Dual Geometric Approach to QCD and strongly interacting systems

Transcription

1 2012 Kobe, March 13-14, 2012 Beyond the Standard Model and The Origin Of Higg s Dual Geometric Approach to QCD and strongly interacting systems Bum-Hoon Lee Center for Quantum Spacetime Sogang University Seoul, Korea

2 Contents I. Introduction Motivation and Basics as a tool for the strongly interacting systems II. AdS/QCD Brief Review 1. Top-down & Bottom up approach 2. Dual Geometry at finite temperature III. Holographic QCD 1. Gluon Condensate Background 2. Dual geometry for finite chemical potential 3. Chiral Condensate Effect 4. AdS/QCD in D3/D-instanton geometry VI. Summary

3 I. Introduction : Motivation & Basics T Holography idea of AdS/CFT applied to QCD is called AdS/QCD With AdS-QCD, how to explain properties such as confinement, χ-symm breaking, phases, etc.? Quark-Gluon Hadron Exists holographic model studies Not well und erstood AdS-CFT or HolographyParameter in extra dimension of Energy or radiu 3+1 dim. QFT (large Nc) s with running coupling constants <-> 4+1 dim. Effective Gravity description Ex) 4d QFT(on boundary open string) <-> 5d Gravity (in bulk,close string) Nc of D3 branes N=4 SU(Nc) SYM SUGRA on AdS5 x S5 (, ) (, ) AdS-CFT Holography : Useful tool for strongly interacting QFT such as QCD, Condensed Matter, composite Higgs from strongly coupled theory etc

4 Main idea on holography through the Dp branes * Dp branes carry tension (energy) and charge (source for p+2 for m) Gravity in AdS space (dim = ((p+1)+1) ) * Dp brane s low energy dynamics by fluctuating open strings Yang-Mills in (p+1) dim. (CFT)

5 (Closed string picture) : Dp branes carry energy AdS ((p+1)+1) dim) and RR-charge source for p+2 form flux H (harmonic function) Ex) D3 brane : = constant (conformal symmetry) The near horizon limit gives AdS x S5 the radius of = the radius of AdS5 x S5 Isometry : SO(4, 2) x SO (6) SUGRA approach is reliable

6 Anti-de Sitter (AdS) Geometry : AdS_(p+2) Gloabal Geometry as Hyperboloid Metric (Poincare Patch) Isometry : SO(p+1,2) Euclidean AdS y x

7 (open string picture of Dp branes ) Low Energy Dynamics --> p+1 dim. SUSY SU(Nc) YM Theories Neumann boundary condition (0, 1,, p) (1) (i) (j) (Nc) ij A µ ij X I (p+1,, 9) (p+1) dim. SYM (#SUSY = 16 =32/2) 1 1 µ u 1 Σ µ I L = Tr( Fµ uf + Dµ Φ D Φ + [ Φ 4g 4 2 #Dp-branes = Nc I J 2, Φ ] ) 2 YM I, J 0,1,... p X I, I = p + 1,..., q Tension=Charge 1 * BPS state ) 1,, 6 0, 1,, 3 : Nc Nc matirices, adj. repn. of U(Nc) + Fermions Conformal symmetry R-symmetry : global symmetry : SO(4,2) x SU(4) = SO(6) If << 1, then perturbative A µ, µ = * Preserves 1/2 of 2x16 SUSY EL QL + ER + QR, R Dirichlet boundary condition * dynamical 10 dim. N=1 SYM 1 MN L = TrFMNF + L 2 fer 4gYM Ex) # Nc D3 branes dim I 2 p 3 I X A µ, Φ I, Ψ : reduction g YM = gsls, Φ = 2 l s E L ( = p+ 1 g s l s brane 0 1 = ±Γ Γ... Γ P E R Anti brane rotation perpendicular t o D3 branes

8 Toward the more realistic models : AdS/CFT with flavors - Intersecting D-Branes Figure from Erdmenger et.al, EPJA (2008 ) Boundary QF T Bulk Gravity Still far from QCD!

