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A Dual Geometry and Holographic QCD in medium. Bum-Hoon Lee Center for Quantum Spacetime Sogang University Seoul, Korea. Excited QCD 2010, 31 Jan.-6 Feb., 2010 Tatra National Park (Slovakia). Contents. I. Introduction – Idea of Holography - PowerPoint PPT Presentation
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Excited QCD 2010, 31 Jan.-6 Feb., 2010
Tatra National Park (Slovakia)
Bum-Hoon Lee
Center for Quantum SpacetimeSogang University
Seoul, Korea
A Dual Geometry and Holographic QCD in medium
I. Introduction – Idea of Holography as a tool for the strongly interacting systems
II. D-branes - Low Energy Dynamics - Gauge Theories - Spacetime Geometry – AdS space
III. AdS/CFT – Holography Principle IV. AdS/QCD
1. Top-down & Bottom up approach2. Dual Geometry at finite temperature3. Dual Geometry at finite density
V. Summary
Contents
• AdS-CFT or Holography 3+1 dim. QFT (large Nc) with running coupling constants <-> 4+1 dim. Effective Gravity description
Ex) 4 d QFT <-> 5d Gravity (on “boundary” D-brane, open string) (in “bulk”,
close string) with (conf. inv.) in Anti-deSitter
Space with (asym. freedom ) in asymptotic
AdS QCD in ??
• AdS-CFT : useful for strongly interacting QFT Ex) (Excited) QCD, Heavy Ion Phys., Condensed Matter
Systems, etc. (* ) AdS/CFT duality applied to QCD is called as AdS/QCD
I. Introduction Parameter in extra “dimension”
of Energy or “radius”
– Idea on Holography
as a tool for the strongly interacting systems
http://cquest.sogang.ac.kr
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#N D3 ㅡ ㅡ ㅡ ㅡ X X X X X X
#M D7 ㅡ ㅡ ㅡ ㅡ ㅡ ㅡ ㅡ ㅡ X X
#N
#M
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1) Low energy dynamics
Strings
3-3 : Aμ, Φ, λ, χ
II. D-Branes
3-7 : Q, ~ , ψ, ~Q 4
0,1,2,3
CPX
(N=2 Vector multiplet)
Adjoint representation
(N=2 hyper multiplet ) Matter in
Fundamental
Still far from QCD !
Yang-Mills in 3+1 dim SU(Nc), #M flavors,N=2 SUSY
Ex) D3-D7
• Low Energy Dynamics of Nc D3 branes (+ others, etc.) -> 3+1 dim. SU(Nc) YM theory (DBI action) + (matter etc.)
• Spacetime Geometry by large Nc D3 branes (+ others, etc.) -> AdS5 x S5 ( etc. … )
( for D-brane )
2) D branes - Spacetime Geometry
(string frame)
Ex) D3 brane :
Dp-brane solution in Supergravity
the radius of = the radius of ( ( )
near horizon limit : reducdes to AdS x S5 geometry
Remark :
AdS5 x S5
= constant (conformal symmetry)
Note : For , we can trust this supergravity solution
Anti-de Sitter (AdS) Geometry : AdS_(p+2)
Metric (Poincare Patch)
Gloabal Geometry as Hyperboloid
Isometry :
SO(p+1,2)
Euclidean AdSy
x
III. AdS/CFT Holography
Extension of the AdS/CFT• the gravity theory on the asymptotically AdS space -> modified boundary quantum field theory (nonconformal, less SUSY, etc.) QCD ?
• the gravity theory on the black hole background -> corresponds to the finite temperature field theory
- Gravity on the 5D bulk - QFT on the 4D boundary SUGRA on AdS5 x S5 N=4 SU(Nc) SYM- Isometries of AdS x S5 - Conformal x R-symmetry SO(4,2) x SO(6) SO(4,2) x SO(6)- the classical gravity theory - strongly coupled QFT
( , ) ( , )
Parameters
Partition function (semi-classial)
boundary value of the bulk field
Generating functional forthe boundary operators
: the source of the boundary operator
• 5D bulk field Operator 5D mass Operator dimesion • 5D gauge symmetry Current (global
symmetry)• small z Large Q • Confinement (IR) cutoff zm (in
Hard Wall)• Kaluza-Klein states Excited, Resonant
spectrum
AdS/CFT DictionaryWitten 98; Gubser,Klebanov,Polyakov 98
: Conformal dimension : mass (squared)
(Operator in QFT) <-> (p-form Field in 5D)
Note : the fluctuation field on the bulk space corresponds to a source for the QCD Operator .
