1. Integrating out Holographic QCD Models to Hidden Local Symmetry Masayasu Harada (Nagoya University) Dense strange nuclei and compressed baryonic matter @ Yukawa Institute, Kyoto, Japan (April 21, 2011) MH, S.Matsuzaki and K.Yamawaki, Phys. Rev. D 74, 076004 (2006) MH, S.Matsuzaki and K.Yamawaki, Phys. Rev. D82, 076010 (2010) MH and M. Rho, arXiv:1102.5489

2. ＱＣＤ (Strong Coupling Gauge Theory) Hadron Phenomena

3. Q C D Low Energy hadron Phenomena Lattice ＱＣＤ Effective models

4. ☆ Holographic QCD Models Effective models of QCD ・ Large Nc limit QCD ⇒ weakly interacting theory of mesons Baryons are given as solitons. ○ infinite number of mesons ・ large λ = Nc g 2 (’t Hooft coupling) limit Correspondence in real-life QCD ? Contribution from infinite tower can be included In holographic models Example : Short distance behavior of N-N potential ∝ 1/r 2 (with infinite tower contribution summed up) Hashimoto-Sakai-Sugimoto, PTP122, 427 (2009)

5. ☆ Predictions of hQCD models ○ momentum independent quantities e.g., mass, coupling, … It is not difficult to compare model predictions with experiments. ○ Momentum dependent quantities e.g., form factors, scattering cross sections, … It seems difficult to compare model predictions with experiments, since it is difficult to add up contributions from infinite number of mesons.

6. Integrating out heavy modes ☆ Proposal hQCD models HLS model Most general effective model for  and  Low Energy hadron Phenomena This may give an interpretation of hQCD results in terms of lowest lying mesons (  and  ). This may give a clue to understand what we can learn from hQCD on real life QCD ?

7. Outline 1.Introduction 2.Hidden Local Symmetry 3.A Method for Integrating Out 4.Form Factors in Sakai-Sugimoto Model 5.Application to Nucleon Form Factors 6.Summary

8. M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985) M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988) M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003)

9. based on chiral symmetry of QCD ρ ・・・ gauge boson of the HLS ◎ Hidden Local Symmetry Theory ・・・ EFT for  and  M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985) M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988) M.H. and K.Yamawaki, Physics Reports 381, 1 (2003

10. ☆ Chiral Lagrangian Non-Linear Realization of Chiral Symmetry SU(N ) ×SU(N ) → SU(N ) f ff LRV ◎ Basic Quantity U = e → g U g R 2 i π T /F a a π L † ; g ∈ SU(N ) L,R f ◎ Lagrangian L = tr ［ ∇ U ∇ U ］ F π 2 4 μ μ† ∇ U ≡∂ U － i L U + i U R μ μ μμ L, R ; gauge fields of SU(N ) μ μfL,R

11. ☆ Hidden Local Symmetry U = e = ξ ξ 2 i π/ F π L † R F, F ・・・ Decay constants of π and σ πσ h ∈ ［ SU(N ) ］ fV local g ∈ ［ SU(N ) ］ f L,R global ・ Particles ρ μ = ρ μ a T a ・・・ HLS gauge boson π=π a T a ・・・ NG boson of ［ SU(N f ) L ×SU(N f ) R ］ global symmetry breaking σ=σ a T a ・・・ NG boson of ［ SU(N f ) V ］ local symmetry breaking ◎ 3 parameters at the leading order F  ・・・ pion decay constant g ・・・ gauge coupling of the HLS a = (F  /F  ) 2 ⇔ validity of the  meson dominance m = a g F π ρ 22 2

12. Maurer-Cartan 1-forms Lagrangian

13. based on chiral symmetry of QCD ρ ・・・ gauge boson of the HLS ◎ Chiral Perturbation Theory with HLS H.Georgi, PRL 63, 1917 (1989); NPB 331, 311 (1990): M.H. and K.Yamawaki, PLB297, 151 (1992) M.Tanabashi, PLB 316, 534 (1993): M.H. and K.Yamawaki, Physics Reports 381, 1 (2003) Systematic low-energy expansion including dynamical  loop expansion ⇔ derivative expansion ◎ Hidden Local Symmetry Theory ・・・ EFT for  and  M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985) M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988) Leading order Lagrangian is counted as O(p 2 ), Next order terms are of O(p 4 ).

