Table Of ContentChiral Symmetry Breaking in the Dynamical Soft-Wall Model
Qi Wang1 , An Min Wang2
∗ †
1Department of Modern Physics, University of Science and Technology of China, Hefei 230026, PR China
2Department of Modern Physics, University of Science and Technology of China, Hefei 230026, PR China
Inthispaperamodelincorporatingchiralsymmetrybreakinganddynamicalsoft-wallAdS/QCD
is established. The AdS/QCD background is introduced dynamically as suggested by Wayne de
Paulaetcalandchiralsymmetrybreakingisdiscussedbyusingabulkscalarfieldincludingacubic
term. The mass spectrum of scalar, vectorand axial vector mesons are obtained and a comparison
with experimentaldata is presented.
PACSnumbers:
2
Keywords:
1
0
2
I. INTRODUCTION breaking in this model is not QCD-like. Some further
n researches try to incorporate chiral symmetry breaking
a andconfinementinthis modelbyusing highorderterms
Quantum Chromodynamics (QCD) is an non-abelian
J
in the potential for scalar field [10] [11].
gaugetheoryofthestronginteractions. Ithasacoupling
6
1 constantthathighlydependsontheenergyscale,andthe But the dilaton field and scalar field in the soft-wall
perturbation theory doesn’t work at low energies. The model are imposed by hand to and cannot be derived
] Anti-de Sitter/conformal field theory (AdS/CFT) corre- fromanyequationofmotion. Someworksonthedilaton-
h
spondence[1]providesanovelwayto relateQCDwitha gravitycoupledmodelshowthatalinearconfiningback-
p
- 5-D gravitationaltheory and calculate many observables ground is possible as a solution of the dilation-gravity
p more efficiently. This leads to two types of AdS/QCD coupled equations and Regge trajectoriesof mesonspec-
e models: the top-down models that are constructed by trum is obtained [7] [8].
h
branes in string theory and the bottom-up models by
[ In this paper, we try to incorporate soft-wall
the phenomenological aspects. In the top-down model,
AdS/QCDand dynamicaldilaton-gravitymodel. We in-
1 mesons are identified as open strings with both ends on
v flavorbranes[2]. Themostsuccessfultop-downmodelin troducethe mesonsectoractionwithacubic termofthe
9 bulk field in the bulk scalarpotential under the dynami-
reproducing various facts of low energy QCD dynamics
4 calmetricbackground. Wederiveanonlineardifferential
is D4-D8 system of Sakai and Sugimoto [3] [4].
3 equation related the vacuum expectation value(VEV) of
3 The bottom-up approachassumes that QCD has suit- bulk field, wrap factor and dilaton field. An analysis of
. able 5D dual and construct the dual model from a
1 the asymptotic behavior of the VEV is presented. We
small number of operators that are influential. The first
0 obtain that the VEV is a constant in the IR limit that
2 bottom-upmodelis the so-calledhard-wallmodel[5] [6]; suggests chiral symmetry restoration exists. The cou-
1 it simply place an infrared cutoff on the fifth dimension
plingconstantcanbedeterminedbyotherparametersin
: to break the chiral symmetry. But in hard-wall model
v this model.
the spectrum of mass square m2 grow as n2, which is
i n
X contradictwith the experimentaldata [12]. The analysis This paper proceeds as follow: A brief review of dy-
r of experimental data indicates a Regge behavior of the namical gravity-dilaton background is presented in Sec.
a highly resonance m2 n. To compensate this difference II. In Sec. III we introduce the bulk field and meson
between QCD andnAd∼S/QCD, soft-wall model was pro- sector action under dynamical metric background and
posed [9]. In this model the spacetime smoothly cap off obtain the differential equation of the bulk field VEV.
instead of the hard-wall infrared cutoff by introducing a An analysis of asymptotic behavior is presented and a
background dilaton field Φ: parametrized solution is given. Using the parametrized
solution we calculate the scalar, vector and axial vector
mass spectrums in Sec. IV. A conclusion is given in Sec.
S = d5x√ ge Φ(z)L,
5 − − − V.
