AdS/CFT and Its Application to QCD

11th, 12th August, 2008
SI 2008 @ Chi-Tou, Taiwan
AdS/CFT
and
Its Application to QCD
Koji Hashimoto （橋本幸士）
RIKEN （理研）
Plan of the talk
Part 1 : Why is AdS/CFT important?
10 pages
Part 2 : “Derivation” of AdS/CFT
30 pages
Review of D-branes
AdS/CFT : equivalent two descriptions of D-branes
What can be computed in AdS/CFT
Part 3 : Holographic QCD
What is holographic QCD?
Construction of QCD by D-branes
Sakai-Sugimoto model
Glueball decay
Baryon
Thermal holographic QCD
Regge behavior
Quark mass in Sakai-Sugimoto model
80 pages
1.
Why is AdS/CFT
important?
The claim of “AdS/CFT correspondence”
The weakest form of the AdS/CFT claim is :
[Maldacena(97)]
4d large N SYM : “CFT”
N=4 supersymmetric SU(N) Yang-Mills theory
at large N, large ‘tHooft coupling in 4 dimensions
||
Classical type IIB supergravity in 10 dimensions
on [AdS5 x S5 + 5-form flux] background
10d SUGRA : “AdS”
What can be computed
4d large N SYM : “CFT”
Strong
coupling
Correlation functions of
gauge invariant operators
||
10d SUGRA : “AdS”
Fluctuations of supergravity fields
on the AdS5 x S5 background
Weak
coupling
[Gubser,Klebanov,Polyakov(98)]
[Witten(98)]
Importance of the AdS/CFT?
For phenomenologists :
We can use it for computing
strongly coupled gauge theories!
For string theorists :
We can use it for defining string theory?
Use of AdS/CFT in phenomenology?
AdS5 has been used extensively :
Randall-Sundrum braneworld models
Warped throat, brane inflation
Examples of applications of the AdS/CFT :
Technicolor, Higgs(less)
CFT interpretation of RS
[Csaki,Grojean,Pilo,Terning(03)]
[Contino,Nomura,Pomarol(03)]
[Arkani-Hamed, Porrati, Randall (00)]
Unparticle?
Blackhole physics?
= “bottom-up approaches”
[Tanaka (02)]
“Holographic QCD” = Application of AdS/CFT to QCD
First extensive participation of string theorists
to application of AdS/CFT
Bottom-Up approach
[Erlich, Katz, Son, Stephanov(05)]
[DaRold, Pomarol(05)]
Top-Down approach
[Kruczenski, Mateos, Myers, Winters(03)]
[Sakai, Sugimoto(04)]
Superstring theory, as a technology, is truly useful!
A new kind of significance of string theory
D-brane Æ Soliton physics, Blackhole entropy, math, …
Strong form of the AdS/CFT
History of changes of the naming
AdS/CFT correspondence
Gauge / gravity correspondence
[Maldacena(97)]
[Witten(98)]
Gauge / string duality
[Berenstein, Maldacena, Nastase(02)]
String excitations in 10d SUGRA background
have their dual operators in 4d SYM
Defining string theory by AdS/CFT?
String theory on curved backgrounds can be
defined non-perturbatively by gauge theories
Implication to ….
Uniqueness?
String field theory?
M-theory?
Landscape?
Brief history of string theory : Recurrence time?
1960’s～70’s： String theory was born in hadron physics
Regge trajectory, s-t channel duality,
’tHooft large N, ….
？ 1970’s～80’s：String as quantum gravity and unification
Standard model and superstrings, supergravity
Late in 1990’s～：Revolution by D-branes and duality
Toward non-perturbative definition
AdS/CFT (gauge/string duality）
2. “Derivation” of
AdS/CFT
Review of D-branes
AdS/CFT : equivalent but different
descriptions of D-branes
What can be computed in AdS/CFT
2-1
Introducing D-branes
Step 1 : What is string theory?
It is a theory of 1+1 dim. worldsheet.
