Mesons as Open Strings

Transcription

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

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

3 7 (P,C)=(+,+) (P,C)=(+, ) (P,C)=(,+) (P,C)=(, ) 6 ( ) mass 2 [GeV 2 ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Spin Mesons are strings!? 3

4 Consistent in 10 dim space Jme massless parjcles with J = 1 and J = 2 (open) (closed) (for open string) Not consistent with meson spectrum!? 4 dim gauge theory 10 dim string theory massive parjcles in 4 dim dual dual (in a certain curved background) massless parjcles in 10 dim The above difficulties can be solved!! 4

5 4 dim QCD 10 dim string theory dual (in a certain curved background) holographic QCD (in some approximation: low energy, large Nc, 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 Q: What about the other mesons? Consider massive modes (excited strings) 5

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

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

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

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

10 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 invariant under Z 2 sym generated by I y9 ( 1) F L D8 I y9 : (y,x 9 ) ( y, x 9 ) ( τ τ ) z y Consider invariant states 10

11 3 Meson spectrum Consider open strings attached on D8 Strategy t Hooft coupling 1 Consider flat space Jme, ( justified when ) and quanjze the open strings a]ached on D8. space-time: (topology) x 0 3 (z,y) x 6 9 D8-brane: (x μ,z) x S 4 In the flat space-time limit, S 4 R 4, SO(5) rotation and translation of x Pick up the invariant states. reduced to 5 dim: (x μ,z) 3 Recover the z dependence of the background (perturbajvely). 11

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

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

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

15 z 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) J PC x 2 lowest modes: J PC x x x x x x 6 15

16 (naive shortcut) Flat space-time limit: : excitation level Massive particle in curved space-time D8 y 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

17 4 Comparison with data [T.Sakai and S.S. 04] used to fix MKK=949 MeV B μ (1) B μ (2) B μ (3) B μ (3)... φ (0) J PC mass (MeV) ρ a 1 ρ a 1... π [776] experiment:? 17

18 mass 2 [GeV 2 ] α =0.45 GeV Spin (P,C)=(+,+) (P,C)=(+, ) (P,C)=(,+) (P,C)=(, ) α =1.1 GeV 2 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

19 J PC x 3 (N=1, n=0) 2 ++, 1 +-, 0 -+, 0 ++ cannot be N=0 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

20 (N=2, n=0) J PC x x x x x 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

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

22 5 Discussion 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