Holographic model for hadrons in deformed AdS5 background

K. Ghoroku (Fukuoka Tech.)N. Maru (U. Rome)M. Yahiro (Kyushu U.)M. Tachibana (Saga U.) Phys.Lett.B633(2006)606

1. Introduction and motivation2. An example of holographic model for QCD3. Analysis in deformed AdS5 background4. Summary and discussion

2006/03/01 @KEK

Plan of talk

1. Introduction and motivation2006/03/01 @KEK

QCD, the theory of strong nuclear force, is nortoriouly intractable, due to the nature of strong coupling.

Then one needs appropriate effective models for QCD. Chiral Lagrangian is one of them, based on chiral symmetry.

Meanwhile, vector mesons play a significant role in hadron physics, though their interactions are not constrained by low-energy theorems.

How to incorporate vector mesons into field theoretical frameworks? What is the EFT involving vector mesons? What symmetry behind?

Holographic QCD via AdS/CFT

Original idea of AdS/CFT correspondence (Maldacena ‘98)

Nc D3-branes

N=4 U(Nc) Super Yang-Mills in 4d

Type IIB SUGRA on AdS5×S 5

(gauge theory on the boundary)

(gravity theory in the bulk)

Gubser-Klebanov-Polyakov ‘98Witten ‘98The field operator correspondence

Correlation functions in the gauge theory can be computed by differentiatingwith respect to a field living in the bulk (= source for the boundary operator)

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Field theory at strong coupling ⇔ Classical (super)gravity (string)

Let’s apply this Idea to Hadron Physics!

First applications to QCD (pure Yang-Mills)

1. Wilson loop operator in N=4 SYM (Maldacena ‘98)

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E(L) = −4π 2 2gYM

2 Nc( )1/ 2

Γ 1/4( )4L

∝ gYM2 Nc

2. Glueball mass spectra (Csaki et al., Koch et al., Zyskin, Minahan, Ooguri et al ‘98 )

Two point function of glueball operators : (e.g., )

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O

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O = Tr(Fμν2 )

Solving the field eq. (e.g., for dilaton) in some curved background

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∂μ ggμν ∂ν Φ( ) = 0

These are Yang-Mills staffs…..2006/03/01 @KEK

Adding flavor (Karch-Katz ‘02)

Applications into flavor physics

Nc D3-branes Nf D7-branes (flavor brane)€

q fc

Probe approximation:

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N f << Nc

Karch-Katz-Weiner(‘02), Myers et al.(‘03), Sakai-Sonnenschein(‘03), Nunez et al.(‘03),Babington et al(‘03), Ghoroku-Yahiro(‘04),Sakai-Sugimoto(‘04), etc.

Chiral symmetry breakingMeson spectraQuark mass effect etc

2006/03/01 @KEK

“flavor quark”

Supersymmetric/Non-supersymmetric

“Bottom-Up approach”

“One starts from four dimensional QCD, and attempts to guess its five dimensional holographic dual.”

Erlich-Katz-Son-Stephanov (EKSS) (‘05)Da Rold-Pomarol (DRP) (‘05)Brodsky-Teramond (‘05) Hirn-Sanz (‘05) Katz-Lewandowski-Schwartz (‘05)Hambye et al. (‘05)(see also N. Evans, Nature 439, 921 (‘06))

AdS/QCD

Various quantities being computed:

Shifman, hep-ph/0507246

Holographic models for hadrons

meson and baryon spectra (e.g., vector, axial-vector mesons) decay constants coupling among mesons some QCD formula (sum rules)

2006/03/01 @KEK

Non-supersymmetric models

2. An example of holographic model for QCD -- The EKSS/DRP model --

The model

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m5( )2

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4D : Ο(x)

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5D : φ(x,z)

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p

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ΔΟ

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q Lγ μ t aqL

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q Rγ μ t aqR

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q Rα qL

β

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ALμa

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ARμa

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2 /z( )X αβ

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1

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1

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0€

3

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−3

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3

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3

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0

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0

5D geometry

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ds2 =1

z2−dz2 + dx μ dxμ( )

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0 < z < zm

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zm : IR cutoff

AdS5

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Δ 0 + p( ) Δ0 + p − 4( ) = m52

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m5 : 5D mass of the p-form field

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ΔΟ : conformal dim. of operator O

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“A minimal set of operators associated with chiral dynamics”

The action

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S = d4 x∫ dz gTr DX2

+ 3 X2

−1

4g52

FL2 + FR

2( )

⎧ ⎨ ⎩

⎫ ⎬ ⎭0

zm∫

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DM X = ∂M X − iALM X + iXARM