Map
defines string location.
・ Open string:
・・・・
→ massless elemag field, massive fields
・ Closed string → graviton, tensor fields, massive fields
Scattering of these excitations can be computed.
→ You can write
“low energy effective action”
of these fields.
Step 2 : Low energy SUGRA and Blackholes
Closed string theory at low energy is 10dim. supergravity
Graviton
Dilaton
Fields in
SUGRA
Tensor gauge fields
Generalization of Blackholes
In this 10 dim. SUGRA,
BPS black “p-brane” solutions
exist for p=1,3,5,7.
・ Extending along p+1 directions
・ Emitting closed string modes
・ p-brane couples to (p+1)-form
p=1
p=3
Step 3 : “D” for Dirichlet
D-branes ＝ Hypersurface on which strings can end.
When varying worldsheet action,
instead of Neumann boundary condition
open
closed
consider Dirichlet boundary condition!
D-brane
[Dai,Leigh,Polchinski(89)]
Worldvolume dimension of Dp-brane is p+1
Type IIB theory has BPS D-branes with p=1,3,5,7,9
(Type IIA: p=0,2,4,6,8)
Step 4 : Black-branes = D-branes!
deform
D-branes
||
Source of
closed strings
In 1995, Polchinski showed that D-branes have tensor
charges, and can be identified with the black p-branes.
Perturbative description of black holes!
Leading to the second revolution of string theory.
Step 5 : Fields living on D-branes = Yang-Mills fields
Each field lives only on the D-brane.
Massless:
2 D-branes → strings are labeled as
Low energy effective action of N Dp-branes
＝ p+1 dim. SU(N) Yang-Mills + matter
Summary of this section:
1. Strings can end on D-branes
2. D-branes emit closed strings
3. D-branes are charged blackholes
4. SYM lives on coincident D-branes
2-2
AdS/CFT : equivalent but different
descriptions of D-branes
AdS/CFT： Equivalence of two ways to describe D-branes
[Maldacena(97)]
Open string （gauge theory）
Low energy effective action of
open strings on N D3-branes
4dim. gauge theory
Closed string （gravity）
Closed string in blackbrane
background of N D3-branes
Corresp.
10 dim. supergravity
in curved backgrounds
Close look at the theories on both sides
Open string side
At the low energy limt
Supersymmetric YM theory
Symmetry of the theory
Conformal symmetry due to susy
Generators of conformal group :
Lorentz, translation,
conformal boost, dilatation
R-symmetry rotates 6 adjoint scalars
SO(2,4)
SO(6)
Closed string side
Low energy (massless) effective (super)gravity :
BPS black 3-brane solution ：
We need to take a near horizon limit
The energy scale of interest in SYM :
where X is the distance between the (N-1) D3 and a D3
W-boson mass :
(N-1)
D3
D3
Geometry of
Isometry of this spacetime is SO(2,4) × SO(6)
which is identical with the symmetry of SYM.
4d N=4 SYM
“boundary”
corresp.
10d supergravity
on
“bulk”
A concrete example of “holography”
Equivalence between theories in different dimensions
Parameter region for valid correspondence
Gravity solution has a scale
so the classical solution is valid when
,
(1) quantum gravity correction can be ignored,
(2) stringy correction can be ignored,
is required
Therefore, corresponding gauge theory should
be at large N and at strong coupling
2-3
What can be computed in AdS/CFT
SYM Ù gravity : dictionary
Correlation functions
Coulomb branch
Quark-antiquark potential
YM instanton
Correlation functions in correspondence
Nc D3-branes
Black
brane
Gluon
Propagation of graviton
in near-horizon geometry
of black 3-brane
Propagation of
SU(Nc) gauge theory
composite states
(Glueball)
Large Nc
Large λ
Correlation functions in CFT can be computed in gravity
[Gubser,Klebanov,Polyakov (98)] [Witten (98)]
CFT operator
corresponds to bulk field
which can be read from open-closed coupling :
Gauge invariant composite
operators in YM theory
Bulk fields in
supergravity
Conformal dimensions and bulk mass
AdS5 geometry :
Suppose a bulk field
has a mass
, then
it is in exponential form asymptotically :
: invariant length
where
Dilatation
is determined by bulk Laplace equation
Ù
Î Conformal dimension of
Î
Ù
Quark-antiquark potential
Insert two fundamental strings onto D-branes
Infinitely
long string
＝
Infinitely heavy
charged particle
＝
＝
External
quark field
String is bent
and connected
in the curved
background.
