstateinthechiralsigmamodel. At ρ , φ su
sufficiently large µ , however, chiral or meson conden- su d
Y
sates involving the s-quark become disfavored and the 0.5
chiral symmetry in the s-quark sector will presumably ρ , φ φ
ud s s
berestored. Thenthereisapossibilityfortheremaining
u- and d-quarks to form the pion condensate ρud = 0. 0
6
We thus find a competition between the kaon condensed
0 0.5 1 1.5 2
phase (ρ = 0, φ = 0) and the pion condensed phase
us 6 d 6 µ
(ρ =0, φ =0)atlargeµ . We willsee shortlyinthis
ud s Y I
6
section that the pion-condensed phase is indeed favored
at large µ . This competition exists also in the NJL
Y
model[12],andweexpectthatthismechanismiscommon 2
for the models exhibiting chiralsymmetry restorationat
large chemical potentials.
(III) anomaly effect—Mixingoftheorderparameters 1.5 ρ
duetotheanomalytendstounifychiralphasetransitions ud
into a single one as is seen in the case of N =2. In the
f
afobrseNnfce=of3threedmuceessontocolongd(eαnσsautσedsσ,sth+e1a)n,owmhailcyhtceormupΩleas µY 1 ρsu, φd ρ
threechiralcondensates. Whenoneofthechiralconden- su
satesmeltsaway,theflavormixingamongthechiralcon-
0.5
densates becomes small. With finite meson condensates,
however, the anomaly term induces mixing among the ρ , φ φ
ud s s
chiralandthemesoncondensates. Forexample,oncethe
pion condensate is formed ρ =0 with rotating φ and 0
ud u
6
φ away,the mixing term σ σ σ becomes small but the 0 0.5 1 1.5 2
d u d s
term φ ρ2 in Eq. (29) generates new flavor mixing. Be-
s ud µ
causeofthismixing,thechiralandpioncondensatestend I
to vary cooperatively as the chemical potentials change.
Furthermore, the anomaly term makes the ground state
with nonzero φs and ρud condensates more stable as the FIG. 2. Phase diagram for Nf = 3 on µI–µY plane with
termappearswithaminussignintheeffectivepotential, µq =0inthechirallimit(top)andwithnonzeroquarkmasses
which results in the extension of the meson condensed mu = md = 0.02, ms = 0.1 (bottom). In the bottom panel,
phase compared to the case without anomaly. the lower (upper) vertical narrow area represents the phase
with nonzero φu, φd and φs (φu and φd). Other parameters
and notations are thesame as in Fig. 1.
1. µ –µ plane
I Y
B. Numerical result
Wefirstpresentthephasediagramontheµ –µ plane
I Y
with zero quark chemical potential µ = 0 in the chiral
q
We calculated the phase diagram with model param- limit(top)andatfinitequarkmasses(bottom)inFig.2.
eters Σ = 1, α = 0.5 and γ = 1 in the chiral limit Note that the effective potential has the symmetry of
m = m = m = 0 as well as at finite quark masses, µ µ and, moreover, µ µ if we change u
u d s Y Y I I
↔ − ↔ −
m =m =0.02 and m =0.1. We show the phase dia- and d flavorssimultaneously. We then present the result
u d s
gram on the µ –µ plane with µ = 0, the µ –µ plane only in the first quadrant.
I Y q q I
with µ =0, and the µ –µ plane with µ =0. Let us first focus on the case in the chiral limit. In
Y q Y I
9
the smallchemicalpotentialregion,wefindthepionand
2.5
kaon condensed phases. These two phases are separated
bythefirst-orderphasetransitionline µ =µ ;the pion
I Y
condensed phase appears when µ > µ and otherwise φ
I Y 2
s
thekaoncondenses. Thisisalsofoundinthechiralsigma
model[7]. When chemical potentials are increased, we
ρ
findthattheregionsoftwomesonicphasesareexchanged 1.5
ds
onthe diagram;the pioncondensedphase appearswhen
µI
µ < µ and otherwise the kaon condenses. As already
I Y φ
explained,thispeculiarbehavioristriggeredbythechiral 1 d
restoration; these meson condensed phases at the larger
chemical potentials are accompanied by the melting of
0.5
the chiral condensate of the remaining flavor. When µY ρ , φ
isfurtherincreasedatsmallµ ,thepioncondensedphase ud s
I
continuesindefinitely becauseρud becomes insensitive to 0
µ once φ disappears. On the other hand, ρ continu-
Y s ud 0 0.5 1 1.5 2
ouslyvanishesatsomepointasµ increases,becauseρ
I ud µ
which has the isospin charge is affected by µ , whereas
I q
nonzero chiral condensate φ survives irrespective of µ
s I
for small µ because it has no isospin.