14. ◎ Typical examplesof O(p 4 ) terms

15. M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 74, 076004 (2006) M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010)

16. ☆ hQCD models which include 5-dimensional gauge field at intermediate step In the following, I will use Sakai-Sugimoto model Kinetic term in 4-Dim.Mass term in 4-Dim 5-D gauge transformation Residual gauge symmetry = Hidden Local Symmetry Gauge fixing :

17. ◎ Mode expansion of the gauge field Eigenvalue equation Axial-vector mesonsVector mesons SS model input ・・・ Pions as NG bosons

18. ☆ m(  ) ～ E ≪ m(a 1 ) ・ Integrate out vector and axial-vector mesons other than  meson. Solve the equations of motions with kinetic terms neglected. HLS with a particular choice of parameters Note that z 1 ～ 8 are all determined by the model, which include effects of heavy mesons. ≠ truncation of heavy mesons

19. M.H., S.Matsuzaki and K.Yamawaki, Phys. Rev. D 82, 076010 (2010)

20.  EM form factor In Sakai-Sugimoto model infinite tower of  mesons contributes. k=1 :  meson k=2 :  ’ meson k=3 :  ” meson … = 1.31 + (-0.35) + (0.05) + (-0.01) + … ’’   ’’  ’’’  meson dominance ⇒ ; In the Hidden Local Symmetry  EM form factor is parameterized as ・ Reduction of Sakai-Sugimoto model ⇒

21. ◎ Alternative way to relate hQCD to HLS k=1 :  meson k=2 :  ’ meson k=3 :  ” meson … m ～ Q2 ≪ m’m ～ Q2 ≪ m’ Sum Rules

22.  EM form factor  meson dominance  2 /dof = 226/53=4.3 ; SS model :  2 /dof = 147/53=2.8 best fit in the HLS :  2 /dof=81/51=1.6 Exp data : NA7], NPB277, 168 (1996) J-lab F(pi), PRL86, 1713(2001) J-lab F(pi), PRC75, 055205 (2007) J-lab F(pi)-2, PRL97, 192001 (2006) Infinite tower works well as the  meson dominance ! MH, S.Matsuzaki, K.Yamawaki, PRD82, 076010 (2010) cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003)

23.  transition form factor MH, S.Matsuzaki, K.Yamawaki, arXiv:1007.4715 cf : MH, K.Yamawaki, Phys.Rept 381, 1 (2003) best fit in the HLS  2 /dof=24/30=0.8 Sakai-Sugimoto model :  2 /dof=45/31=1.5 ・  meson dominance  2 /dof=124/31=4.0 Violation of  /  meson dominance may indicate existence of the contributions from the higher resonances.

24.  form factor    m  = m  is used. SS model :  2 /dof = 63/5 = 13  meson dominance  2 /dof = 4.8/5=1.0 best fit in the HLS  2 /dof=3.0/4 = 0.7

25.  →       decay A B A/B ≪ 1 ⇒  meson dominance is well satisfied in SS model

26. MH and M. Rho,. arXiv:1010.1971 [hep-ph]

27. ☆ Hong-Rho-Yee-Yi hQCD Model 5-D effective model including 5-D baryon field + 5-D gauge field ⇒ 4-D effective model with baryon (nucleon) and an infinite tower of vector and axial-vector mesons Nucleon form factos PRD76, 061901 (2007) JHEP 0709, 063 (2007)

28. Example 3: Proton EM form factor M.H. and M.Rho, arXiv:1102.5489 [hep-ph]  meson dominance :  2 /dof=187 best fit in the HLS :  2 /dof=1.5 a = 4.55 ; z = 0.55 Violation of the  meson dominance (well-known) can explained by the existence of the infinite tower Hong-Rho-Yee-Yi model :  2 /dof=20.2 a = 3.01 ; z = -0.042

29. ◎ We relate holographic QCD models to the HLS model by integrating out heavier mesons in hQCD models. ・ Showed that the infinite tower of vector mesons can contribute even to pion EM form factor → can fit the data well as the  meson dominance ・ Violation of  meson dominance in wp transition form factor can be explained by the existence of infinite tower ・ Violation of r/w meson dominance in the proton form factor is well explained by the existence of infinite tower 6. Summary