Ex)
Gluon cond . dilaton 0 4 0baryon density vector w/ U(1) 1 3 0
operator of QCD• gluon condensation• chiral condensation
• mesons in the
flavor group
field in gravity• massless dilaton• scalar field with
• m=0 vector field in
the
gauge group
dual
Goal : Try to understand QCD using the 5 dimensional dual gravity theory (AdS/CFT correspondence)
Need the dual geometry of QCD.1. Approaches :
Top-down Approach : rooted in string theory
Find brane config. for the gravity dual
Bottom-up Approach : phenomenological Introduce fields, etc. as needed based on the AdS/CFT * Hard Wall Model - Introduce IR brane for confinement * Soft Wall Model – dilaton running
Light-Front : radial direction of AdS <-> Parton momenta
(Brodsky, de Teramond, 2006)
IV. AdS/QCD
QFT with Asymptotic AdS SUGRA Duals
• N=1* Polchinski & Strassler,hep-th/0003136
• Cascading Gauge Theory Klebanov & Strassler, JHEP 2000
• D3-D7 system Kruczenski ,Mateos, Myers, Winters JHEP 2004
• D4-D8 system Sakai & Sugimoto 2005
-> closely related to QCD
etc.
Top-Down Approach
Observation :• Nc of D3 branes : AdS5 x S5 <-> N=4 SUSY YM• Nc of D3 / orbifold, etc. : AdS5 x X <-> N=2, 1 YM
QFT with Asymptotic AdS SUGRA Duals• N=1* Polchinski & Strassler,hep-th/0003136
• Cascading Gauge Theory Klebanov & Strassler, JHEP 2000
• Nc of D3 + M of D7 system Kruczenski ,Mateos, Myers, Winters 2004
• Nc of D4 + M of D8 system Sakai & Sugimoto 2005
-> closely related to QCD
etc.
D3-D7 system
• large-Nc N=2 SYM with quarks• Flavor branes: Nf D7-branes with flavor symmetry: U(Nf)• Quarks are massive (in general): mq
• Probe approximation (Nc>>Nf) (~ quenched approx. )
No back reaction to the bulk gometry from the flavor branes. • Free energy ~ Flavor-brane action
Kruczenski ,Mateos, Myers, Winters JHEP 2004
D4-D8 system (Sakai-Sugimoto Model)• Type IIA in Nc D4 Background (Witten) + Nf Probe D8 Branes along (x,z) x S4 (Nc >> Nf) as dual description to 4D QCD w/ Nf m=0 quarks• Topology of the background : R(1,3) x R2 x S4• Glueballs from closed strings• Mesons from the open strings on D8• The Effective Action : 5D U(Nf) YM CS theory
Sakai-Sugimoto, 2004
Bottom-Up Approach to AdS/QCD
• Introduce the contents (fields, etc.) as needed based on the AdS/CFT principle
• phenomenological Kaluza-Klein modes - radial excitations of
hadrons identified by the symmetry properties of the modes
• Hard Wall Model Introduce IR brane for confinement
• Soft Wall Model - dilaton
Bottom-Up Approach Hard Wall model
Ex) Hard wall ModelErlich, Katz, Son, Stephanov, PRL (2005)Da Rold, Pomarol, NPB (2005)
Infrared Brane at Confinement
(Polchinski & Strassler ’00)
Metric – Slice of AdS metric
Introduce the contents (fields, etc.) as needed based on the AdS/CFTConfinement realized in * Soft Wall Model – by dilaton running * Hard Wall Model – by introducing IR brane for confinement
1) Low T (confining phase : tAdS (thermal) AdS space, no stable AdS
black hole
2) High T (deconfining phase) : AdS BH Schwarzschild-type AdS black hole
: cosmological
constant
: AdS radius
2. The Dual Geometry for the pure Yang-Mills theory (without quark matters)
Witten ‘98
Transition of bulk geometry at temperature β(=1/T).
Thermal AdS (Low T)
AdS-BH (High T)
“confinement” phase “de-confinement” phase
Hawking-Page transition
This geometry is described by the following action
zOpen string
z=0 (Boundary)
• According to the AdS/CFT correspondence,
the on-shell string action is dual to the
(potential) energy between quark and anti-
quark
• Using the result of the on-shell string
action, obtain the Coulomb potential
• There is no confining potential.
[Maldacena, Phys.Rev.Lett. 80 (1998) 4859 ]
AdS metric :
Wick rotation
the boundary located
at z=0 with the
topology
1) tAdS
tAdS :The periodicity of :
confinement in tAdS
z
Open string
z=0 (Boundary)
IR cut-off
To explain the confinement, we introduce the hard wall ( or IR cut-off) at by hand, which is called `hard wall model’ .
II
I
I
When the inter-quark distance is sufficiently
long,
•In the region I,
the energy is still the Coulomb-like
potential.
• In the region II,
the confining potential appears.
• So, the tAdS geometry in the hard wall
model corresponds to the confining phase of
the boundary gauge theory.
: String tension
In the real QCD at the low temperature, there exists the confinement.
2) AdS BH
z
z=0 (Boundary)
black hole
horizon
• an event horizon at
• The Hawking temperature :
identified with the temperature
of the
boundary gauge theory.
• This black hole geometry
corresponds to
the deconfining phase of the
boundary
gauge theory, since there is no
confining
potential.
black hole
3) the Hawking-Page phase transition
[ Herzog , Phys.Rev.Lett.98:091601,2007 ]
QCD gravity theory
deconfinement phase transition
Hawking-Page transitiondual
The regularized on-shell action
1) for the tAdS,
2) for the AdS BH,
To investigate the Hawking-Page transition, we should calculate
the free energy, which is proportional to the gravity on-shell
action.
