Enhancement of Chiral Symmetry Breaking from the Pion condensation at finite isospin chemical potential in a holographic QCD model

0.26 MB·

by  Hiroki Nishihara

Enhancement of Chiral Symmetry Breaking from the Pion condensation at finite
isospin chemical potential in a holographic QCD model
Hiroki Nishihara ∗1 and Masayasu Harada†1
1 Department of Physics, Nagoya University, Nagoya 464-8602, Japan
(Dated: January 14, 2014)
We study the pion condensation at finite isospin chemical potential using a holographic QCD
model. By solvingtheequationsof motion forthepionfieldstogetherwith thosefortheiso-singlet
scalar and iso-triplet vectormeson fields, weshow thatthephasetransition from thenormalphase
to the pion condensation phase is second order with the mean field exponent, and that the critical
value of the isospin chemical potential µI is equal to the pion mass, consistently with the result
obtained by the chiral effective Lagrangian at O(p2). For higher chemical potential, we find a
4 deviation, which can be understood as a higher order effect in the chiral effective Lagrangian. We
1 investigatetheµI-dependenceofthechiralcondensatedefinedbyσ˜ ≡phσi2+hπai2. Wefindthat
0 σ˜ is almost constant in the small µI region, while it grows with µI in the large µI region. This
2 implies that the strength of the chiral symmetry breaking is not changed for small µI: The isospin
chemicalpotential playsarole torotatethe“vacuumangle” of thechiral circletan−1 hπai2/hσi2
n p
a with keeping the “radius” σ˜ unchanged for small µI. For large µI region, on the other hand, the
J chiral symmetry breaking is enhanced by theexistence of µI.

3
PACSnumbers: 11.30.Rd,21.65.Cd,11.25.Tq,14.40.Be
1

]
h I. INTRODUCTION mid µI region. Actually, many analyses were done by
p using the Nambu-Jona-Lasinio (NJL) model [8–13], the
- random matrix model [14], the strong coupling lattice
p Quantum ChromoDynamics (QCD) at finite isospin
analysis[15],Ginzburg-Landauapproach[16],thehadron
e chemical potential is an interesting subject to study. It
h resonance gas model [17], holographic QCD models [18–
combined with the finite baryon number chemical po-
[ 28].
tential will provide a clue to understand the symmetry
Althoughtherearesomanyworksonthepionconden-
1 energy which is important to describe the equation of
v state inside neutron stars [1]. In addition, it may give sation at finite isospin chemical potential, there are not
8 some informations on the structure of the chiral symme- many works for studying the strength of the chiral sym-
2 metry breaking. Namely, it is interesting to ask whether
try breaking [2].
9 or not the chiral symmetry is partially restored in the
When we turn on the isospin chemical potential µ
2 I isospin matter.
. at zero baryon number density, the pion condensation is
1 InRefs.[10–13],basedonthe NJLmodelanalysis,the
expectedtooccuratacriticalpoint. SonandStephanov
0 [3]showedthat,usingthechiralLagrangianatO(p2),the authors seemed to conclude that the reduction of q¯q
4 h i
inthe isospinmatterimplies thepartialchiralsymmetry
1 phase transitionto the pioncondensationphase is ofthe
restoration. In Ref. [16], the Ginzburg-Landauapproach
v: second order and the critical value of µI is equal to the is used to study the the q¯q condensate together with
pion mass. It was also shown [4] that q¯γ q condenses
i h 5 i thepioncondensation. Hohwevier,theabsolutestrengthof
X in the high isospin density limit. Then a conjecture of
the chiral symmetry breaking, which is characterized by
r no phase transitionfrom the pion condensation phase to
a q¯γ q condensation phase was made. The structure in the chiral condensate σ˜ = σ 2+ πa 2, is not clearly
5 h i h i
h i studied. The analysis using the strong coupling lattice
the mid µ region is highly non-perturbative issue, so
I p
in Ref. [15] shows that σ˜ decreases in the high isospin
that it is not easy to understand such a region.
chemical potential associated with the decreasing pion
Apureisospinmatterwithzerobaryondensitycanbe
condensation. The decreasing pion condensation might
simulated by the lattice analysis. References [5–7] shows
beaspecialfeatureinthestrongcouplinglatticeanalysis,
thatthe phasetransitionisofthe secondorder,andthat
so that it would be interesting to study the behavior of
the critical chemical potential is equal to the pion mass.
σ˜ using the various ways.
