Hadron Spectra and Quark Mass Dependence in Holographic QCD

1. 9th Feb. 2010 @ NFQCD workshop (YITP) Hadron Spectra and Quark Mass Dependence in Holographic QCD Koji Hashimoto (RIKEN) arXiv/0803.4192 (JHEP) Hirayama, Lin, Yee, KH arXiv/0906.0402 Hirayama, Hong, KH arXiv/0910.1179 Iizuka, Ishii, Kadoh, KH

2. Holographic QCD ? Action of hadrons QCD action Hadron specturm, interactions Large Nc, strong coupling Low energy effective field theory on D-branes Graviry description of those D-branes AdS/CFT correspondence superstring theory

3. Plan 1. A Holographic QCD 2. Mesons and Quark masses 3. Holographic Baryons 4. Baryons and Quark masses

4. 1. A Holographic QCD [Sakai,Sugimoto(04)]

5. Sakai-Sugimoto model U(Nf) Yang-Mills-Chern-Simons theory in curved space This is the holographic dual of massless QCD, with large and large ‘tHooft coupling Important features: z Extra “holographic” dimension . The gauge group is flavor U(Nf). Chiral rotation is defined at

6. KK modes of Vector mesons A KK mode of  Massless pion Kaluza-Klein decomposition gives mesons Eigen values correspond to masses of vector mesons. Comparison with experimental data

7. Derivation of chiral lagrangian The action with Going to gauge, integration Chiral lagrangian

8. - Skyrm term is derived ! Good agreement with experimental data : - Inclusion of the vector mesons, easy  Natural realization of hidden local symmetry

9. 2. Mesons and Quark masses arXiv/0803.4192 (JHEP)Hirayama, Lin, Yee, KH

10. Worldsheet instanton gives quark masses Put a D6-brane as a probe. D6 charge is (1,-1) under the chiral symmetry :  Explicit breaking of the chiral symmetry D6 spike Worldsheet instanton for the quark mass is given in flat spacetime 

11. Derivation of the quark mass term in Chiral lagrangian Worldsheet instanton looks same. D6 spike D4 throat : effective coupling

12. 2D8 Chiral condensate, flavor dependence D6 2D8 On each D8, D6 ends Standard chiral lagrangian !

13. Numerics ( just for illustration ) Pion / quark mass From pion decay constant and rho meson mass, We substitute , Cf) Experiments : Chiral condensate Cf) Lattice :

14. Other terms Vector / axial vector mesons : Mass shift is suppressed by Pion mass differences ?  two worldsheet instantons

15. 3. Holographic Baryons [Sakai,Sugimoto(04)] [Hata,Sakai,Sugimoto,Yamato(07)] Cf. [Hong, Rho, Yee, Yi (07)]

16. Baryons = YM instantons Baryon = D4-brane wrapping S4 = Instanton in of SS model [Witten(98), Gross,Ooguri(98)] [Sakai, Sugimoto(04)] Instanton charge sources U(1)v Quantization of instantons  Baryon spectrum [Hata, Sakai, Sugimoto, Yamato (07)] [Hong, Rho, Yee, Yi (07)]

17. Baryon solution : dyonic instanton Small instanton localized at The background can be approximated by flat space Solution : BPST instanton + electrostatic potential Inserting this back to the action leads to a potential U(1) Coulomb self-repulsion Effect of curved space Size is stabilized to be small,

18. Quantization of the instanton Moduli space approximation : Moduli with small potentials Moduli : , Lagrangian : acted by : isospin + spin  Harmonic-like potential  Baryons labeled by Baryon states are given by wave function of the QM : Proton :

19. Mass spectrum of baryons Baryon mass formula : PDG: (tables taken from [Hata et.al])

20. 4. Baryons and quark masses [Hirayama, Hong and KH, 0906.0402] [Sakai, Sugimoto and KH, 0806.3122]

21. 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

22. 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

23. Results and Comparisons Pion mass dependence of nucleon mass higher order. Our result : [Hirayama, Hong and KH, 0906.0402] Lattice QCD : [Brommel et al., 0804.4706] [Bernard, 0706.0312] [QCDSF-UKQCD hep-lat/0312030] [Walker-Loud et al., 0806.4549] [PACS-CS 0810.0351] Pion mass dependence of Roper mass Our result :

24. 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

25. Decouplet / Octet mass shifts Our results :

26. Comparison with experiments Our inputs : Mass splittings of baryons with Mass splittings of baryons with

27. 5. Conclusion & Discussions

28. 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]

29. Other quantities computed We compute also static properties of baryons, including meson-baryon coupling.  Long range nuclear force They nicely match exp. data [Sakai, Sugimoto and KH, 0806.3122] We compute Nuclear force at short distances =First analytic result reproducing repulsive core, from strongly coupled QCD [Sakai, Sugimoto and KH, 0901.4449] Three-body nuclear force. [Iizuka, Nakatsukasa, KH, 0911.1035]

30. Work in progress 3-body nuclear force  N-body in general ? Nuclear force among hyperons/nucleons Chiral properties of hyperons Color-flavor locking in holographic QCD [Chen, Matsuura, KH, 0909.1296]

32. Static quantities of nucleons derived input :

33. Glueball sector

34. Closed string side (gravity) Witten’s geometry for pure YM without SUSY Nc D4-branes wrapping S1 Gaugino : antiperiodic  4d bosonic YM at low energy Gravity solution Double-Wick rotated AdS7 x S4 blackhole [Witten (98)] (written with 11 dim. supergravity notation) What is cleaver about this geometry : How to break susy is specified  field theory dual is clear

35. Closed string side (gravity) Confinement in AdS/CFT Spacetime is smoothly “truncated” at the core  Confinement Quark antiquark potential No spacetime Linear potential

36. Computing Glueball spectrum via AdS/CFT Witten’s geometry : Supergravity fluctuation corresponding to the lightest glueball [Constable,Myers(99)] [Brower,Mathur,Tan (03)] ：Eigenfunction in higher dim ：Glueball field

37. Glueball spectrum obtained in AdS/CFT Lattice calculation (SU(3) pure YM) [Morningstar,Peardon (99)] [Brower,Mathur,Tan (03)]

38. Glueball decay

39. ID of QCD glueball? : Scalar glueball Lattice prediction of lightest glueball mass: 1600MeV 0++ state We need theoretical description of glueball decay! Chiral perturbation Perturbative QCD Difficult Mixings? Lattice QCD Holographic QCD can compute the interaction  identification of the glueball! [Terashima, C-I Tan and KH (0709.2208)]

40. Computing the coupling between glueballs and mesons Glueball　→　Gravity Meson →Gauge (on D8) ①correspondence: ②In string theory, all the interactions between gravity and D8 gauge fields are encoded in D8-brane action We substitute gravity and D8 gauge fluctuations representing the mesons and glueballs, and perform the integration of higher dimensional space We obtain interacting lagrangian of glueball / meson fields

41. Result Glueball , Pion ,ρmeson Kinetic terms No mixing between mesons and lightest glueball Interaction terms （This expression is for a single flavor, for simplicity）

42. 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 the decay and so are negligible. Possible decay processes are These reproduces decay products of f0(1500)

43. Decay width and branching ratios Decay width of each branch ： not produced If we tune the glueball mass and eta mass by hand so that it can fit the experimental data, then Comparison：　In experiments, f0(1500) decays as The results are consistent with f0(1500)