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

23 April 2010 at NTUShoichi Kawamoto2 Phase diagram of “ real ” QCD [hep-ph/ ]

23 April 2010 at NTUShoichi Kawamoto3 massless QCD [Rajagopal-Wilczek hep-ph/ ] 1 st order

23 April 2010 at NTUShoichi Kawamoto4 Large N QCD and chiral density wave Quark “Cooper pair” are not color singlet and then it is suppressed in large N limit. Instead, in large N limit, another spatially modulated phase will be favored. [Deryagin-Grigoriev-Rubakov] In the large N limit, there will appear clear confining/deconfinement transition. No color superconductivity (or CFL) in large N limit. For high density, low temperature “chiral density wave” phase

23 April 2010 at NTUShoichi Kawamoto5 large N QCD phase diagram??? CDW ? Another confinement/deconfinement transition?? quarkyonic? [McLerran, Pisarski, …]

23 April 2010 at NTUShoichi Kawamoto6 Phase diagram of holographic QCD

23 April 2010 at NTUShoichi Kawamoto7 Holographic Realization of Pure YM (1) Nc D4-brane compactified on S1 with SUSY breaking spin structure (Scherk-Schwarz circle) N c D4 o o o o o Fermions : tree level massive (anti-periodic boundary condition) 5 scalars : 1-loop massive (no supersymmetry) 3+1D pure Yang-Mills theory (with KK modes) low energy theory on D4

23 April 2010 at NTUShoichi Kawamoto8 Holographic Realization of Pure YM (2) confining geometry deconfined geometry [Witten]

23 April 2010 at NTUShoichi Kawamoto9 Confinement/Deconfiment transition Compactify on a thermal circle, we compare thermodynamic free energy. tEtE tEtE x4x4 x4x4 At a critical temperature, we need to switch these two geometries (phase transition) quark potential linear screened [Aharony-Sonnenschein-Yankieowicz] ConfinementDeconfined

23 April 2010 at NTUShoichi Kawamoto10 Phase diagram (1) deconfined confining (This phase transition is leading and will not be changed by introducing flavors)

23 April 2010 at NTUShoichi Kawamoto11 Adding Quarks (Sakai-Sugimoto model) N c D4 o o o o o o o o o o o o o o L Symmetry: To add the quark degrees of freedom, we introduce N f probe D8-branes. [Sakai-Sugimoto] 4-8 open strings give chiral (from D8) and anti-chiral (from anti-D8) fermions in the fundamental representation. N f flavor massless U(Nc) QCD in 3+1D In the gravity dual, this symmetry is broken down to the diagonal U(N f ).

23 April 2010 at NTUShoichi Kawamoto12 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. A B L The one (A) breaks the chiral symmetry, while for the other configuration ending on the horizon (B) the chiral symmetry is restored.

23 April 2010 at NTUShoichi Kawamoto13 Chiral symmetry restoration [Aharony etal. hep-th/ ] The restoration depends on the position of U T (the Hawking temperature) and the asymptotic separation L. separation L temperature T We will consider a fixed L. There is a critical temperature T. Chiral symmetry restored Chiral symmetry breaking

23 April 2010 at NTUShoichi Kawamoto14 Phase diagram (2) ?

23 April 2010 at NTUShoichi Kawamoto15 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

23 April 2010 at NTUShoichi Kawamoto16 Baryons in Sakai-Sugimoto model D4-brane wrapping on S 4 is a baryon vertex. [Witten] electric charge on a compact space To cancel charge, need to attach N c 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 a 0 However, this configuration is unstable. [Callan-Guijosa-Savvidy-Tafjord] D4 brane is attracted to D8 and becomes an instanton on it.

23 April 2010 at NTUShoichi Kawamoto17 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

23 April 2010 at NTUShoichi Kawamoto18 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 n b is proportional to instanton density. D8 profile is the same as before except U=U c (tip). The new configuration is determined by minimizing the total action with respect to U c Uc cc L For given L and n b, U c is uniquely fixed and the angle at the tip is

23 April 2010 at NTUShoichi Kawamoto19 Chiral symmetry restoration due to n b In the deconfinement geometry, chiral symmetry can be restored by having baryon density. Large baryon number density (n b ) is “heavy” due to the tension of D4, and is pilling the tip of D8 towards the horizon. n b large

23 April 2010 at NTUShoichi Kawamoto20 Phase diagram (3)

23 April 2010 at NTUShoichi Kawamoto21 Fluctuations on D8-brane Dictionary of gauge/gravity correspondense bulk field leading sub-leading nonnormalizable mode normalizable mode boundary source term Finally, we will consider the fluctuation on D8-brane and see that it suggests an instability.

23 April 2010 at NTUShoichi Kawamoto22 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 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.

23 April 2010 at NTUShoichi Kawamoto23 Fluctuations U(1) gauge field: D8-brane embedding: Take quadratic order in fluctuations 6 Linearlized equation of motion Using expansion:

23 April 2010 at NTUShoichi Kawamoto24 Boundary conditions (Coupled) euqations of motion take the form of 2 nd order ordinary linear differential equations. With the boundary condition (m=0), this is an eigenvalue equation and a solution exists for specific  2. Need to specify the boundary condition for the other “end” U=U c. : Dirichlet or Neumann y: Dirichlet (fixing the position of the tip) : Neumann (fixing the electric source)

23 April 2010 at NTUShoichi Kawamoto25 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.

23 April 2010 at NTUShoichi Kawamoto26 “ Shooting ” to find solutions First, look at the marginal case (  2 =0). We tune k to find a normalizable solution (shooting method). Solution starts to exist. Large n b (instanton density) and low temperature tend to develop the instability.

23 April 2010 at NTUShoichi Kawamoto27 Result of the numerical analysis k -2-2 The solution is confirmed to represent actual unstable mode.  2 =0 solution means onset of instability. Only a i modes develop unstable modes. vector current unstable for nonzero k Spatially modulated!

23 April 2010 at NTUShoichi Kawamoto28 Phase diagram of holographic QCD

23 April 2010 at NTUShoichi Kawamoto29 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.