Due to the existence ofthe signproblem,itis difficult to
Inthiswork,westudythepioncondensationphaseina
apply the lattice analysis for studying the hadron prop-
holographicQCDmodel[29,30]bysolvingtheequations
erty at the finite baryon number density. In this sense,
of motion for mean fields corresponding to π, σ and the
analysis by models may give some clues to understand
time component of ρ meson. Our results show that the
the phase structure and the relevant phenomenon in the
phase transition is of the second order consistently with
theoneobtainedintheO(p2)chiralLagrangian[3],while
it is contrary to the result in Ref. [24]. It is remarkable
∗e-mail: [email protected] thatthechiralcondensatedefinedbyσ˜ σ 2+ πa 2
≡ h i h i
†e-mail: [email protected] is almost constant in the small µI region, while it grows
p
2
with µ in the large µ region. This implies that the analysis we adopt the L = R = 0 gauge, and use the
I I 5 5
chiral symmetry breaking is enhanced by the existence IR-boundary condition FL =FR =0.
5µ|zm 5µ|zm
of the isospin chemical potential. In the vacuum the chiral symmetry is spontaneously
This paper is organized as follows: In section II we broken down to U(2) by the vacuum expectation value
V
introduce some basic point of the model which we use in of X. This is given by solving the equation of motion
the present analysis. Section III is devoted to the main as [29, 30]
partofthispaper,wherewestudythepioncondensation
1 1
together with the chiral condensation. In section IV we X (z)= m z+σz3 v(z) , (II.9)
0 q
2 ≡ 2
analyze our result in terms of the chiral Lagrangian at
O(p4). Finally, we give a summary and discussions in wheremq corresponds(cid:0)tothecurren(cid:1)tquarkmassandσto
section V the quark condensate. They are related with each other
by the IR-boundary condition:
v
II. MODEL zm∂5v IR = λv2 2m2 . (II.10)
| −2 − IR
(cid:12)
The fields are parameter(cid:0)ized as (cid:1)(cid:12)
In the present analysis we use the hard-wall holo- (cid:12)
graphic QCD model given in Refs.[29, 30]. The action X = 1 S0σ0+Saσa eiπbσb+iη , (II.11)
is given by 2
Lµ(cid:0)+Rµ (cid:1)
zm Vµ = , (II.12)
S = d4x dz , (II.1) 2
5 5
Z Zǫ L A = Lµ−Rµ , (II.13)
µ
2
where ǫ and z are the UV and the IR cutoffs. The
m VA = Tr V σA , (II.14)
5-dimensional Lagrangianis µ µ
1 AAµ = Tr(cid:2)Aµ σA(cid:3) (II.15)
=√gTr DX 2+3X 2 F2+F2 + BD,
L5 | | | | − 4g2 L R L5 whereσa (a=1,2,3)(cid:2)arePau(cid:3)limatricesandσ0 =1,and
(cid:20) 5 (cid:21)
(cid:0) (cid:1) (II.2) the superscript index A runs over0, 1, 2 and 3. The iso-
where the metric is given by singletscalarpartS0isseparatedintoabackgroundfield
partandafluctuationpartasS0 =v+S˜0. A parameter
ds2 =a2(z) η dxµdxν dz2 (II.3) g2 is determined by matching with QCD as
µν − 5
with a(z) = 1/z. The co(cid:0)variant derivative(cid:1)and the field g2 = 12π2. (II.16)
strength are given by 5 N
c
The pionis describedas alinear combinationofthe low-
D X = ∂ X iL X +iXR , (II.4)
M M − M M esteigenstateofπa andthelongitudinalmodeofAa,and
FMLN = ∂MLN −∂NLM −i[LM,LN] . (II.5) the ρ meson is the lowest eigenstate of Vµ. The µvalues
of the m and z together with that of σ are fixed by
whereM =(µ,5)is the5thdimensionalindices. BD in q m
L5 fittingthemtothepionmassmπ =139.6MeV,theρme-
Eq. (II.2) is the boundary term introduced as [31]
son mass m = 775.8 MeV and the pion decay constant
ρ
f =92.4 MeV as [29, 30]
BD = √gTr λz X 4 m2z X 2 δ(z z ) , π
L5 − m| | − m| | − m
(II.6) m =2.29MeV, z =1/(323MeV), σ =(327MeV)3 .
where zm in the (cid:8)coefficients of the X 4(cid:9)term and the q m (II.17)
X 2 term are introduced in such a w|ay| that λ and m2 By using this value of σ and a scalar meson mass as
| |
carry no dimension. This model has a chiral symmetry inputs, the values of the parameters m2 and λ in the
correspondingto U(2)L U(2)R. There exists the Chern- boundary potential are fixed. [31] It was shown[31] that
×
Simonsterminadditiontotheaboveterm. However,the there is an upper bound for the scalar meson mass as
CS term does not contribute to the pion condensation 1.2GeV, but the dependence on the mass on the value
when the spatial rotational symmetry is assumed as in of λ is small. So in the present analysis, we use the a
0
this paper. meson mass m =980 MeV as a reference value, which
a0
The scalar field X and the gauge fields LM and RM fixes m2 = 5.39 and λ = 4.4, and see the dependence of
transform under the U(2)L U(2)R as our results on the scalar meson mass.
