Mesons as Open Strings in Holographic QCD Shigeki Sugimoto (IPMU, - PowerPoint PPT Presentation

Mesons as Open Strings in Holographic QCD Shigeki Sugimoto (IPMU, Univ of Tokyo) arXiv:1005.0655 based on: with T. Imoto and T. Sakai Seminar @ Cambridge 2/17/2011 1 ／ 22

Introduction 1 mesons ( N f = 2, Isovector) parity charge conjugaJon mass (MeV) spin △ ... not established Q: How can we understand these numbers? 2

(P,C)=(+,+) Hint: Regge trajectory (P,C)=(+, − ) (P,C)=( − ,+) 7 (P,C)=( − , − ) 6 ( ) ( ) ( ) 5 mass 2 [GeV 2 ] ( ) ( ) 4 ( ) ( ) 3 ( ) ( ) ( ) 2 1 0 0 1 2 3 4 5 6 Spin Mesons are strings !? 3

Difficulties in the old days Consistent in 10 dim space‐Jme ∃ massless parJcles with J = 1 and J = 2 (open) (closed) (for open string) Not consistent with meson spectrum !? Gauge/String duality suggests 4 dim gauge theory 10 dim string theory (in a certain curved background) dual massive parJcles massless parJcles in 4 dim in 10 dim dual The above difficulties can be solved !! 4

Holographic QCD 4 dim QCD 10 dim string theory dual (in a certain curved background) holographic QCD (in some approximation: low energy , large N c , large λ , ... ) D4/D8-branes in type IIA string theory [T.Sakai and S.S. 04] mesons are open strings on D8 etc. are obtained from the massless mode 1 ++ 0 -+ 1 -- Q: What about the other mesons ? Consider massive modes (excited strings) 5

Plan Introduction 1 Brief review of the model 2 Meson spectrum 3 Comparison with data 4 Discussion 5 6

Brief review of the model 2 Brane configuration [T. Sakai and S.S. 04] x 0 x 1 x 2 x 3 τ x 5 x 6 x 7 x 8 x 9 D4 x N C O O O O O D8-D8 x N f O O O O O O O O O S 1 with SUSY b.c. D8 τ 4 dim U(N C ) QCD with N f massless quarks (at low energy) D4 D8 r ( radial direcJon of x 5 ～ 9 ) 7

Holographic description replace D4 with the corresponding curved background [Witten 98] D8 are treated as probe brane (assuming ) D8 τ τ D8 D4 r D8 r (topology) x 0 ～ 3 ( r , Θ 1 ～ 4 ) τ x 0 ～ 3 ( r , τ ) Θ 1 ～ 4 τ S 4 r y D8 z 8

Hadrons in the model 5dim with D8 extended along ( x μ ,z ) x S 4 x 0 ～ 3 ( z , y ) ( μ =0 ～ 3) particles in closed strings glueballs today’s open strings on D8 mesons topic D4 wrapped on S 4 baryons 9

QCD mesons vs artifacts Our brane config. is invariant under SO (5) S 4 quarks and gluons are invariant under SO (5) (non‐invariant states are massive KK modes) mesons Bound states of quarks and gluons are SO (5) invariant (non‐invariant states are arJfacts made by unwanted massive modes) Similarly, we can show that quarks and gluons are y invariant under Z 2 sym generated by I y9 (‐1) F L D8 I y9 : ( y,x 9 ) → (‐ y,‐x 9 ) ( τ → ‐τ ) z Consider invariant states 10

Meson spectrum 3 Consider open strings attached on D8 ’t Hooft coupling Strategy ( justified when ) Consider flat space‐Jme, 1 and quanJze the open strings a]ached on D8. space-time: D8-brane: ( x μ ,z ) x S 4 x 0 ～ 3 ( z , y ) x 6 ～ 9 (topology) In the flat space-time limit, S 4 ⇒ R 4 , SO (5) ⇒ rotation and translation of x 6 ～ 9 Pick up the invariant states. 2 reduced to 5 dim: ( x μ ,z ) Recover the z dependence of the background (perturbaJvely). 3 11