9 Radial coord. r in the bulk is proportional to the energy scale E of QFT AdS/CFT Dictionary Witten 98; Gubser,Klebanov,Polyakov 98 Parameters (, ) (, ) Partition function of bulk gravity theory (semi-classial) bdry value of the bulk field : scalar, Generating functional of bdry Q FT for operator : source of the bdry op. Correlation functions b y 5D bulk field Operator w/ 5D mass w/ Operator dimesion 5D gauge symmetry Current (global symmetry)

10 Ex) (Operator in QFT) <-> (p-form Field in 5D) : Conformal dimensio n : mass (squared) Gluon cond. dilaton baryon density vector w/ U(1) (Composite) Higgs?? fields in gravity massless dilaton scalar field with m=0 vector field gauge group in the dual operators of QCD gluon condensation chiral condensation mesons in the flavor group

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12 II. AdS/QCD Goal : Using the 5 dim. dual gravity, study 4dim. QCD properties such a s spectra & Phases (confining, deconfining, etc.), etc., in terms of parameters (Nc, Nf, mq, T and μ, χ-symm., gluon condensation, etc.) Needs the dual geometry of QCD. - Holography idea of AdS/CFT applied to QCD Approaches : Top-down Approach : rooted in string theor y Find brane config. or SUGRA solution givin gex) Nc of D3(D4) + M of D7(D8), the 10Dim. gravity dual SUGRA (May solution put the etc. probe brane) Bottom-up Approach:phenomenological Introduce fields, etc. (as needed based on AdS/CFT) 5D setup 4D effective Lagrangian * Kaluza-Klein modes radial excitations of hadrons * Confinement in Hard/Soft Wall Model - by IR brane/dilaton running Figure from Erdmenger et.al, EPJA(2008)

13 Ex) Hard wall Model Erlich,Katz,Son,Stephanov PRL(2005), IR Brane at Metric Slice of AdS metric Da Rold, Pomarol NPB(2005) Confinement 5D action (Nf=2) (for χ-symm breaking) Parameters (# of colors = input parameter)

14 AdS/CFT at finite temperature Witten 98 (for the pure Yang-Mills theory without quark matters) 1) Low T (confining phase) : tads (thermal) AdS space(w/ir cutoff ) at β(=1/tc) QCD Phase dual Hawking-Page no stable AdS transition black hole transition =Transition of bulk geometry 2) High T (deconfining phase) : AdS BH Schwarzschild AdS blackhole is stable Hawking-Page phase transition [ Herzog, Phys.Rev.Lett.98:091601,2007 The geometry is described by the following ] action : cosmological constant The regularized on-shell actio n 1) for the tads, 2) for the AdS BH, the period in the t-direction of tads = the period in t-direction of AdS BH at z. : arbitrary The geometry with smaller action is the stable one for given T. > 0 for T < T c < 0 for T>Tc

15 1) tads w/o IR cutoff no confinement AdS metric : the boundary located at z=0 with the topology Wick rotation tads : The periodicity of : Open stri ng z the Coulomb potential -> no confining potential. [Maldacena, Phys.Rev.Lett. 80 (1998) 4859 ] tads with IR cut-off (hard wall model) confinement I IR cut-off In region I, the Coulomb-like potential. I Open string II z In region II, the confining potential.

16 2) AdS BH deconfining phase For z<<, AdS5 (x S5) z= Boun dary) black hole black hole horizon z an event horizon at smooth geometry if t periodic with The Hawking temperature : identified with the temperature of the boundary gauge theory. This black hole geometry corresponds to the deconfining phase of the boundar y gauge theory, since there is no confinin g potential.

17 III. Holographic QCD - gluon condensation, finite density effects, etc - towards the dual geometry of AdS/QCD 1. Gluon Condensate Background 4dim gluon condensate the dilaton in 5 di Action m. Dilaton wall solution (cf. dilaton black hole solution) Csaki & Reece, hep-th/ , singular at Perturbative expansion near the boundary z 0 Gluon condensate T-independent

18 General solution with metric back reaction Kim,BHL, Park, Sin, hep-th/ (JHEP 09(2007) ) Note : For a=0, the solution reduces to the dilaton-wall solution. For c=0, becomes the AdS Schwarzschild black hole solution. with T by Hence, describes the finite temperature with the gluon condensation with the metric having an essential singularity at Thermodynamics with gluon condensation Gluon condensate is sensitive to the QCD deconfinement transitio n. The heavy quark potential becomes deeper as the gluon condensate value decreases. Kim, BHL, Park, Sin, arxiv: (PRD80,2009).