×
X X′ =g Xg† , (II.7)
→ L R
L L′ =g L g† ig†∂ g , (II.8) III. PION CONDENSATION PHASE
M → M L M L− L M L
where g U(2) are the transformation matrices Inthissectionwestudythepioncondensationforfinite
L,R L,R
∈
of the chiral U(2) U(2) symmetry. In the following isospin chemical potential µ in the holographic QCD
L R I
×
3
model explained in the previous section. Since the pion doesnotcondensebyitself. However,theexistenceofSa
mass exists in the present model. We will have a phase condensation(a mesoncondensation)togetherwith the
0
transition from the normal phase to the pion condensa- pion condensation triggers the η condensation. The a
0
tion phase for increasing µ . In the present paper we meson condensationwill occur for µ m(a ). Since we
I I 0
≥
are interested in the pion condensation phase for small study the region of µ m , we expect that both η and
I ρ
≤
isospin chemical potential, so that we assume that the a condensations vanish in this region. Actually, we can
0
rotationalsymmetryO(3)isnotbrokenbye.g. theρme- check that η = Sa = 0 is a solution of the equations of
son condensation. We also assume the time-independent motion in the following way: By using the parametriza-
condensate, then the vacuum structure is determined by tion of Eq. (II.11), the terms including η and Saof La-
studying the mean fields of five-dimensional fields which grangian(II.2) are written as
do not depend on the four-dimensional coordinate.
UVW-beoiunntdroadryucvealtuheeoifsotshpeintimcheemcoimcaplopnoetnetnotifatlhµeIgaausgae 5 a3Tr SS U∂5U† 2+L0L0+R0R0+a2
L ∼ 4
field of SU(2)V symmetry as ( S0h2+nS(cid:0)S)(∂ η)(cid:1)2+4iS0(∂ η)S U∂ U†o
5 5 5
−
V03(z)|ǫ =µI , (III.1) [S(cid:0),(∂(cid:1)5S)] U∂5U† (∂5S)2 2SL(cid:0)0SUR0(cid:1)U†
− − −
wherethesuperscript3indicatesthe thirdcomponentof +8S0A00S L(cid:0)0−UR(cid:1)0U† (III.3)
the isospin corresponding to the neutral ρ meson. The
(cid:0) (cid:1)(cid:3)
assumptionof rotationalinvariance implies Li =Ri =0. where S = Saσa and U = eiπaσa. It is easy to confirm
ThenthewavefunctionsofV0a(z)(a=1,2,3),π0a(z)and that η = Sa = 0 together with A0 = 0 is a solution
Aa(z)aredeterminedbysolvingtheequationsofmotion. 0
0 of the equations of motion for them. Then, in following
In the present analysis, the CS term, given as analysis, we take η =Sa =0.
zm For writing the equations of motion for mean fields, it
S d4x dz is convenient to express
CS
∝
Z Zǫ
×ǫMNPQSL0MTr FNLPFQLS −(L→R) , (III.2) eiπaσa =cosb1+isinb(naσa) , (III.4)
vanishes because this ter(cid:2)m must p(cid:3)roportional to La or
i
Ra,whereL0 isagaugefieldcorrespondingtoU(1) i.e. where
i M L
L0 =Tr[L ]. The gauge field corresponding to U(1) ,
M M V πa
which includes the ω meson and its radial excitations, b=√πbπb , na = . (III.5)
does not show up in our analysis, because it couples to b
other fields only through the CS term.
TheX fieldconsistsofeightdegreesoffreedom,which We include the condition (na)2 = 1 into the Lagrangian
includeη andSa (a=1,2,3). Sincethe η isisosinglet,it using a Lagrange multiplier λ˜.
Now, the coupled equations of motion are given as
∂ a3 S0 2sin2b∂ n3 a3 S0 2 sin2bn3(nbVb)+sin2b A3 nbAb +sinbcosbǫb3cAbVc +2λ˜n3 =0,
5 − 5 − 0 0 0 0 0
(cid:16) (cid:0) ∂(cid:1) a ∂ A3(cid:17) a3(cid:0)S0(cid:1)2 (cid:2)cos2bA3+sin2bn3 nbAb (cid:0) sin(cid:1)bcosbǫbc3Vbnc =0, (cid:3)
5 g2 5 0 − 0 0 − 0
(cid:18) 5 (cid:19)
∂ a ∂ V1 a3(cid:0)S0(cid:1)2 (cid:2)sin2bV1 sin2bn1(cid:0)nbVb(cid:1)+sinbcosbǫbc1Abnc(cid:3)=0,
5 g2 5 0 − 0 − 0 0
(cid:18) 5 (cid:19)
∂ a ∂ V2 a3(cid:0)S0(cid:1)2 (cid:2)sin2bV2 sin2bn2(cid:0)nbVb(cid:1)+sinbcosbǫbc2Abnc(cid:3)=0 , (III.6)
5 g2 5 0 − 0 − 0 0
(cid:18) 5 (cid:19)
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3)
where the summation over the indices b and c are understood.