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AL,R = AL ,Ra t a

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F = dA

Symmetry

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X → LXR+

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FL → LFLL+

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FR → RFR R+

Gauged flavor symmetry

Chiral symmetry breaking (Classical solution for X field)

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X0(z) =1

2Mz +

1

2Σz3

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SU(N f )L ⊗ SU(N f )R

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M = μ,z( )

Free parameters

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Σαβ = q α qβ

where

Chiral condensate

We take

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Σαβ =σδαβ

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M = mq1N f ×N fQuark mass

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mq

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σ

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zm

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g5

(cf. compared to QCD: )

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mq

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ΛQCD

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Nc

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Below analysis is done in the case of Nf=2 light flavors

taken from thetalk by M.Stephanov

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Vμ ~ ALμ + ARμ

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Aμ ~ ALμ − ARμ

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X = X0 exp(i2π a t a )

vector/axial-vector mesons

: pseudo scalar mesons

*:input

3. Analysis in deformed AdS background

3. Analysis in deformed AdS background

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Sg = d4 xdz −g1

2κ 2R − 2Λ( ) −

1

2∂φ( )

2−V φ( )

⎧ ⎨ ⎩

⎫ ⎬ ⎭∫

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φ : bulk scalar field

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ds2 = A2(z) η μν dx μ dxν + dz2( )

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A(z) : warp factor

Solutions of Einstein’s eq.

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μ =−˜ Λ

6

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λ =1

3κ 2α 2

Bulk action

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A(z) =1

μ

λ

sinh λ z[ ]

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λ =0

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1

μz(AdS5)

EKSS/DRP€

φ(z) = αz

Mass of the adjointfermions of N=4 SYM

2006/03/01 @KEK

In this sense, λis the modification parameter, whichcorresponds to the supersymmetry breaking and givessome change to the system of our interest at the infrared.

How this parameter affects the physical observables (Mv, Fv) including some higher excited states.

Singlet scalar excitation (σ) which was not taken into accountbefore (but see, Da Rold-Pomarol, hep-ph/0510268).

How much the results depend on the selected 5D background?

In short, our interests here are

2006/03/01 @KEK

Scalar and vector meson part

Chiral symmetry breaking

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v(z) =1

A3(z)c + mq −λ ⋅sinh−1 A(z) / λ( ) + A(z) A2(z) + λ[ ] ⎛ ⎝ ⎜ ⎞

⎠ ⎟

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λ → 0

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v(z) = mqz + cz3

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mq :

c :

quark mass

chiral condensatethus

Singlet scalar meson (σ)

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1

A2m2 + ∂z

2 + 3∂z A

A∂z

⎡

⎣ ⎢ ⎤

⎦ ⎥σ = −3σ

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∂μ∂μσ =m2σ( )

with

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σ(z)z=ε

= ∂zσ (z)z= zm

= 0

2006/03/01 @KEK

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mnV

( )2

+ A−1∂z A∂z[ ] fnV = 0

Vector meson ( gauge )

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Vz = 0

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Vμ (x,z) = Vμ(n )(x) fn

V (z)n

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

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mnV: the n-th excited vector meson mass

The 1st excited vector meson mass, , is obtained as the function of and . By utilizing the experimental value of , can beexpressed as the function of . As the result, the 2nd and all highlyexcited vector meson masses depend on only .

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mρ

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λ

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zm

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mρ

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zm

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λ

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λ

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λ

We find that the 1st and the 2nd excited vector mesonare fitted by the parameters and .

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zm = 4

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λ =0.8

2006/03/01 @KEK

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Fρ1/ 2€

m2V

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mσ

A comment (a bit long, but important)

In the sector of scalar and vector mesons, the masses (also the decay constants)are independent with (or equivalently, the quark mass and the chiralcondensate ). This is contrasted with the results from, e.g., the QCD sum rule,where the vector meson masses are the function of them. This is due to the factthat spontaneous chiral symmetry breaking is introduced by hand in this model,which was done by the choice of the profile of the scalar field X and this featureseems to be common to many models of holographic QCD.

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v(z)

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mq

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c

(For QSR, Krasnikov-Pivovarov-Tavkhelidze ‘83, Reinders-Rubinstein-Yazaki ‘85)

Axial vector sector Vector and axial vector sectors

a: decay const. b: meson mass c: pion decay const.

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a1

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a1 d=a, e=b, g=c h: ρ meson massf: 2nd excited vector mass

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m2V

As for a) and b), the agreement of thetheoretical results with experimentBecomes better as λ decreases. Onethen sees that λ=0 case, i.e., the AdScase, yields a best fit.

