Hadron Spectra and Quark Mass Dependence in Holographic QCD

　 Koji Hashimoto (RIKEN)

9th Feb. 2010@ NFQCD workshop (YITP)

arXiv/0803.4192 (JHEP) Hirayama, Lin, Yee, KH

arXiv/0906.0402 Hirayama, Hong, KHarXiv/0910.1179 Iizuka, Ishii, Kadoh, KH

Low energy effective field

theory on D-branes

Holographic QCD

QCD action

Graviry descriptionof

those D-branes

superstring theory

AdS/CFTcorrespondence

Action of hadronsHadron specturm,

interactions

?

Large Nc, strong coupling

Plan

1. A Holographic QCD

2. Mesons and Quark masses

3. Holographic Baryons

4. Baryons and Quark masses

1. A Holographic QCD

[Sakai,Sugimoto(04)]

Sakai-Sugimoto model

U(Nf) Yang-Mills-Chern-Simons theory in curved space

zImportant features:

The gauge group is flavor U(Nf).

Extra “holographic” dimension .

Chiral rotation is defined at

This is the holographic dual of massless QCD, with large and large ‘tHooft coupling

Eigen values correspond to masses of vector mesons.

Comparison with experimental data

KK modes of 　　　　 Vector mesonsA KK mode of Massless pion

Kaluza-Klein decomposition gives mesons

Going to gauge, integration

Chiral lagrangian

Derivation of chiral lagrangian

The action with

- Skyrm term is derived !

Good agreement with experimental data :

- Inclusion of the vector mesons, easy

Natural realization of hidden local symmetry

2. Mesons and Quark masses

arXiv/0803.4192 (JHEP) Hirayama, Lin, Yee, KH

Worldsheet instanton gives quark masses

Put a D6-brane as a probe.

D6 charge is (1,-1) underthe chiral symmetry : Explicit breaking of the chiral symmetry

Worldsheet instanton for the quark mass is given in flat spacetime

D6 spike

Derivation of the quark mass term in Chiral lagrangian

D6 spikeD4 throat

Worldsheet instantonlooks same.

: effective coupling

Chiral condensate, flavor dependence

D6

On each D8, D6 ends

Standard chiral lagrangian !

2D8

2D8

Numerics ( just for illustration )

From pion decay constant and rho meson mass,

We substitute ,

Cf) Experiments :

Pion / quark mass

Chiral condensate

Cf) Lattice :

Other terms

Vector / axial vector mesons :

Mass shift is suppressed by

Pion mass differences ? two worldsheet instantons

3. Holographic Baryons

[Sakai,Sugimoto(04)][Hata,Sakai,Sugimoto,Yamato(07)]

Cf. [Hong, Rho, Yee, Yi (07)]

Baryon = D4-brane wrapping S4

= Instanton in of SS model

Baryons = YM instantons

[Witten(98), Gross,Ooguri(98)]

[Sakai, Sugimoto(04)]

Instanton charge sources U(1)v

[Hata, Sakai, Sugimoto, Yamato (07)][Hong, Rho, Yee, Yi (07)]

Quantization of instantons Baryon spectrum

Baryon solution : dyonic instanton

The background can be approximated by flat space

Solution : BPST instanton + electrostatic potential

Inserting this back to the action leads to a potential

Size is stabilizedto be small,

Small instanton localized at

U(1) Coulomb self-repulsion Effect of curved space

Quantization of the instanton

Moduli space approximation : Moduli with small potentials

Moduli :

Lagrangian :

,

: isospin + spin

Harmonic-like potential Baryons labeled by

Baryon states are given by wave function of the QM :

Proton :

acted by

Mass spectrum of baryons

Baryon mass formula :

PDG:

(tables taken from [Hata et.al])

4. Baryons and quark masses

[Hirayama, Hong and KH, 0906.0402]

[Sakai, Sugimoto and KH, 0806.3122]

Quark mass term and baryon

Quark mass is introduced by worldsheet instantons

[Hirayama, Lin, Yee and KH, 0803.4192] [Aharony, Kutasov]

Pion mass is induced as

We substitute the BPST instanton in the singular gauge

Baryon mass shift

Then we obtain

The baryon mass shift is given by

Using the wave function for rho coordinate,

for nucleons

for Roper etc

Results and Comparisons

Pion mass dependence of nucleon mass

higher order.