We can easily check that π3 = A3 = 0 together with At finite isospin chemical potential this theory has
0
V1 =V2 =0givesasetofsolutionsfortheabovecoupled the U(1) symmetry which is a subgroup of the isospin
0 0
equations of motion, which implies that the neutral pion SU(2) . Using this U(1) symmetry, we rotate away the
V
does not condense. Then, in the following we assume condensation of the π2 field to keep only the π1 conden-
that this set of solution is physically realized, and take sation.
π3 =A3 =0 and V1 =V2 =0.
0 0 0
4
By setting π3 =A3 =0 and V1 =V2 =0 and writing
0 0 0
eiπaσa = cosb1+isinbσ1,
Aa = θ (cosζ,sinζ,0) ,
0
V3 = ϕ+µ ,
0 I
the Lagrangianis rewritten as
= a3 ∂ S0 2 S0 2(∂ b)2
5 5 5
L 2 − −
ah3 (cid:0)S0 2 (cid:1) (cid:0) (cid:1) i
+ sin2b (ϕ+µ )2 θsin2bsinζ (ϕ+µ )+θ2 θ2sin2bsin2ζ
I I
2 − −
(cid:0) (cid:1)
+ 3a5 S0 2(cid:2) + a (∂ ϕ)2+(∂ θ)2+θ2(∂ ζ)2 . (cid:3) (III.7)
2 2g2 5 5 5
5
(cid:0) (cid:1) h i
From the above Lagrangian,the equations of motion are obtained as
∂ a3∂ S0 +a3S0(∂ b)2 3a5S0
5 5 5
− −
(cid:0) a3(cid:1)S0 sin2b(ϕ+µI)2 θsin2b sinζ (ϕ+µI)+θ2 θ2sin2b sin2ζ =0,
− − −
∂ a3 S0 2∂(cid:2)b a3 S0 2 sin2b (ϕ+µ )2 θ2sin2ζ 2θcos2b sinζ(cid:3)(ϕ+µ ) =0,
5 5 I I
− − 2 − −
(cid:0) (cid:1)
(cid:16) a (cid:0) (cid:1) a (cid:17) a3(cid:2)S0 2 (cid:8) (cid:9) (cid:3)
∂ ∂ θ θ(∂ ζ)2 sin2b sinζ (ϕ+µ )+2θ 1 sin2b sin2ζ =0,
5 g2 5 − g2 5 − 2 − I −
(cid:18) 5 (cid:19) 5 (cid:0) (cid:1)
a a3 S0 2 (cid:2) (cid:8) (cid:9)(cid:3)
∂ θ2∂ ζ + θsin2b cosζ (ϕ+µ ) θ2sin2ζ =0,
5 g2 5 2 − I −
(cid:18) 5 (cid:19) (cid:0) (cid:1)
a a3 S0 2 (cid:2) (cid:3)
∂ ∂ ϕ 2sin2b(ϕ+µ ) θsin2b sinζ =0. (III.8)
5 g2 5 − 2 I −
(cid:18) 5 (cid:19) (cid:0) (cid:1)
(cid:2) (cid:3)
Usingthe boundaryconditionslistedinTableI,wesolve We show the resultant relation between the isospin
the above coupled equations of motion to determine the chemical potential and the isospin density in Fig. 1 for
isospin chemical potential as an eigenvalue. Then, we λ = 1, 4.4 and 100 corresponding to m = 610MeV,
a0
calculate the isospin number density from the following 980MeV and 1210MeV.This shows that the phase tran-
formula obtained from the Lagrangian(III.7): sition is of the second order and the critical chemical
potential is predicted to be equalto the pion mass. This
∂ 5 is consistent with the result obtained by the chiral La-
n = dz L
I
∂µ grangianapproach in Ref. [3], but contrary to the result
Z I
a3 S0 2 in Ref. [24]. Furthermore, our result on the relation be-
= dz 2sin2b(ϕ+µ ) θsin2bsinζ . tweenisospinnumberdensityandisospinchemicalpoten-
I
2 −
Z (cid:0) (cid:1) tial for small µI agrees with the following one obtained
(cid:2) (II(cid:3)I.9) by O(p2) chiral Lagrangian[3]:
m4
n = f2µ 1 π . (III.10)
variables UV IR I π I(cid:18) − µ4I (cid:19)
S0 Sz0|ǫ =mq ∂5S0|zm = −2Szm0 (cid:16)λ(cid:0)S0(cid:1)2−2m2(cid:17)(cid:12)(cid:12)zm For µ >500MeV,there is a difference betweenour pre-
(cid:12) I
b b|ǫ =0 ∂5b|zm =0 dictionsandtheonefromO(p2)chiralLagrangian,which
θ θ|ǫ =0 ∂5θ|zm =0 canbeunderstoodasthehigherordercontributionaswe
ζ ζ|ǫ = π2 ∂5ζ|zm =0 will show in the next section.
