1. Baryons with Holography Hideo SUGANUMA ( Kyoto Univ. ) Toru KOJO ( Kyoto Univ. ) Kanabu NAWA ( RCNP ) in collaboration with

2. holography According to the concept of ‘holography’, non-perturbative aspects of QCD (meson dynamics) can be analyzed by a tree-level superstring theory. Sakai-Sugimoto model : Holograhic QCD [T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 (2005)] INTRODUCTION Meson spectra of (axial) vector mesons. Hidden local symmetry. Vector meson dominance. KSRF relation. GSW model. [Bando, et al.,1985] [Sakurai, 1969] [Kawarabayashi, Suzuki, Riazuddin, and Fayyazuddin, 1966] [GellMann, Sharp, and Wegner, 1962] Many phenomenological aspects of mesons are uniquely derived, starting from one superstring model.

3. INTRODUCTION Holographic description is reliable for large- limit. Large- QCD becomes equivalent with the weak coupling system of mesonsbaryon mesons (and glueballs); baryon does not directly appear as a dynamical degrees of freedom. How to describe a baryon in the large- holographic model ? * Chiral soliton picture. Baryon as a topological soliton in 4dim. meson effective action Brane-induced Skyrmion induced by Holographic QCD (Brane-induced Skyrmion). [‘tHooft, Nucl. Phys. B72, 461(1974); B75, 461(1974)] holography According to the concept of ‘holography’, non-perturbative aspects of QCD (meson dynamics) can be analyzed by a tree-level superstring theory. Sakai-Sugimoto model : Holograhic QCD [T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 (2005)]

4. Holography Holographic QCD (Sakai-Sugimoto model) Baryons in holographic QCD * * * CONTENTS

5. Gauge theory on D-branes Superstring theory has 10 dim. space-time to avoid ‘anomaly’. D-brane is introduced as fixed edges of open strings. (‘D’ comes from ‘Dirichlet boundary condition’) Dp-brane = (p+1) dim. membrane open string ‘parallel ’ ‘perpendicular ’ ; scalar (NG) field. ; gauge field. (p+1) dim. gauge theory appears ‘on’ Dp-brane !!! 10dim.

6. Holography 10dim. D p- brane×N c 1 space dim. with finite curvature. (p+1) dim. gauge theory ([p+ ]+1) dim. super gravity 1 ; low energy ; weak coupling ; strong coupling Strong-weak duality (S-duality) D-brane = black hole (black brane) [Polichinsky (1995)] [Maldacena (1997)]

7. Holographic QCD Construction of massless QCD (quarks and gluons) on D4/D8/D8-branes. [T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 (2005)] D4-brane×N C D8-brane×N f, L 4-4 4-8 D8-brane×N f, R Index on D4 brane ( color ) Index on D8 brane ( flavor ) : quark (L) : quark (R) : gluon 10dim.

8. Probe approximation with Mass of Dp-brane is proportional to its sheets number. D4×Nc D8, D8×N f ‘‘probe’’ SUGRA background light probe heavy object gravitational background m N f ∝ D4-brane × N C D8×N f 10dim. space time D8 light probe 4-4 4-8 8-8 M N C ∝

9. [‘tHooft, Nucl. Phys. B72, 461(1974); B75, 461(1974)] D4-brane×N C D8×N f 4-4 4-8 Chiral symmetry breaking and color confinement 10dim. space time ； Chiral symmetry breaking. Colored particles are located behind horizon and become invisible from outside. quark (L) quark (R) gluon 8-8 D8 horizon L R ； Color confinement. Same critical temperature (Tc=170MeV) in lattice QCD. Mesons appear from 8-8 strings on the residual D8-branes, without explicit baryons ; property of large-Nc QCD.

10. ~ DBI action of probe D8-brane with D4 SUGRA background ： surface tension 9 4 = 5 5dim. Yang-Mills theory D8-brane is 9dim. membrane. ： dilaton field D4 SUGRA backgroud Integration out symmetric 4dims. Unified theory of mesons with pion and infinite number of (axial) vector mesons !!!

11. 0 0 1 Mode expansion : vector : axial-vector mass : parity : Z Z pion (axial) vector mesons : axial-vector complete basis : 5dim. gauge field Pion mainly couples with lightest vector mesons (ρ) Vector and axial-vector mesons alternately appear in the excitation spectra.