Z
where the backgroundfieldis parametrizedto reproduce
Regge-like mass spectrum. In spite of success in re-
producing Regge-like mass spectrum, chiral symmetry
II. BACKGROUND EQUATIONS
∗Email:[email protected] We start from the Einstein-Hilbert action of five-
†Email:[email protected] dimensional gravity coupled to a dilaton Φ as proposed
2
by de Paula, Frederico, Forkel and Beyer [7]: In order to describe chiral symmetry breaking, a bi-
fundamental scalar field X is introduced and we add an
cubictermpotentialinsteadofthequartictermasshown
1 1
S = d5x√ g R+ gMN∂ Φ∂ Φ V(Φ) in [10]
2κ2 − − 2 M N −
Z (cid:18) (cid:19)
(1)
where κ is the five-dimensional Newton constant and V S5 =− d5x√−ge−Φ(z)Tr[|DX|2+m2X|X|2
Z
is a still general potential for the scalar field. Then we
restrict the metric to the form
1
λX 3+ (F2+F2)] (11)
gMN =e−2A(z)ηMN (2) − | | 4g52 L R
wfahcteorer aηMs N is the Minkowski metric. We write the warp 4wπh2e.reTmheXfie=lds−F3, λ iasrae dcoefinnsteadntbyand g52 = 12π2/Nc =
L,R
A(z)=lnz+C(z) (3) FMN =∂MAN ∂NAM i[AM ,AN ],
L,R L,R− L,R− L,R L,R
Variationoftheaction(1)leadstotheEinstein-dilaton here AMN =AMNta, Tr[tatb]= 1δab, and the covariant
L,R L,R 2
equations for the backgroundfields A and Φ: derivative is DMX =∂MX iAMX +iXAM.
− L R
1
6A2 Φ2+e 2AV(Φ) = 0 (4)
′ ′ −
− 2
A. Bulk scalar VEV solution and dilaton field
1
3A 3A2 Φ2 e 2AV(Φ) = 0 (5)
′′ ′ ′ −
− − 2 −
The scalar field X is assumed to have a z-dependent
dV
Φ′′ 3A′Φ′ e−2A = 0 (6) VEV
− − dΦ
v(z) 1 0
Here the prime denotes the derivative with respect to z. <X >= (12)
2 0 1
By adding the two Einstein equation ones can get the (cid:18) (cid:19)
dilaton field directly:
We can obtain a nonlinear equation related A, Φ and
v(z) from the action (11)
Φ =√3 A2+A (7)
′ ′ ′′
3
and by substituting (IIIpA) in (4) or (5): ∂z(e−3Ae−Φ∂zv(z))+e−5Ae−Φ(3v(z)+ λv2(z))=0
4
(13)
3e2A(z) If A and Φ are given, this equation can be simplified
V(Φ(z))= A (z) 3A2(z) (8)
′′ ′
2 − to a second order differential equation of v(z)
(cid:2) (cid:3)
If A(z) = log(z)+zλ is set for simplicity, the asymp- 3
v (z) (Φ +3A)v (z)+e 2A(3v(z)+ λv2(z))=0 (14)
toticbehaviorofdilationfieldΦinUVandIRlimitscan ′′ − ′ ′ ′ − 4
be obtained
where Φ =√3√A2+A .
′ ′ ′′
Firstly, we consider the limit z . If wrap factor
Φ(z)= z →0 2 3(1+λ−1)zλ2 (9) C(z) has a asymptotic behavior of→z2∞as z →∞, we can
−z−−→ √3zλ determine that v(z) constant. This means that chiral
( →∞ p symmetry restoratio→n in the mass spectrum. But there
−−−−→
aremanycontroversiesonwhether sucha restorationre-
ally exists [13] [14].
III. THE MODEL
Secondly, we analyze the asymptotic behavior of v(z)
as z 0. As shown in [15] [16], the VEV as z 0 is
In this section, we will establish the soft-wall → →
required to be
AdS/QCDmodelfirstintroducedin[9]underthedynam-
ical background mentioned in Sec. II. The background v(z)=m ζz+ σz3 (15)
q
geometry is chosen to be 5D AdS space with the metric ζ
ds2 =gMNdxMdxN =e−2A ηµνdxµdxν +dz2 (10) where mq is the quark mass, σ is the chiral condensate
and ζ = √3.
whereAisthewarpfactor,and(cid:0)ηµν isMinkowski(cid:1)metric. We ass2uπme that C(z) takes the asymptotic form as
Unlike[9]weintroduceabackgrounddilatonfieldΦthat z 0
is coupled with A and Φ can be determined in terms of →
A as shown in Sec. II. C(z)=αz2+βz3 (16)
3
Then we can get the asymptotic form of Φ in the IR
′
limit. Considering the coefficients of 1 and constant
z 1 1
terms at the left side of Equation (14) should be elim- −∂z2Vn(z)+(4ω′2− 2ω′′)Vn(z)=m2VnVn
inated as z 0, we can conclude that the cubic term of
→
bulk fieldis necessaryanda relationbetween α, β andλ
can be obtained
1 1
∂2A (z)+( ω2 ω +g2e 2Av2(z))A (z)=m2 A (z)
λm ζ =4√2α. (17) − z n 4 ′ −2 ′′ 5 − n An n
q
(23)
where the prime denotes the derivative with respectto z
B. A parametrized solution and parameter setting andω =Φ(z)+3A(z)forscalarmesons,ω =Φ(z)+A(z)
for vector and axial vector mesons.