19 Meson spectra in the gluon condensate backgroun d Action Ko, BHL, Park, JHEP 1004, (2010) (arxiv: ) dilaton wall solution Eq. of motion (with axial gauge ) deformation of AdS corresponding to the confining phase with gluon condensation. becomes IR cutoff for Confined phase by 1) Hard wall at zc or 2) braneless approach

20 hard wall approach Various meson masses depending on the gluon condensation - As the gluon condensation increases mass of the vector meson decreases slightly while masses of the axial vector meson and pion increase very slowl y

21 Braneless approach - singlularity identified with the IR cutoff Boundary condition at z=0, at z=zc Fixing zc by mρ= 776 MeV gives zc = 1/325 MeV Gluon condensation (for Nc = 3) Cf. Lattice calculation decreases as T increases Meson masses Miller, hep-ph/ (Phys. Rept, 2007) Rho-meson masses - meson spectra well defined in spite of the singularity - meson masses similar to those in EKSS model for gluon condensation larger than that of lattice calculation. - Meson spectra significantly depend on the gluon condensate - As the gluon condensation becomes large, masses of the vector meson and pion increase while masses of the axial vector mesons decrease

22 2. Dual geometry for finite chemical potential bulk field boundary dual operator ( quark number density ) Chamblin-Emparan-Johnson-Myers,1999 Cvetic-Gubser, dimensional action dual to the gauge theory with quark matters Euclidean Equations of motion 1) Einstein equation 2) Maxwell equation Ansatz :

23 Solutions S.-J. Sin, 2007 most general solution, which is RNAdS BH (RN AdS black hole) black hole mass black charge corresponds to the deconfining p hase (quark-gluon plasma) quark chemical potential quark number density Note 1)The value of at the boundary ( ) corresponds to the quark chemical potential of QCD. 2) The dual operator of is denoted by, which is the quark (or baryon) number density operator. 3) We use

24 What is the dual geometry of the confining (or hadronic) phase? find non-black hole solution (BHL, Park,Sin JHEP 0907,(2009) ) baryonic chemical potential We call it tcads (thermal charged AdS space) baryon number density Note : Solutions in both phases are valid for arbitrary densities Hawking-Page phase transition (in dense matter) Phase : hadronic phase deconfined phase Geometry: tcads RNAdS BH (q = 0) Geometry : thermal AdS AdS BH (Geometries of Hawking-Page Tr. w/o chemical potential)

25 Hawking-Page transition The difference of the on-shell actions for RN AdS BH and tcads When, Hawking-Page transition occurs Suppose that at a critical point 1) For deconfining phase 2) For, tcads is stable. confining phase

26 For the fixed chemical potential dimensionless variables For the fixed number density Legendre transformation, the Hawking-Page transition occurs at

27 Light meson spectra in the hadronic phase Jo, BHL, Park,Sin JHEP 2010, arxiv: Turn on the fluctuation in bulk corresponding the meson spectra in QC D X is the dual to the quark bilinear operator <qbar q >. thermal charged AdS space : zero temperature with finite density in confining phase (due to a hard wal l) the chiral symmetry breaking is obtained with nonzero vev of S π becomes the Nambu-Goldstone boson.

28 Light meson spectra in the hadronic phase 1. Vector meson 2. Axial vector meson 3. pion

29 3. Chiral Condensate Effect (w/ & w/o Density Effect) The gravity action in the bulk is C. Park, B-HL, S. Shin, arxiv: We set The background geometries of our interest are obtained from a gravity actio n with the massive scalar field and U(1) gauge interaction. an ansatz for the asymptotic AdS metric in the Fefferman-Graham coordinate a s M = 1, 2, 3, z We take R = 1.