ϕ ϕ|ǫ =0 ∂5ϕ|zm =0 We next study the µI dependences of the “σ”-
condensate corresponding to q¯q and the π-condensate
to q¯γ σaq . For obtaining htheise condensate through
5
Table I: Boundary conditions of variables. thehAdS/CiFT correspondence, we introduce the scalar
5
1.6
1
1.4
0.8 1.2 〈π1〉/〈σ〉0
] 1
3
-fm 0.6 0.8
[
n I 0.4 0.6
0.4
0.2 〈σ〉/〈σ〉0
0.2
0 0
0 mπ 200 400 600 800 0 mπ 200 400 600 800
µI [MeV] µI [MeV]
Fig. 1: Relation between theisospin numberdensitynI and Fig. 2: µI dependences of the π-condensate (red curve) and
the isospin number chemical potential µI. The green, red the“σ”-condensate (green curve).
and blue curves show our results for λ = 1, 4.4 and 100,
respectively. Thepinkdashed-curveshowstheresultgivenby
thechiralLagrangianinRef.[3]. Eachchoiceofλcorresponds
This implies that the phase transition here is the mean
to ma0 =610MeV, 980MeV and 1210MeV, respectively. field type.
Wealsoshowthe“chiralcircle”inFig.3. Itisremark-
source s and the pesudoscalar source pa as
δX δX† s 1.4
= = 1,
1.2
z z 2 400MeV
δX δX† ipa 1 300MeV
= = σa. (III.11) σ〉0 0.8 200MeV
z − z 2 〈/
1〉π 0.6
The UV-boundary term of X is written as 〈
0.4
0.2
δX†
δSUV = d4x Tr a2(∂5X)+h.c. 0
z
Z (cid:20) (cid:21)ǫ -1 -0.5 0 0.5 1
= d4x Tr δX†a ∂ X + X +h.c. . 〈σ〉/〈σ〉0
z 5 z z2
Z (cid:20) (cid:18) (cid:19) (cid:21)ǫ
(III.12)
Fig.3: Thechiralcircleisshowedastheredcurve. Theblack
We neglect the second term in the last line of the above curveis an unit circle.
equation. Then the π-condensate and “σ”-condensate
are defined as 1
able that the value of the “chiralcondensate”defined by
1 X
q¯γ5σaq Tr iσaa ∂5 +h.c. = πa , σ˜ = σ 2+ πa 2 (III.14)
h i≡2 (cid:20) (cid:18) z (cid:19) (cid:21)ǫ h i h i h i h i
1 X p
q¯q Tr a ∂5 +h.c. = σ . (III.13) is constant for increasing isospin chemical potential µI
h i≡2 z h i for µ .300MeV, and that it grows rapidly in the large
(cid:20) (cid:18) (cid:19) (cid:21)ǫ I
µ region.
I
We show the µ dependences of these condensate in
I
Fig. 2, where σ is the “σ”-condensate at µ = 0.
0 I
h i
Thisshowsthatthe“σ”-condensatedecreasesrapidlyaf-
ter the phase transition where the π-condensate grows IV. COMPARISON WITH THE CHIRAL
LAGRANGIAN
rapidly. The “σ”-condensate becomes very small for
µ & 400MeV, while the π-condensate keeps increasing.