Energy of the connected string
Æ Quark-antiquark potential
Computation of the quark antiquark potential
Nambu-Goto action in this background :
Redefinition :
Energy :
We fix the asymptotic location of the string and minimize
the energy. Conserved quantity for shifting
is
→
Substituting this back
to the energy,
Subtracting the infinite energy of single quark x 2,
the potential energy is finite,
Inter-quark distance is
The potential is
・ Inter-quark potential is proportional to
(purely non-perturbative effect)
・ Proportional to
, because of conformality
Coulomb branch of SYM
N D3
D3
Large
(IR in the bulk)
Large
“UV / IR relation”
(UV of CFT)
[Douglas, Taylor (98)]
[Peet, Polchinski (98)]
YM instantons = D-instanton in D3
Since D-branes are source for bulk tensor fields,
low energy effective action in the presence of a Dp-brane is
Open string excitations on the Dp-brane introduces
additional terms with gauge fields :
coupling to instanton charge
Maxwell term
Holographic description of the YM instantons
k D(-1)
Instantons
=
Large N
in 4d SU(N) YM
N D3
[Balasubramanian, Kraus,
Lawrence, Trivedi (98)]
[Douglas (95)] [Witten (95)]
instanton
size
“UV / IR relation”
Progress in AdS / CFT
Note : this AdS/CFT has not been proven yet.
・ Analysis on the gauge theory side is difficult, due to
its strong coupling.
・ Conformal dimension of BPS composite operators
has been analyzed to support the correspondence.
Æ Composite long operators are found to correspond
to strings in the curved background on the gravity side
“gauge / string duality”
3.
Application to QCD
What is holographic QCD?
Construction of QCD by D-branes
Sakai-Sugimoto model
Glueball decay
Baryon
Thermal holographic QCD
Regge behavior
Quark mass in Sakai-Sugimoto model
3-1
What is holographic QCD?
Holographic QCD (“AdS/QCD”)
Philosophy
AdS/CFT deals with strongly coupled gauge theories;
Then why not applying it to QCD?
Present status：
It reproduces various characteristics of
low energy hadron physics and provides
a new viewpoint (paradigm). It has real
predictions.
Difficulties：
・ Large N
・ Decoupling of higher dimensional DoF
Two approaches in holographic QCD
・“top-down” type ： Find a D-brane configuration
which has QCD matter content and symmetry,
then take a near horizon limit to obtain a gravity
dual of the large N QCD.
・“bottom-up” type : According to the philosophy of
AdS/CFT, in view of some sectors of low energy
QCD, give a gravity dual by hand. Compute
various quantities from the gravity theory and
compare them with real data. Model building.
“One starts from QCD and attempts to guess its
5d holographyc dual.” [Shifman, hep-ph/0507246]
Æ “AdS/QCD”
A famous bottom-up model
[Erlich, Katz, Son, Stephanov(05)] [DaRold, Pomarol(05)]
In AdS, itroduce 5d fields
corresponding to QCD
operators
and see their spectra
and interactions.
・ 5d gauge sym. = chiral sym.
・ AdS is cut-off at “IR” to incorporate confinement.