12. (pions and ρ-mesons) 4dim. meson effective action from holographic QCD,, pion field : 1-form of pion field : axial vector current : vector current : pion decay constant : ρ-meson mass : Holographic QCD has just two parameters. All the couplings are uniquely determined !!! [K. Nawa et al., Phys. Rev. D75, 086003 (2007)]

13. (pions and ρ-mesons) 4dim. meson effective action from holographic QCD,, pion field : 1-form of pion field : axial vector current : vector current : Traditional phenomenologies can be discussed in the action. Hidden local symmetry. Vector meson dominance. KSRF relation. [Bando, et al.,1985] [Sakurai, 1969] [Kawarabayashi et al., 1966] [K. Nawa et al., Phys. Rev. D75, 086003 (2007)]

14. hidden local symmetry * [M. Bando, T. Kugo, and K. Yamawaki, Prog. Theor. Phys. 73, 1541 (1985); Phys. Rep. 164, 217 (1988)] Comparison with phenomenology Hidden local symmetric terms : Couplings remain as parameters. All the couplings are uniquely determined. Holographic QCD is not just the low-energy effective theory only with the constraint of symmetries in 4 dim. chiral perturbation theory * ： ：,, [I. Zahed and G.E. Brown, Phys. Rep. 142, 1 (1986)] 2-derivative term 4-derivative term Instability of Skyrme soliton. 5dim. YM theory from holographic QCD: Topological picture for baryon is also supported by superstring theory. L 2 L 4,,

15. Baryons in Holographic QCD Large- QCD becomes equivalent with the weak coupling system of mesonsbaryon mesons (and glueballs); baryon does not directly appear as a dynamical degrees of freedom. [‘tHooft, Nucl. Phys. B72, 461(1974); B75, 461(1974)] How to describe a baryon in large-Nc holographic model ? Skyrme term naturally appears ; (topological picture for baryon is also supported by superstring theory). * Baryon as a chiral soliton [K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, 086003 (2007)]

16. Baryon as Chiral Soliton [T.H.R. Skyrme, Proc. R. Soc. A260, 127 (1961)] π ; ρ ; Pauli matrix Hedgehog soliton with B=1 Shrinkage Pions are slightly replaced by ρ-mesons through interactions in holographic QCD. Shrinkage of pion amplitude F(r). Stable hedgehog soliton solution exists !!! * * ( pion ) (ρ -meson ) [K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, 086003 (2007)]

17. -meson components in the core of a Baryon (Skyrmion) (Comparison of total enegy density and each energy density in -meson channels) -mesons are actively affect the total mass of a baryon in the deeper interior of it by using various interaction channels.

18. Mass and radius of baryon two Holographic QCD has just two parameters ; ; ultraviolet cut-off scale. ; large-N c effective theory. pion decay constant ρ-meson mass Experimental inputs for pion decay constant and ρ-meson mass ; Mass :, MeV Exp. ： Radius : Isospin rotation as semi-classical quantization. Dynamical pion cloud. Increase of total size !!! (for charge radius)

19. Summary and outlook ‘Holography’ gives the duality between (p+1) dim. gauge theory without gravity and ([p + 1 ]+1) dim. supergravity. S-duality gives powerful analysis for each side from the other side. S-duality makes difficult the exact proof of gauge/gravity duality. Non-perturbative aspects of QCD : glueball mass : baryon properties : meson properties : Non-perturbative aspects of gravity : critical exponent in blackhole formation : Hawking temperature of blackhole : * * [E. Witten, Adv. Theor. Math. Phys. 2 (1998), 505] [T.Sakai and S.Sugimoto, Prog. Theor. Phys. 113, 843 ; 114, 1083 (2005)] [K. Nawa, H. Suganuma, and T. Kojo, Phys. Rev. D75, 086003 (2007)] [L. Alvarez-Gaume et al., PLB 649, 478 (2007)] [D.K.Hong et al. (M. Rho group), Phys. Rev. D76 : 061901 (2007)] [H. Hata et al.(Particle Theory Group, Kyoto Univ.), hep-th/0701280] [S. Mukherji et al., JHEP 0205 (2002) 026] [E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998)]