Dynamical Instability of Holographic QCD at Finite Density

1. 23 April 2010 at National Taiwan University Dynamical Instability of Holographic QCD at Finite Density Shoichi Kawamoto National Taiwan Normal University Based on arXiv:1004.0162 in collaboration with Wu-Yen Chuang (Rutgers), Shou-Huang Dai, Feng-Li Lin (NTNU) and Chen-Pin Yeh (NTU)

2. Phase diagram of “real” QCD [hep-ph/0503184] Shoichi Kawamoto

3. massless QCD [Rajagopal-Wilczek hep-ph/0011333] 1st order Shoichi Kawamoto

4. Large N QCD and chiral density wave In the large N limit, there will appear clear confining/deconfinement transition. Quark “Cooper pair” are not color singlet and then it is suppressed in large N limit. No color superconductivity (or CFL) in large N limit. Instead, in large N limit, another spatially modulated phase will be favored. For high density, low temperature [Deryagin-Grigoriev-Rubakov] “chiral density wave” phase Shoichi Kawamoto

5. large N QCD phase diagram??? CDW ? quarkyonic? [McLerran, Pisarski, …] Another confinement/deconfinement transition?? Shoichi Kawamoto

6. Phase diagram of holographic QCD Shoichi Kawamoto

7. Holographic Realization of Pure YM (1) Nc D4-brane compactified on S1 with SUSY breaking spin structure (Scherk-Schwarz circle) 0 1 2 3 4 5 6 7 8 9 o o o o o Nc D4 Fermions : tree level massive (anti-periodic boundary condition) 5 scalars : 1-loop massive (no supersymmetry) low energy theory on D4 3+1D pure Yang-Mills theory (with KK modes) Shoichi Kawamoto

8. Holographic Realization of Pure YM (2) confining geometry [Witten] deconfined geometry Shoichi Kawamoto

9. Confinement/Deconfiment transition Compactify on a thermal circle, we compare thermodynamic free energy. [Aharony-Sonnenschein-Yankieowicz] x4 tE tE x4 quark potential screened linear Confinement Deconfined At a critical temperature, we need to switch these two geometries (phase transition) Shoichi Kawamoto

10. Phase diagram (1) deconfined confining (This phase transition is leading and will not be changed by introducing flavors) Shoichi Kawamoto

11. Adding Quarks (Sakai-Sugimoto model) [Sakai-Sugimoto] To add the quark degrees of freedom, we introduce Nf probe D8-branes. 0 1 2 3 4 5 6 7 8 9 o o o o o Nc D4 L o o o o o o o o o 4-8 open strings give chiral (from D8) and anti-chiral (from anti-D8) fermions in the fundamental representation. Nf flavor massless U(Nc) QCD in 3+1D Symmetry: In the gravity dual, this symmetry is broken down to the diagonal U(Nf). Shoichi Kawamoto

12. Chiral symmetry breaking in SS model In this cigar geometry, D8 and anti-D8 need to connect. Geometrical realization of chiral symmetry breaking U(1)B subgroup is counting the number of quarks. Later we will introduce the chemical potential for this conserved quantity. In the deconfined geometry, there will be two configurations for the same boundary condition of D8. B The one (A) breaks the chiral symmetry, while for the other configuration ending on the horizon (B) the chiral symmetry is restored. A L Shoichi Kawamoto

13. Chiral symmetry restoration The restoration depends on the position of UT (the Hawking temperature) and the asymptotic separation L. temperature T Chiral symmetry restored Chiral symmetry breaking [Aharony etal. hep-th/0604161] separation L There is a critical temperature T. We will consider a fixed L. Shoichi Kawamoto

14. Phase diagram (2) ? Shoichi Kawamoto

15. Introducing Baryon chemical potential U(1) part of chiral U(Nf) symmetry: The conserved charge is the ordinary fermion number. The corresponding gauge field will be turned on. Temporal component of the gauge field is electric: we need to have a source. Then we will introduce the source for the gauge field on D8-brane. The Baryon vertex Shoichi Kawamoto