From AdS/CFT to QCD : brief history
Getting closer to QCD：
・ Running gauge coupling constant (Non-conformal cases)
[Freedman-Gubser-Pilchi-Warner hep-th/9906194]
[Polchinski-Strassler hep-th/0003136]
[Klebanov-Strassler hep-th/0007191] etc
・ Eliminating supersymmetries
[Witten hep-th/9803131]
・ Introduction of flavors (quarks)
[Karch-Katz hep-th/0205236] [Grana-Polchinski hep-th/0106014]
[Kruczenski-Mateos-Myers-Winters hep-th/0311270]
・ Chiral symmetry and its breaking
[Sakai-Sugimoto hep-th/0412141]
Computing quantites relevant for QCD :
・ Wilson loop
[Maldacena hep-th/9803002] [Rey-Yee hep-th/9803001] etc
・ Finite temperature
[Witten hep-th/9803131] etc
・ Chiral condensate
[Kruczenski-Mateos-Myers-Winters hep-th/0311270]
・ Mesons spectrum and chiral symmetry breaking
[Sakai-Sugimoto hep-th/0412141, hep-th/0507073]
・ Description of baryons
[Witten hep-th/9805112] etc
[Hata-Sakai-Sugimoto-Yamato hep-th/0701280]
[Hong-Rho-Yee-Yi hep-th/0701276]
・ Finite baryon density, chemical potential
[Chamblin-Emparan-Johnson-Myers hep-th/9902170]
[Kim-Sin-Zahed hep-th/0608046] etc
3-2
Construction of QCD by D-branes
ClosedWitten’s
string side
(gravity)
geometry
for pure YM without SUSY
Nc D4-branes wrapping S1
Gaugino : antiperiodic
Æ 4d bosonic YM at low energy
Gravity solution
Double-Wick rotated
AdS7 x S4 blackhole
[Witten (98)]
(written with 11 dim. supergravity notation)
What is cleaver about this geometry :
How to break susy is specified Æ field theory dual is clear
Closed string side
(gravity) in AdS/CFT
Confinement
Spacetime is smoothly “truncated” at the core
Æ Confinement
Quark antiquark potential
No spacetime
Linear
potential
Computing Glueball spectrum via AdS/CFT
Witten’s geometry :
Supergravity fluctuation corresponding to the lightest glueball
[Constable,Myers(99)] [Brower,Mathur,Tan (03)]
：Glueball field
：Eigenfunction in higher dim
Glueball spectrum
obtained in AdS/CFT
[Brower,Mathur,Tan (03)]
Lattice calculation
(SU(3) pure YM)
[Morningstar,Peardon (99)]
Introducing quarks by “flavor D-branes”
D3-D7 model
D3
Flavor D7
[Karch, Katz (0205236)]
- Nc D3: 0123
- Nf D7: 0123 4567
N=2, N=1 SQCD
distance ~ quark mass
D4-D6 model
[Kruczenski, Mateos,
Myers, Winters (0311270)]
- Nc D4 : 0123 4
- Nf D6 : 0123 567
For D4, take Witten’s geometry
D4
Flavor D6
Taking a large N limit, one replaces the color D-branes
by their geometry, while keeping the flavor D-branes as
probes (“probe approximation”) :
This geometry terminates
smoothly at
Flavor
D-brane
meson
AdS/CFT dictionary
Mesons = gauge fields on flavor D-brane
Sakai-Sugimoto model
The best model of top-down type.
Nc D4-branes (Witten’s)
+ flavor Nf D8-branes (left sector)
+ flavor Nf D8bar branes (right sector)
gluon + massless quark
・ Spontanesou chiral symmetry breaking is described
in dual geometry.