As for d), e) and g), λ=0 is the bestfit. But once we take into account as well as , the best value will beshifted to some small value of λ.

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m2V

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mσ

4. Summary and discussion

Application of the idea of AdS/CFT duality to QCD dynamics

“ Bottom-up” approach = holographic model for hadrons (gauged flavor symmetry)

Some properties of scalar, vector and axial vector mesons(comment on v(z) independence of scalar and vector sector)

2006/03/01 @KEK

Outlook

Glueballs, baryons and U(1) vector mesons

Strange mesons

Finite temperature and/or density = phase transition(for finite T, Ghoroku-Yahiro, hep-ph/0512289)

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U(N f )L ⊗U(N f )R = U(1)L ⊗U(1)R ⊗ SU(N f )L ⊗ SU(N f )R( )

2006/03/01 @KEK

Vector meson

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Vμ =1

2ALμ + ARμ( )

Linearlized eq. of motion ( gauge)

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Vz = 0

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∂z

1

z∂zV⊥(q,z)

⎛

⎝ ⎜

⎞

⎠ ⎟+

q2

zV⊥(q,z) = 0

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V⊥(q,z)

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V0(q)

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z

The action on the solution

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S5 = −1

2g52

d4 x1

z∫ Vμ

a∂zVaμ

z=ε

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V0μa (q) :

source of thevector current

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Jμa = q γ μ t aq

then

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V μ (q,z) = V (q,z)V0μ (q) with

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V (q,ε) =1

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ε : UV cutoff

Current correlator

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e iqx Jμa (x)Jν

b (0) = δ abx

∫ qμqν − q2gμν( )ΠV (Q2)

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ΠV (−q2) = −1

g52Q2

∂zV (q,z)

zz=ε

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Q2 = −q2

since

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V (q,z) = (Qz)K1(Qz) =1+Q2z2

2log(Qz) +L

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ΠV (Q2) = −1

2g52

logQ2c.f. In QCD,

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ΠV (Q2) = −Nc

24πlogQ2

Thus

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g52 =

12π 2

Nc2006/03/01 @KEK

“Hadrons”Normalizable modes of the 5D fields

Eigenvalue of normalizable mode : Squared mass of meson Derivative of the mode : Decay constant of the meson

and

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ψρ (ε) = 0,

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∂zψ ρ (zm ) = 0,

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(dz /z)ψ ρ (z)2∫ =1.

then

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ΠV (−q2) = −1

g52Q2

∂zV (q,z)

zz=ε

= −1

g52

′ ψ ρ (ε) /ε[ ]2

(q2 − mρ2 )mρ

2ρ

∑

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Fρ =1

g5

′ ′ ψ ρ (0) decay constant

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ψρ (z)Let be a solution of eq. of motion for vector meson with

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q2 = mρ2

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Axial vector and pion

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Aμ =1

2ALμ − ARμ( ),

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X = X0 exp(i2π a t a )

Eqs. of motion

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∂z (z−1∂z A⊥) + z−1q2A⊥− z−3g5

2v 2A⊥ = 0;

∂z (z−1∂zϕ ) + z−3g5

2v 2(π −ϕ ) = 0;

−q2∂zϕ + z−2g52v 2∂zϕ = 0.

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Aμ = Aμ⊥+ ∂μϕ

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v(z) = mqz + σz3

GOR relation

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2mq q q = −mπ2 fπ

2

is reproduced.

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Nc

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g5 = 12π 2 /Nc = 2π

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mρ = 2.405 /zm

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zm = (323MeV )−1

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fπ

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mπand

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σ =(327MeV )3

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mq = 2.29MeVand

taken from thetalk by M.StephanovInput

What they have not done

2006/03/01 @KEK

(i) Glueball spectrum

(ii) OPE and higher order terms

(iii) strange mesons

(iv) Chiral anomaly (WZW)

(v) Baryons (Brodsky-Teramond)

(vi) running of the gauge coupling

3.2 Axial vector meson and pion part

Unlike the previous case, the masses and decay constants of axial vectorand pion depend on four parameters and .

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mq,

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c,

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zm

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λ

Gell-Mann-Oakes-Renner (GOR)

quark mass dependence

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a1 meson mass

pion mass

GOR relation

Solid line: direct calc. of

Dotted circle: obtained from calculated through the GOR

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mπ2

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Fπ €

λ =0.1,

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c = (325MeV )3

Two results agree with each other!

In the right figure, when calculated pion mass and the decay constantwell reproduces the experimental values.

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mq = 2.41MeV