I
Using the form πa (µ µc)ν near the phase transi- Inthis section,we compareourpredictiononthe rela-
tionpoint,wefiththie∝criticIa−lexIponentν toobtainν = 1. tion between the isospin number density and the isospin
2
chemical potential shown in Fig. 1 as well as the µ -
I
dependencesoftheπ-condensateandthe“σ”-condensate
in Fig. 2, with the ones from the chiral Lagrangian in-
1 Notethatweusehq¯qi=hu¯ui+hd¯di. cludingtheO(p4)terms. Hereweusethefollowingchiral
6
Lagrangianfor two flavor case [32, 33]: where β 1 cosα and
≡ −
LChPT = 14F02Tr DµUDµU† + 41F02Tr χ†U +U†χ A ≡ L(12)+L(22) ,
B 2L(2)+L(2) ,
+L(2) T(cid:2)r DµU†D(cid:3)U 2 (cid:2) (cid:3) ≡ 6 8
1 µ C L(2) . (IV.6)
+L(22)T(cid:0)r D(cid:2)µU†DνU T(cid:3)(cid:1)r DµU†DνU ≡ 4
We determine the value of β by minimizing the above
+L(42)Tr(cid:2)DµU†DµU(cid:3)Tr(cid:2)χ†U +U†χ(cid:3) V , and then calculate the isospin number density, the
eff
+L(2) T(cid:2)r χ†U +U†(cid:3)χ 2(cid:2) (cid:3) “σ”-condensate and the π-condensate through
6
+L(72)(cid:0)Tr(cid:2)χ†U −U†χ(cid:3)(cid:1)2 nI = ∂Veff
+L(2)T(cid:0)r χ(cid:2)†Uχ†U +χ(cid:3)U(cid:1)†χU† − ∂µI
8 =f2µ (2 β)β+16Aµ3(2 β)2β2
+iL(92)Tr(cid:2) FµRνDµU†DνU +F(cid:3)µLνDµUDµU† π1I6Cm−2µ (2 β)β2I, − (IV.7)
+L(2)Tr (cid:2)U†FLUFRµν (cid:3) − π I −
10 µν σ µ2 m2
h i =1 κ 1 8C I(2 β) 16B π (1 β) β ,
+H(2)Tr(cid:2)FLµνFL +F(cid:3)RµνFR σ − − f2 − − f2 −
1 µν µν h i0 (cid:20) (cid:18) π π (cid:19) (cid:21)
(IV.8)
+H(2)Tr(cid:2)χ†χ , (cid:3) (IV.1)
2
π µ2 m2
whereU isparametriz(cid:2)edb(cid:3)ythepseudoscalarmesonfields hσi =κ 1+ 8CfI2(2−β)−16B f2π β β(2−β) ,
as h i0 (cid:20) (cid:18) π π (cid:19) (cid:21)p (IV.9)
U =ei2π/fπ , π =πaT . (IV.2)
a where
χ includes the scalar and pseudoscalar source fields, and f2m2
κ= π π . (IV.10)
the covariant derivative D U is expressed as
µ m σ
q 0
h i
D U ∂ U i U +iU , (IV.3)
µ ≡ µ − Lµ Rµ We set κ 1.04 2, which is given by using mπ = 139.6
≃
MeV, f =92.4 MeV and (II.17).
where and areexternalgaugefieldscorresponding π
µ µ
L R We fit the values of the coefficients A, B andC to our
to SU(2) .
IntheLfo,Rllowinganalysis,wewillstudytherelationbe- result on the µI dependence of nI , hhσσii0 and hhπσi10i shown
tween the isospin number density and the isospin chem- in Figs. 1 and 2, by minimizing
ical potential as well as the condensates using the above
chiralLagrangianattreelevel. Intheordinarychiralper- n n (A,B,C) 2
I I
ttuerrmbastaiorne othfetohrey,satmheetorredee-rleavselthcoenotnrieb-ulotoiopncofrnotmribOu(tpio4n) dXate"(cid:18)fπ2µI(cid:12)(cid:12)result− fπ2µI (cid:12)(cid:12)ChPT(cid:19)
of O(p2), so that both contributions should be included. + hσi (cid:12)(cid:12) hσi 2 (cid:12)(cid:12)
However, because one-loop corrections are counted as σ − σ
(cid:18)h i0(cid:12)result h i0(cid:12)ChPT(cid:19)
next to leading order in 1/Nc expansion, we neglect the π1 (cid:12)(cid:12) π (cid:12)(cid:12) 2
one-loop corrections in the present analysis. Then, one + h i(cid:12) h i(cid:12) . (IV.11)
cansimplyintroducetheisospinchemicalpotentialµI as (cid:18)hσi0(cid:12)result− hσi0(cid:12)ChPT(cid:19) #
vacuum expectation values of these external gauge fields (cid:12) (cid:12)
(cid:12) (cid:12)
as 3 = 3 = µIδ . In this case, µ appears only Weshowthebest(cid:12)fittedvalues(cid:12)ofA, B andC forλ=4.4
thrhoLuµgih thehRcoµviarian2t d0eµrivative as I in Table II together with a set of empirical values. 3 We
µ
D U =∂ U i I[σ3,U] (IV.4) A×103 B×103 C×103
0 0
− 2
Fitting result 0.093 1.01 0.63
where σ3 is the third component of the Pauli Empirical values 0.4±1.2 1.2±0.9 1.1±0.6
matrices. Parameterizing U as U = cosα +
isinα σ1cosφ+σ2sinφ , we get the effective poten-
tial as
(cid:0) (cid:1)
2 Thedeviationofκfrom1expressesthedeviationfromtheGell-
Veff = −LChPT Mann-Oakes-Renner relation due to the contribution of H2(2)−
= fπ2µ2(2 β)β f2m2 (1 β) 2L(82).
− 2 I − − π π − 3 We calculate the empirical values of A, B and C through
4Aµ4(2 β)2β2+8Cm2µ2(2 β)β2 Eq. (IV.6), where L(i2) are determined from the values of low
− I − π I − energyconstant inthetwoflavorChPT[34]withtherenormal-
−8Bm4πβ2+(const.) (IV.5) izationscaleequaltoMη.