・ Meson, baryon spectra,
Skyrm term, … can be
computed. Various
chiral dynamics can be
derived
3-3
Sakai-Sugimoto model
D-brane configuration for massless QCD
Open string side (D-branes)
[Sakai, Sugimoto (hep-th/0412141)]
・ Nc D4-branes wrap
, and gauginos satisfy
anti-periodic boundary condition
→ 4d pure Yang-Mills at low energy
・ Nf D8-branes intersect with the D4s
→ Nf left-handed massless quarks
・ Nf anti-D8-branes intersect with the D4s
→ Nf right-handed massless quarks
[Witten]
Massless QCD is brane-engineered at low energy
Closed string side (gravity)
D8s are kept as probes
(probe approximation, valid at
Witten’s geometry :
Relation to gauge theory parameters :
Free parameter of the theory :
)
Spontaneous chiral symmetry breaking
Chiral ＝ Gauge symmetries on
the D8 and anti D8
symmetry
Massless QCD
(weak coupling description)
Replacing the D4
by its gravity solution
Spontaneous chiral SB
at strong coupling
D8s are connected, and gauge symmetry is
Once the correspondence is applied ・・・
・ gravity → bound states of gluons（Glueballs）
・ D8 → bound states of quarks（Mesons, Baryons）
KK modes of gauge fields on the D8 → Meson
D8-brane action → Chiral lagrangian
＝
D4
D8
Computing meson sector
D8-brane action on the curved background
Induced metric on the D8 :
Redefinition of a coordinate：
z
z
Action is evaluated as
・ KK modes of gauge field :
Eigen equation for the modes :
・ Pion is given by the zeroth mode of the decomposition :
・ Higher modes are absorbed by field redefinition :
Final lagrangian quadratic in KK modes is
KK modes of
A KK mode of
Æ Vector mesons
Æ Massless pion
Eigenvalues correpond to masses of vector mesons.
Comparison with experimental data
Derivation of chiral lagrangian
D8-brane action with
Going to
Chiral lagrangian
gauge, integration
- Skyrm term is derived !
Good agreement with experimental data :
- Inclusion of the vector mesons, easy
Æ Natural realization of hidden local symmetry
3-4
Glueball decay
ID of QCD glueball?
: Scalar glueball
Lattice prediction of lightest glueball mass: 1600MeV
0++ state
We need theoretical description of glueball decay!
Chiral perturbation
Perturbative QCD
Lattice QCD
Difficult
Mixings?
Holographic QCD can compute the interaction
Æ identification of the glueball!
[Terashima, C-I Tan and KH (0709.2208)]
Computing the coupling between glueballs and mesons
① correspondence:
Glueball → Gravity
Meson → Gauge (on D8)
② In string theory, all the interactions between gravity
and D8 gauge fields are encoded in D8-brane action
We substitute gravity and D8 gauge fluctuations
representing the mesons and glueballs, and perform
the integration of higher dimensional space
We obtain interacting lagrangian of glueball / meson fields
Result
Kinetic terms
Glueball
, Pion
,ρmeson
No mixing between mesons and lightest glueball
Interaction terms
（This expression is for a single flavor, for simplicity）
Possible decay process of the lightest glueball
Interaction terms obtained via AdS/CFT :
YM
← CS
Among these, （ii）（iii） includes more than 5 pions after the
decay and so are negligible. Possible decay processes are
These reproduces decay products of f0(1500)
Decay width and branching ratios
Decay width
of
each branch
： not produced
If we tune the glueball mass and eta mass by hand so that
it can fit the experimental data, then
Comparison： In experiments, f0(1500) decays as
The results are consistent with f0(1500)
3-5
Baryon
Baryon = instanton on D8
Baryon = D4 wrapping S4
= Instanton in
[Witten (98)] [Gross, Ooguri (98)]
of D8
[Sakai, Sugimoto(04)]
Embodying the Atiyah-Manton ansatz :
(instanton) = Skyrmion
In fact, the Chern-Simons term shows the equivalence :
Instanton charge sources U(1)v
Construction of the electrically charged instanton
Self energy of electric charge + Potential in z direction
Æ Stabilization of the instanton size
[Hata, Sakai, Sugimoto, Yamato(07)]
SU(2) part is the BPST instanton,
Rescaling :
The YM part of the action is expanded for large
as
The eom of the U(1) part is
Æ solved as
The total energy of the soliton is
self electric energy
Æ minimized by the size
: Baryon mass
Quantization of instantons Æ Baryon spectrum
Static properties of baryons
[Sakai, Sugimoto, KH (0806.3122)]
Chiral currents
Static properties
Charge radius
Magnetic moments
Axial radius, coupling
Chiral symmetry = D8 gauge symmetry at
ex) Baryon number current
Isoscalar mean
square radius
Static quantities of nucleons derived
input :
3-6
Thermal Holographic QCD
Deconfinement transition
To introduce temperature, we Euclideanize the time
direction and put anti periodic b.c. for fermions.