General rules for light-cone quantization (NS-sector) ( light‐cone direcJon x ± = x 0 ± x 1 ) Fock vacuum fermion boson creation op. physical state odd mass No SO (5) invariant states in R-sector. Parity and Charge conjugation: P : C : 12

Massless mode ( N =0) μ =0,1,2,3 5 dim gauge field not invariant under KK decomposition along z direction Recovering the curved background, we obtain 5 dim U ( N f ) YM-CS theory in a curved space-time. B μ (1) B μ (2) B μ (3) φ (0) φ (1) ... ... ( μ =0,1,2,3) J PC 1 -- 1 ++ 1 -- 0 - + ... eaten eaten ρ a 1 ρ’ ... π complete sets [T.Sakai and S.S. 04] : eigenvalue ⇒ mass 2 of B μ ( n ) : eigenfunction 13

First excited massive modes invariant states: SO (4) li]le gr 5dim field 1 1 S 4 14

KK decomposition along z direction etc. lowest modes: ( i , j , k = 1, 2, 3 ) h ij (0) h iz (0) h zz (0) A ijk(0) A ijz (0) φ [1,2](0) 1 + - 0 - + 1 -- 0 ++ x 2 J PC 2 ++ 0 ++ Second excited massive mode lowest modes: 3 -- 2 ++ 2 -- 2 - + x 2 1 -- x 7 1 ++ x 3 1 + - x 4 1 - + 0 ++ x 2 0 - + x 6 J PC 15

Mass formula (naive shortcut) Flat space-time limit: : excitation level Massive particle in curved space-time y D8 z particle in potential: harmonic oscillator approx. More careful analysis shows that the O (1) term is not affected by the RR-flux, α ’ correction, etc. 16

Comparison with data 4 Massless mode [T.Sakai and S.S. 04] ... B μ (1) B μ (2) B μ (3) B μ (3) φ (0) J PC ... 1 -- 1 -- 0 -+ 1 ++ 1 ++ ... ρ a 1 ρ’ a’ 1 π mass ... [776] 1189 1607 2024 0 (MeV) used to fix M KK =949 MeV ? experiment: 17

Regge trajectory (P,C)=(+,+) (P,C)=(+, − ) (P,C)=( − ,+) α’=0.45 GeV ‐2 7 (P,C)=( − , − ) 6 5 mass 2 [GeV 2 ] α’=1.1 GeV ‐2 4 3 2 1 0 0 1 2 3 4 5 6 Spin If we use f π to fix α’, we obtain α’ = 0.45 GeV ‐2 . This is unfortunately too small. If we set α’=1.1 GeV ‐2 we get very good fit. 18

First excited states ( N =1, n =0 ) 2 ++ 1 + - 0 - + 0 ++ x 3 2 ++ , 1 + - , 0 - + , 0 ++ cannot be N =0 J PC 1 -- ⇒ good candidates for N =1 ? : N =0 * : N =1 degenerate around 1300 MeV a 0 (980) is considered to be a four quark state. * 19

★ ★ ★ Second excited states ( N =2, n =0 ) J PC 3 -- 2 ++ 2 -- 2 - + x 2 1 -- x 7 1 ++ x 3 1 + - x 4 1 - + 0 ++ x 2 0 - + x 6 ? ? ? ? ? : N =0 : N =1 : N =2 * degenerate around 1700 MeV ★ : predicJon ? π 1 (1400) is claimed to be a four quark state. (could be hybrid) * 20

Summary : N =0 : N =1 : N =2 : 4 quarks I think this is non-trivial. What do you think? 21

Discussion 5 Mesons are Strings Wikipedia says: Problems and controversy Although string theory comes from physics, some say that string theory's current untestable status means that it should be classified as more of a mathematical framework for building models as opposed to a physical theory. ..... Yet, for all this activity, not a single new testable prediction has been made, not a single theoretical puzzle has been solved. .... Don’t criticize string theory in this way anymore ! 22