: arbitrary
To remove the divergence at
introduce a UV cut-off
the period in the t-direction of tAdS = the period in t-direction of AdS BH at z = . Using this, the difference of two actions :
• The Hawking-Page (or deconfinement) transition occurs at
• At the low temperature
the tAdS space is stable (confining phase) .
• At the high temperature the AdS BH is more stable (deconfining phase)
or
Introduce new variables
Complex scalar field
Action for light meson spectra
vector meson axial vector
meson : break the chiral symmetry
: pseudoscalar meson
Meson spectra (in Hard Wall) [ EKSS ,
Phys.Rev.Lett.95:261602,2005 ]
dual operator of QCD• chiral condensation
• chiral
group
of global flavor symmetry
field in gravity• scalar field with
• gauge
group
of local symmetry
dual
According to the AdS/CFT correspondence,
Vector meson ( )
In the gauge , equations of motion for the axial vector meson and pion
• Boundary conditions
- at the UV cut-off , impose the Dirichlet BC - at the IR cut-off , impose the Neumann BC
• If requiring that the rho meson mass becomes 776 MeV,
we obtain
• : transverse part of
the
axial vector field
• : pseudoscalar meson
(pion)
Axial vector meson
• Boundary conditions
at the UV cut-off and at the IR cut-off
• After numerical calculation,
Pion
• Boundary conditions
at the UV cut-off and at the IR
cut-off
•
• We obtain numerically.
3. AdS/QCD with quark matters
bulk boundaryfield dual operator
Dual geometry for quark matter
( quark number density )
5-dimensional action dual to the gauge theory with quark matters
in the Euclidean version ( using )Ansatz :
Equations of motion1) Einstein equation
2) Maxwell equation
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
most general solution, which is RNAdS BH (RN AdS black hole)
Solutions
black hole mass
black charge
quark chemical potential quark number density
corresponds to the deconfining phase
( QGP, quark-gluon plasma)
What is the dual geometry of the confining (or hadronic) phase ?
find non-black hole solution
• baryonic chemical potential
• baryon number densityWe call it tcAdS (thermal
charged AdS space)
RNAdS BH (QGP)
• black hole horizon :
• Hawking temperature
• For the norm of at the black hole horizon
to be regular, we should impose the Dirichlet boundary condition
• From this, we can obtain a relation between and
• Using these relations, we can rewrite as a function of and
• After imposing the Dirichlet boundary condition at the UV cut-off
the on-shell action is reduced to
• Note that the above action has a divergent term as
• So, we need to renormalize the above action.
• By subtracting the AdS on-shell action,
we can obtain the renormalized on-shell RN AdS BH action
• the grand potential ( micro canonical ensemble )
Free energy ( canonical ensemble)
• For describing the quark density dependence in this system, we
should find
the free energy by using the Legendre transformation
where
• As a result, the thermodynamical free energy is
We can reproduce free energy by imposing the Neumann B.C. at the
UV cut-off
Note that we should add a boundary term to impose the Neumann
B.C. at the
UV cut-off .
• The renormalized action with the Neunmann B.C. becomes
with the boundary action
• Using the unit normal vector and
the boundary term becomes
• Therefore, the free energy is the same as the previous
one.
We can conclude that
1)The bulk action with the Dirichlet B.C. at the UV cut-off
corresponds to
the grand potential, which is a function of the chemical potential.
2) The bulk action with the Neumann B.C. at the UV cut-off
corresponds to
the free energy, which is a function of the number density.
tcAdS ( Hadronic phase )
Impose the Dirichlet boundary condition at the IR cut-off
where is an arbitrary constant and will be determined later.
Using this, we can find the relation between
After imposing the Dirichlet B.C at the UV cut-off, the
renormalized on-shell action for the tcAdS
From this renormalized action, the particle number is reduced to
Using the Legendre transformation, should satisfy the
following relation
where the boundary action for the tcAdS is given by
So, we see that should be
Then, the renormalized on-shell action for the tcAdS
with
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
Introducing new dimensionless variables
the Hawking-Page transition occurs at
For the fixed chemical potential
After the Legendre transformation, the Hawking-Page transition in
the fixed quark number density case occurs at
For the fixed number density
Light meson spectra in the hadronic phaseTurn on the fluctuation in bulk corresponding the meson spectra in QCD
Using the same method previously mentioned, we can investigate the meson spectra depending on the quark number density.
1. Vector meson
2. Axial vector meson
3. pion
V. Summary• Holographic Principles :
(d+1 dim.) (classical) Sugra ↔ (d dim.) (quantum) YM
theories
• AdS/QCD : can be a powerful tool for QCD
- Top-down Approach & Bottom-up Approach
• Holographic QCD using dual Geometry
- confinement/deconfinement phase transition
by the Hawking-Page transition between thermal AdS
↔ AdS BH
• Dual Geometry of YM theories in Dense matter
- U(1) chemical potential baryon density
- In the hadronic phase, the quark density dependence
of the light
meson masses has been investigated.