7
Table II: The best fitted values of A ≡ L(2) +L(2), B ≡ we assumed that the neutral pion does not condense.
1 2
2L(2)+L(2) and C ≡ L(2) compared with a set of empirical We solved the coupled equations of motion for the π-
6 8 4
values. condensateand“σ”-condensatetogetherV3todetermine
0
µ as an eigenvalue.
I
show the µ dependence of n in Fig. 4 and µ depen-
I I I
dence of hσi and hπ1i in Fig.5. These figures showthat Ourresultshowsthatthephasetransitionisofthesec-
hσi0 hσi0
the deviation of our result from the one obtained from ondorderandthecriticalchemicalpotentialispredicted
the O(p2) chiral Lagrangian is actually explained by in- to be equal to the pion mass. This is consistent with
cluding effects of O(p4) terms. the resultobtainedby the chiralLagrangianapproachin
Ref. [3], but contrary to the result in Ref. [24]. Further-
more, our result on the relation between isospin number
1 densityandisospinchemicalpotentialforsmallµ agrees
I
withtheoneobtainedbyO(p2)chiralLagrangian[3]. For
0.8
large µ (> 500MeV), there is a difference between our
] I
-3m 0.6 predictions and the one from O(p2) chiral Lagrangian,
[f which is shown to be understood as the O(p4) contribu-

nI 0.4 tions.
0.2
Wealsostudiedtheµ dependenceoftheπ-condensate
I
0 0 mπ 200 400 600 800 and“σ”-condensate. Ourresultshowsthat,atthephase
µI [MeV] transitionpoint, the π-condensate increasesfromzeroas
πa (µ µc)ν with ν = 1/2 consistently with the
h i ∝ I − I
mean-field type of the phase transition. Furthermore,
Fig. 4: µI dependence of nI obtained from the O(p4) chiral we find that the “σ”-condensate decreases rapidly af-
ter the phase transition where the π-condensate grows
Lagrangian for the best fitted values of A, B and C (green
curver) compared with our result (red curve). rapidly,whilethevalueofthe”chiralcondensate”defined
by σ˜ = σ 2+ πa 2 is constant for µ . 300MeV,
I
h i h i h i
and that it grows rapidly in the large µ region, which
I
p
1.6 is contrary to the result by the strong coupling lattice
shown in Ref. [15]. This indicates that the chiral sym-
1.4
metry restoration at finite baryon density and/or finite
1.2 〈π1〉/〈σ〉
0 temperaturewillbedelayedwhennon-zeroisospinchem-
1
ical potential is turned on. 4
0.8
0.6
Inthepresentanalysis,wedidnotincludetheeffectof
0.4
〈σ〉/〈σ〉 CS term. When the CS term is included, an additional
0
0.2
contribution from the ω-type gauge field should be in-
0 cluded. However, as far as the O(3) spatial rotation is
0 mπ 200 400 600 800
µI [MeV] kept unbroken, the result given in the present analysis
will not be changed.
Fig. 5: µI dependence of hhσσii0 and hhπσ1i0i obtained from the It is interesting to study the ρ-meson condensationby
O(p4)chiralLagrangianforthebestfittedvaluesofA,B and extendingthepresentanalysis. Insuchacase,theω-type
C (green curver) compared with our result (red curve). gaugefieldwillgivea contributionthroughthe CSterm.
Itis alsointerestingtoinclude the explicitdegreesofnu-
cleons,bywhichwecanstudythephasestructureinclud-
ing the baryon number chemical potential as well as the
V. A SUMMARY AND DISCUSSIONS isospinchemicalpotential. Thiswillbedonebyusingthe
“holographicmeanfield”approachdoneinRefs.[35,36].
We leave these analyses in future publications.
We studied the phase transition to the pion conden-
sation phase for finite isospin chemical potential using
the holographic QCD model given in Refs. [29, 30]. We
introduced the isospin chemical potential µ as a UV-
I
boundaryvalue of the time componentofthe gaugefield 4 Ourresultoftheenhancementofthechiralsymmetrybreaking
of SU(2) symmetry as V3(z) =µ . We assumed non- indicates that the critical point for the chiral phase transition
V 0 |ǫ I maybeshiftedtohigherchemicalpotentialduetotheexistence
existence of vector meson condensates since we are in-
of the isospin chemical potential. This is consistent with the
terested in studying the small µI region. Furthermore, resultofRef.[16].
8
Acknowledgements byGrant-in-AidforScientificResearchonInnovativeAr-
eas (No. 2104) “Quest on New Hadrons with Variety of
We would like to thank Shin Nakamura for useful dis- Flavors”fromMEXT,andbytheJSPSGrant-in-Aidfor
cussionsandcomments. Thisworkwassupportedinpart Scientific Research (S) No. 22224003,(c) No. 24540266.