The meson mass spectrum is obtained through calcu-
The equation (14) is a nonlinear differential equation
lating the eigenvalues of these equations. We will calcu-
and it is difficult to solve the VEV v(z), so we choose
late the eigenvalues of these equations numerically using
to select a parametrized form of v(z) that satisfies the
the parametrized solution introduced in Sec. III.
asymptotic constraintinstead ofsolving the equationdi-
Since the potential is complicated, we must solve the
rectly.
eigenvalueequationnumerically. Assuggestin[7],weset
We set the wrap factor as suggested in [7]
ξ =0.58 for scalars and the numerical results are shown
(ξΛ z)2 in Table I.
QCD
A(z)=logz+ (18)
1+exp(1 ξΛ z)
QCD
−
TABLE I: Scalar mesons spectra in MeV.
whereξ isascalefactorandΛ =0.3GeVistheQCD
QCD
scale. n f0(Exp) f0(Model)
Also we assume the VEV v(z) asymptotically behaves 0 550 897
as discussed in previous section 1 980 1135
z γ 2 1350 1348
v(z)= →∞ (19)
−z−−−0→ m ζz+ σz3 3 1505 1540
(cid:26) → q ζ
−−−→ 4 1724 1717
A simple parametrized form for v(z) that satisfies
5 1992 1881
asymptotic constraint (19) can be chosen as
6 2103 2034
z(A+Bz2) 7 2134 2178
v(z)= (20)
1+(Cz)6
The relations between mqp, σ, γ and A, B, C are: Forvectors,the value ofξ is chosento be 0.88andthe
numerical results are listed in Table II.
σ B
A=m ζ,B = , =γ (21)
q ζ C3
TABLE II:Vectormesons spectra in MeV.
Using the data of meson mass data, pion mass and
n ρ(Exp) ρ(Model)
pion decay constant we can get a good fitting of A, B,
C, λ as follows: 0 775.5 988.9
1 1465 1348
2 1720 1650
4√2ξΛ
A=1.63MeV,B =(157MeV)3,C =10,λ= QCD 3 1909 1913
m ζ√1+e
q
4 2149 2147
(22)
In next section we will use this parametrized solution 5 2265 2359
to calculate scalar, vector and axial-vector meson mass
spectra and compare them with experimental data.
Sine ∆m2 =m2 m2 =g2e 2Av2(z) tends to zero
A2n − Vn2 5 −
as z 0, it means the mass of vector and axial-vector
IV. MESON MASS SPECTRUM meso→ns are equal at high values.
Starting from the action (11), we can derive the
Schro¨dinger-like equations that describe the scalar, vec- V. CONCLUSION
tor and axial vector mesons:
1 1 3 In this paper we incorporated chiral symmetry break-
−∂z2Sn(z)+(4ω′2−2ω′′−e−2A(3+2λv(z)))Sn(z)=m2SnSn(zi)ngintothesoft-wallAdS/QCDmodelwithadynamical
4
background. Then a discussion about the asymptotic study,numericallyandanalytically,toobtainamorepre-
behavior of the VEV of the bulk field was presented and ciseformofv(z). Themassspectraofnucleonsandsome
a chiral symmetry restoration was suggested in the IR more complicated properties of mesons and nucleons in
limit. We also introduced a parametrizedsolution of the this model will be our following work.
VEVofthe bulk fieldandfittedthe parametersbyusing
themesonmassspectraandpionmassanditsdecaycon-
stant. Thenumericalsolutionofmesonmassspectraand
comparison with experimental data was also considered.
VI. ACKNOWLEDGEMENT
The agreement between the theoretical calculation and
experimental data is good.
There aresome issuesworthyof further consideration. ThisworkhasbeensupportedbytheNationalNatural
The differential equation of the VEV v(z) needs further ScienceFoundationofChinaunderGrantNo. 10975125.
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