Y
Without the anomalytermthe verticalstraightline of 2.5
the second-order transition boundary seen in the large
µ region in Fig. 2 would extend down to the µ axis.
Y I φ
This is because the effective potential without anomaly 2 s
Ω becomes the sum of two contributions, the φ part
0 s
and the part involving φu,φd and ρud, as mentioned be- 1.5 ρ
fore, unless the kaon condensates have finite values. By ds
comparison, we see that the anomaly coupling between µI φ
φs and ρud makes the pion condensed phase more stable 1 d
and extended to larger µ region.
I φ , φ
When quark masses are set to nonzero, the phase di- d s
agramreceives two qualitative modifications. One is the 0.5
appearance of the chiral condensed phase without any ρ , φ
ud s
meson condensate at small chemical potentials. This is
0
the same result as in the chiral sigma model. The other
0 0.5 1 1.5 2
is the configuration change of the four meson condensed
phases. We find that the pion condensed phase extends µ
q
fromzeroµ tolargeµ ,cuttingthetwokaoncondensed
Y Y
phasesapart. This is becausethe chiralcondensateφ is
s
morefavoredform >m =m andthekaoncondensed
s u d
phasesshrink. Intheextendedpioncondensedphase,we FIG.3. PhasediagramforNf =3onµq–µI planewithµY =
find a first-order phase transition in φ , which creates a 0 in the chiral limit (top) and with nonzero quark masses
s
(bottom). In the bottom panel, the horizontal narrow area
small gap in ρ , too.
ud is the phase with nonzero φu, φd and φs. Parameters and
notations are the same as in Fig. 2.
2. µ –µ plane
q I
We next show the phase diagram on the µ –µ plane Lagrangian analysis. In this phase, the u-quark chiral
q I
with µ = 0 in Fig. 3. In this case the effective poten- condensatemeltsawayφ 0,andtheremainingd-and
Y u
∼
tial is symmetric in µ µ and in µ µ with s-quarks form the kaon condensate, ρ = 0. Note that,
q q I I ds
↔ − ↔ − 6
exchange of the u and d quarks. We present the phase becausem =0,infinitesimaldifference betweenµ and
K d
diagram again only in the first quadrant. µ makesthekaoncondensedphasemorestablethanthe
s
Let us focus on the case in the chiral limit (top panel chirally broken phase without a meson condensate.
of Fig. 3). With infinitesimal µ , the ground state be- The pion and kaon condensed phases vanish contin-
I
comes the pion condensed phase. Unexpectedly, how- uously at large µ , respectively, to the phases with φ
I s
ever,wefindthatakaon-condensedphaseappearsinthe and without any condensate. The kaon condensed phase
region where µ becomes largest among three chemical extends to a larger µ region than the pion condensed
u I
potentials. This phase was not predicted in the chiral phase. This may be understood from the fact that the
10
pioncondensatewithisospincharge1istwiceassensitive
to µ as the kaon condensate with isospin 1/2. 3
I
With finite quark masses we observe that two new
2.5
phases are added, as is seen in the bottom panel of
Fig. 3. The ordinary chirally broken phase appears in
2
the small µI and µq region due to finite mπ, while finite Y φu, φd ρds
m changes lower chemical potential part of the kaon µ 1.5
K
condensed region into the chiral condensed phase with
1 φ
nonzero φ and φ . These two chiral condensed phases s
show the dsecond-osrder phase transitions to the meson 0.5 ρds, φu
condensed phases at larger µ , which is consistent with
I
the chiral Lagrangiananalysis[7]. At small µ , we find a 0
I
-1 -0.5 0 0.5 1 1.5 2
singletransitionfromthe brokentothesymmetricphase
µ
along the µq axis, owing to the anomaly term. q
It wouldbe instructive to comparethis phase diagram
to that of N = 2 on the µ –µ plane. Although the
f q I
phase diagrams are drawn in the same chemical poten-
2.5
tialspace,inclusionofthe thirdquarkflavorchangesthe
phase diagram drastically resulting in a new phase with
2
the kaon condensation. If we take a limit of m in φ , φ ρ
s u d ds
→∞
the Nf = 3 phase diagram, the s-quark should decouple Y 1.5
and the s-quark and kaon condensates disappear. This µ
makes the phase diagram reduce to that of the Nf = 2 1 ρ , φ φs
ds u
case.