30 BHL, Park, Shin JHEP 1012 (2010) arxiv: Chiral condensate dependence of vector-, axial vector-mesons, pion and the pion decay constant. In the right-below figure, we plot the deviation from GOR relation, in which we found that GOR relation is almost satisfied in the chiral limit.

31 With a zero quark mass, the ρ mass increases as the quar k density increases. (consistent wi th the previous results with the qu ark density and the chiral condens ate as parameters separately.) The a1-meson mass and pion ma ss decrease with the quark density. The results are consistent with th e spectra with the chiral condensa te, but contrary to the spectra with the quark density.

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33 4. AdS/QCD based on the D7 embedding in black D3/D-instanton geometry Finite Temperature with Dilaton backgroun d B. Gwak, M. Kim, BHL,Y.Seo, S.-J. Sin, arxiv : Motivation Alternative to the Geometrical phase Transition for in AdS/CFT? Baryon Vertex (phase) and confinement at finite T (Black Hole Backg round)? (Solution of Type IIB SUGRA) Hawking Temperatur e Zero Temperature Limit : becomes near horizon geometry of D3-D( -1) (Liu & Tseytlin ) AdSxS5 at UV Flat at IR (w/ dilaton singular) N=2 (with gluon condensation)

34 Ex) Zero Temperature and without density Background Metric by D3 & D-instantons (Liu & Tseytlin ) D7 Brane as a Probe Induced metric on D7 AdSxS5 at UV Flat at IR (w/ dilaton singular) N=2 (with gluon condensation) DBI action of D7 becomes ( D7 wraps S3 of S5) Embedding solution : Rho->infty, AdS5xS3 Rho->0, rad (S3)->0

35 mq = 0 mq = 5 q = 0, 0.1, 0.3, 0.5, 1,3,5,10 from bottom to top

36 Ex) Finite Temperature and without density Finite Temperature (Black Hole geometry) of D3/D-instanton system T Quark-Gluon Hadron Rewrite in terms of dimensionless parameter or where

37 Induced metric on D7 DBI action of D7 Minkowski and Black Hole embedding (from below) Repulsion by q > attraction by BH Phase transition temperatur e increases as q increases

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39 Finite Temperature and with finite density Turn on U(1) gauge field on D7 brane DBI action of D7 Minkowski and Black Hole embedding

40 Quark Phase Regularity condition As q increases, the repulsion effect on D7 also increases. F1 strings connect BH horizon and probe brane Physical object is freely moving quark

41 In mq 0 limit, we have two phases (Note : If q=0, then the trivial flat embedding is the unique solution for mq 0.) Finite quark mass mq 0

42 Baryon Phase Background metric Induced metric on D5 DBI action F1 s connect spherical D5 & probe D7 Phys. Ob. = b aryon vtx (bd state of Nc q uarks) χ-symm. brok en D7 brane at the tip of D5 with force balance condition

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44 Density dependence of free energy (for mq = 0 and q=1 5) F Baryon phase quark phase (χ sym) Q quark phase (χ broken)

45 Phase Diagram Zero quark mass Finite quark mass

46 Holographic Principles : IV. Summary (d+1 dim.) (classical) Sugra (d dim.) (quantum) YM theories AdS/QCD - Top-down Approach & Bottom-up Approach QCD using Holographic dual Geometry - w/o chemical potential - phase : confined phase deconfined phase transition Geometry : thermal AdS AdS BH Hawking-Page transition - in dense matter - (U(1) chemical potential baryon density ) deconfined phase by RNAdS BH hadronic phase by tcads Hawking-Page phase transition In the hadronic phase, the quark density dependence of the light meson masses has been investigated.

47 IV. Summary - continued Holographic QCD model in D3/D-instanton background Two phases and phase transitions : for given T and density quark phase : physical objects : quarks baryon phase : baryon (vertex) as a physical object We study phase structure with and without quark mass We also study density dependence of chemical potential (eq. of state ) and phase structure in grand canonical ensemble Holography Principle can be quite useful for studying the strongly inte racting system such as composite Higgs models, etc.

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