16. Baryons in Sakai-Sugimoto model D4-brane wrapping on S4 is a baryon vertex. [Witten] electric charge on a compact space To cancel charge, need to attach Nc strings Nc quark bound state (baryon) With dynamical quarks (D8-brane), baryons are charged under flavor symmetry as well Strings are ending on D8 and being a source for a0 However, this configuration is unstable. [Callan-Guijosa-Savvidy-Tafjord] D4 brane is attracted to D8 and becomes an instanton on it. Shoichi Kawamoto

17. Baryons as D4-instantons A nontrivial gauge field configuration on 4-submanifold in 8-brane That gauge fields configuration carries D4-brane charge. Codimension 4 solition (instanton) on D8 is identified with D4-brane inside D8. D8-brane Wess-Zumino term includes the following coupling: Instanton number (D4-charges) density Instantons are indeed a source for U(1) charge. We consider a smeared instanton over 3+1D Shoichi Kawamoto

18. D8-brane profile with D4-instanton For single instanton, an explicit profile is known (Hata-Sakai-Sugimoto-Yamato) and has a finite size. However, the profile for multi-instanton is difficult to determine in general. Consider a small instanton (zero-size) localized at the tip of D8. Then D8 WZ-term (Chern-Simons term) is nb is proportional to instanton density. D8 profile is the same as before except U=Uc (tip). The new configuration is determined by minimizing the total action with respect to Uc L For given L and nb, Uc is uniquely fixed and the angle at the tip is qc Uc Shoichi Kawamoto

19. Chiral symmetry restoration due to nb In the deconfinement geometry, chiral symmetry can be restored by having baryon density. nb large Large baryon number density (nb) is “heavy” due to the tension of D4, and is pilling the tip of D8 towards the horizon. Shoichi Kawamoto

20. Phase diagram (3) Shoichi Kawamoto

21. Fluctuations on D8-brane Finally, we will consider the fluctuation on D8-brane and see that it suggests an instability. Dictionary of gauge/gravity correspondense sub-leading leading bulk field normalizable mode nonnormalizable mode boundary source term Shoichi Kawamoto

22. Dynamical instability Assume that if normalizable solution (A=0) develops growing mode. no source term and tachyonic mode of spontaneous symmetry breaking with order parameter <O> In the bulk side, normalizable modes correspond to small perturbations around the solution. instability of the solution We then look for normalizable tachyonic (growing in time) solution in the bulk. Shoichi Kawamoto

23. Fluctuations U(1) gauge field: D8-brane embedding: Take quadratic order in fluctuations 6 Linearlized equation of motion Using expansion: Shoichi Kawamoto

24. Boundary conditions (Coupled) euqations of motion take the form of 2nd order ordinary linear differential equations. With the boundary condition (m=0), this is an eigenvalue equation and a solution exists for specific w2. Need to specify the boundary condition for the other “end” U=Uc. : Dirichlet or Neumann y : Dirichlet (fixing the position of the tip) : Neumann (fixing the electric source) Shoichi Kawamoto

25. Instability from Chern-Simons term We just look at 3 equations of motion. From Chern-Simons term Domokos-Harvey (and Nakamura-Ooguri-Park) found that with this Chern-Simons term with electric field background the solution can develop unstable modes. Shoichi Kawamoto

26. “Shooting” to find solutions First, look at the marginal case (w2=0). We tune k to find a normalizable solution (shooting method). Solution starts to exist. Large nb (instanton density) and low temperature tend to develop the instability. Shoichi Kawamoto

27. Result of the numerical analysis The solution is confirmed to represent actual unstable mode. -w2 w2=0 solution means onset of instability. k Only aimodes develop unstable modes. unstable for nonzero k vector current Spatially modulated! Shoichi Kawamoto

28. Phase diagram of holographic QCD Shoichi Kawamoto

29. Conclusion • In holographic QCD (Sakai-Sugimoto model), we draw a phase diagram including a spatially modulated phase. • The onset of phase transition is signaled by appearance of unstable mode in the presence of Chern-Simons term. • CS term here is given directly by background baryon density. Shoichi Kawamoto