0.5
φ , φ , φ
u d s
0
3. µ –µ plane
q Y -1 -0.5 0 0.5 1 1.5 2
µ
q
Finally, we address the phase diagram on the µ –µ
q Y
plane with µ = 0. In Fig. 4 we present the phase dia-
I
graminthechirallimit(toppanel)andwithfinite quark
FIG. 4. Phase diagram for Nf = 3 on µq–µY plane with
masses (bottom panel). The effective potential is un-
µI =0inthechirallimit(top)andwithnonzeroquarkmasses
changed under the simultaneous exchanges of µq µq (bottom). ParametersandnotationsarethesameasinFig.2.
↔−
and µ µ . Hence we restrict the phase diagrams
Y Y
↔ −
in µ >0 region. The formation of the kaon condensate
Y
breaks the SU(2) isospin symmetry spontaneously. We
choose arbitrarily the K0-condensed phase (ρds 6= 0) as phases and the φs 6= 0 phase coexist. At the other end,
the meson condensed ground state. Note that no pion we find a critical end point where two kaon condensed
condensed phase appears in this diagram since we have phases and the symmetric phase meet.
set µI =0 mπ. Let us consider the finite mass effect. Because of the
| | ≤
In the chiral limit, the chiral and kaon (pion as well) finite m , there appears the ordinary chirally broken
K
condensed phases are degenerated at µ = µ = 0. phase in the small chemical potential region. The phase
q Y
With infinitesimal µ , the ground state becomes the transitiontothekaoncondensedphaseisofsecondorder.
Y
kaon condensed phase, and therefore the isospin sym- Along the µ axis, we find the phase transition from the
q
metry is broken spontaneously. Assuming the alignment chiralcondensedphase to the symmetric phase occursin
by infinitesimal negative µ , the chiral condensed state one step. The threshold µ for the two-step phase tran-
I Y
is completely rotated into the the K0 condensed state sition is relatively small compared to the corresponding
where φ = φ = 0 and ρ = 0 but the u-quark chiral value of µ found in the µ –µ plane (see Fig. 3).
d s ds I q I
6
condensate remains finite φ = 0. When µ is increased This can be understood as follows: note that without
u q
6
with µ = 0, two order parameters, ρ and φ disap- the flavor mixing, the first order transitions along the
Y ds u
pear at the same time at a critical value. Without the µ axis would occur at different values of µ when the
q q
anomaly effect this phase transition would occur in two quark masses are different. The U(1) anomaly mixing
A
steps. acts to bind them together, and there is a critical value
We note that there is a first-order phase boundary in- for the anomaly strength above which the phase tran-
side of the the kaon condensed phase ρ = 0. Across sitions occur in one step along the µ axis. The larger
ds q
6
this line, φ jumps from a nonzero value to zero and ρ mass difference between u, d-quarks and s-quark makes
u ds
changesalsoitsvaluediscontinuously. Atoneendofthis this binding of the chiral transition more fragile against
line,wefindatriplepointwherethetwokaoncondensed the external field µ . This is the reason why we have a
Y