Our result :[Hirayama, Hong and KH, 0906.0402]

Lattice QCD :

Pion mass dependence of Roper mass

Our result :

[Brommel et al., 0804.4706] [Bernard, 0706.0312] [QCDSF-UKQCD hep-lat/0312030][Walker-Loud et al., 0806.4549] [PACS-CS 0810.0351]

Three flavors

The computation goes exactly the same, except that we have now SU(3) chiral rotations.

We use the following three for the mass parameters,

Then the baryon mass shift is

Decouplet / Octet mass shifts

Our results :

Comparison with experiments

Our inputs :

Mass splittings of baryons with

Mass splittings of baryons with

5. Conclusion & Discussions

Conclusion

String theory technique provides quite a nice description of mesons and baryons. Spectroscopy is one of the quantities which can be computed in holographic QCD.

Derivation of quark mass term in the chiral lagrangian. Chiral condensate, GOR relation

[Hirayama, Lin, Yee, KH, 0803.4192]

Pion mass dependence of nucleon / Roper mass.[Hirayama, Hong, KH, 0906.0402]

Pion / K mass dependence of Octet / Decouplet mass.[Iizuka, Ishii, Kadoh, KH, 0910.1179]

Other quantities computed

We compute also static properties of baryons, including meson-baryon coupling.

We compute Nuclear force at short distances

Long range nuclear forceThey nicely match exp. data

= First analytic result reproducing repulsive core, from strongly coupled QCD

[Sakai, Sugimoto and KH, 0806.3122]

[Sakai, Sugimoto and KH, 0901.4449]

Three-body nuclear force.

[Iizuka, Nakatsukasa, KH, 0911.1035]

Work in progress

3-body nuclear force

Nuclear force among hyperons/nucleons

N-body in general ?

Color-flavor locking in holographic QCD

Chiral properties of hyperons

[Chen, Matsuura, KH, 0909.1296]

Static quantities of nucleons derived

input :

Glueball sector

Closed string side (gravity)

[Witten (98)]

Witten’s geometry for pure YM without SUSY

Nc D4-branes wrapping S1

Gaugino : antiperiodic

4d bosonic YM at low energy

Gravity solution

What is cleaver about this geometry :

How to break susy is specified field theory dual is clear

Double-Wick rotatedAdS7 x S4 blackhole

(written with 11 dim. supergravity notation)

Closed string side (gravity)Confinement in AdS/CFT

Spacetime is smoothly “truncated” at the core Confinement

No spacetim

e

Quark antiquark potential

Linearpotential

Computing Glueball spectrum via AdS/CFT 　　　　

Supergravity fluctuation corresponding to the lightest glueball[Constable,Myers(99)] [Brower,Mathur,Tan (03)]

： Glueball field ： Eigenfunction in higher dim

Witten’s geometry :

[Brower,Mathur,Tan (03)]

Glueball spectrumobtained in AdS/CFT

Lattice calculation(SU(3) pure YM)

[Morningstar,Peardon (99)]

Glueball decay

ID of QCD glueball?

: Scalar glueball

Lattice prediction of lightest glueball mass: 1600MeV

0++ state

Perturbative QCD

Lattice QCD

We need theoretical description of glueball decay!

Chiral perturbation

Mixings?Difficult Holographic QCD can compute the interaction identification of the glueball!

[Terashima, C-I Tan and KH (0709.2208)]

Computing the coupling between glueballs and mesons

Glueball 　→　 Gravity Meson → 　 Gauge (on D8)

② 　 In string theory, all the interactions between gravity and D8 gauge fields are encoded in D8-brane action

We substitute gravity and D8 gauge fluctuationsrepresenting the mesons and glueballs, and perform

the integration of higher dimensional space

We obtain interacting lagrangian of glueball / meson fields

① 　 correspondence:

Glueball , Pion ,ρmeson

（ This expression is for a single flavor, for simplicity ）

Result

Kinetic terms

No mixing between mesons and lightest glueball

Interaction terms

Possible decay process of the lightest glueball 　　　　　　　

Interaction terms obtained via AdS/CFT :

YM

← 　 CS

Among these, （ ii ）（ iii ） includes more than 5 pions after thedecay and so are negligible. Possible decay processes are

These reproduces decay products of f0(1500)

Decay width and branching ratios 　　　　　　　　　　　　　　　　Decay width

of each branch

If we tune the glueball mass and eta mass by hand so thatit can fit the experimental data, then

Comparison ：　 In experiments, f0(1500) decays as

： not produced

The results are consistent with f0(1500)