[1] See, e.g. J. M. Lattimer and M. Prakash, Phys. Rept. [19] A. Parnachev, JHEP 0802, 062 (2008) [arXiv:0708.3170
442 (2007) 109-165 [astro-ph/0612440], and references [hep-th]].
therein. [20] O.Aharony,K.Peeters,J.SonnenscheinandM.Zamak-
[2] See, e.g. B. Friman et al., Lect. Notes Phys. 814 (2011) lar, JHEP 0802, 071 (2008) [arXiv:0709.3948 [hep-th]].
1-980, and references therein. [21] P. Basu, J. He, A. Mukherjee and H. -H. Shieh, JHEP
[3] D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86 0911, 070 (2009) [arXiv:0810.3970 [hep-th]].
(2001) 592-595 [hep-ph/0005225]. [22] M. Ammon, J. Erdmenger, M. Kaminski and P. Kerner,
[4] D. T. Son and M. A. Stephanov, Phys. Atom. Nucl. 64, JHEP 0910, 067 (2009) [arXiv:0903.1864 [hep-th]].
834 (2001) [Yad. Fiz. 64, 899 (2001)] [hep-ph/0011365]. [23] B. -H. Lee, S. Mamedov, S. Nam and C. Park, JHEP
[5] J.B.KogutandD.K.Sinclair,Phys.Rev.D66,014508 1308, 045 (2013) [arXiv:1305.7281 [hep-th]].
(2002) [hep-lat/0201017]. [24] D. Albrecht and J. Erlich, Phys. Rev. D 82, 095002
[6] J.B.KogutandD.K.Sinclair,Phys.Rev.D66,034505 (2010) [arXiv:1007.3431 [hep-ph]].
(2002) [hep-lat/0202028]. [25] F. Rust, Adv. High Energy Phys. 2010, 564 (2010)
[7] W. Detmold, K. Orginos and Z. Shi, Phys. Rev. D 86, [arXiv:1003.0187 [hep-th]].
054507 (2012) [arXiv:1205.4224 [hep-lat]]. [26] Y.Kim,Y.Seo,I.-J.ShinandS.-J.Sin,EPJWebConf.
[8] D. Toublan and J. B. Kogut, Phys. Lett. B 564, 212 20, 03005 (2012).
(2003) [hep-ph/0301183]. [27] K.-I.Kim,Y.KimandS.H.Lee,arXiv:0709.1772 [hep-
[9] A. Barducci, R. Casalbuoni, G. Pettini and L. Ravagli, ph].
Phys.Rev.D 69 (2004) 096004 [hep-ph/0402104] [28] Y.Kim,Y.Seo,I.J.ShinandS.-J.Sin,arXiv:1108.2751
[10] L.-y. He, M. Jin and P.-f. Zhuang, Phys. Rev. D 71 [hep-ph].
(2005) 116001 [hep-ph/0503272] [29] J.Erlich,E.Katz,D.T.SonandM.A.Stephanov,Phys.
[11] L.-y. He, M. Jin and P.-f. Zhuang, Mod. Phys. Lett. A Rev. Lett.95 (2005) 261602 [hep-ph/0501128]
22 (2007) 637-643 [hep-ph/0503249] [30] L. D. Rold and A. Pomarol, Nuclear Physics B 721
[12] L. He, M. Jin and P. Zhuang, Phys. Rev. D 74 (2006) (2005) 79-97 [hep-ph/0501218]
036005 [hep-ph/0604224] [31] L. D. Rold and A. Pomarol, JHEP 0601 (2006) 157
[13] J.O.AndersenandL.Kyllingstad,J.Phys.G37(2009) [hep-ph/0510268]
015003 [hep-ph/0701033] [32] J. Gasser and H. Leutwyler, Nuclear Physics B 250
[14] B. Klein, D. Toublan and J. J. M. Verbaarschot, Phys. (1985) 465-516
Rev.D 68, 014009 (2003) [hep-ph/0301143]. [33] M.HaradaandK.Yamawaki,Phys.Rept.381,1(2003)
[15] Y. Nishida, Phys. Rev. D 69 (2004) 094501 [hep-ph/0302103].
[hep-ph/0312371] [34] J. Gasser and H. Leutwyler, Annals Phys. 158 (1984)
[16] H. Abuki, Phys. Rev. D 87, 094006 (2013) 142
[arXiv:1304.1904 [hep-ph]]. [35] M. Harada, S. Nakamura and S. Takemoto, Phys. Rev.
[17] D. Toublan and J. B. Kogut, Phys. Lett. B 605, 129 D 86, 021901 (2012) [arXiv:1112.2114 [hep-th]].
(2005) [hep-ph/0409310]. [36] B. -R. He and M. Harada, Phys. Rev. D 88, 095007
[18] Y. Kim, C. -H. Lee and H. -U. Yee, Phys. Rev. D 77, (2013) [arXiv:1304.7866 [hep-ph]].
085030 (2008) [arXiv:0707.2637 [hep-ph]].