We have two “thermal circles”:
case 1)
and
case 2)
Depending on the temperature,
SUGRA chooses a solution (with a smaller action)
[Aharony, Sonnenschein, Yankielowicz 0604161]
One can evaluate the onshell actions, to obtain
(i) Low temperature phase
Confinement
(glueballs have massgap)
(ii) High temperature phase
Deconfinement
(glueball spectrum
is continuous)
Hawking-Page transition = deconfinement transition
Chiral symmetry restoration in Sakai-Sugimoto model
In the same manner, on-shell D8brane action determines
one among two possible shapes in the high temperature
phase of the deconfinement :
Case 2-1)
This is realized for intermediate
temperature
Chiral symmetry broken
Case 2-2)
This is realized for high
temperature
Chiral symmetry restored
(Meson has continuous spectrum)
Phase
diagram
Phase
diagram
We have 3 phases.
Low T : confined, χSB
Intermediate T : deconfined, χSB
High T : deconfined, χS restored
3-7
Regge behavior
Linear Regge behavior
Relation between energy and spin (angular momentum)
of hadrons :
In fact, we know a situation similar to this, in the
context of AdS/CFT correspondence :
spinning string / spin chain correspondence
ex) sl(2) sector treats excitations
with spins in AdS
We can naively expect
Reggeons = spinning strings in AdS-like backgrounds
(i) “Derivation” of quark-string model by AdS/CFT
[Kruczenski, PandoZayas, Sonnenschein, Vaman 0410035]
- Similarity to Wilson loops and confinement
- Linearity and deviation from that
(ii) Understanding of “Melting” of (higher spin) mesons
in thermal QCD, by AdS/CFT
- Melting of mesons in thermal holographic QCD
[Peeters, Sonnenschein, Zamaklar 0606195]
“Old” flux-string model
Let me first review the traditional model of
rotating two quarks joined by a string.
We will find a relation between E and J :
i) For massless quarks (
), it
reproduces linear Regge trajectory.
ii) For massive quarks, the relation deviates
from the linear one.
System:
NG string with tension T in flat b.g.
+ quarks with mass m at the ends
Spinning string solution gives linear Regge relation
(i) We start with the lagrangian (2d bulk + 1d boundary)
(ii) With a naïve ansatz
the bulk equations
of motion reduce to
This is solved by any
, so we take
(iii) The equation for the boundary terms is
For given T and m, this gives
.
For massless quarks, the ends move with the speed of light
(iv) The energy and the angular momentum are given by
0
(v) Linear Regge behavior
0
(a) When quarks are massless, we have
Linear Regge behavior
(b) For massive quarks, we have nonlinearlity
Bottomonium
lattice data
[hep-ph/9906293]
“Derivation” of the model via AdS/CFT
We employ D4-D6 model for realizing QCD
[Kruczenski, Mateos, Myers, Winters 0311270]
This geometry terminates smoothly at
probe D6
meson
distance ~ quark mass
Ansatz for spinning solution
The Nambu-Goto action is
and its variation gives
….
The ansatz gives a reduced EoM,
b.c.:
The conserved quantities are
The shape of the spinning string solution
Very similar to the Wilson loop!
Region 1 : String goes down to
1
“end-space wall”
Massive quarks
2
1
Region 2 : String is on the wall
Flux-string with effective
tension
In fact, it is a good approximation to divide the solution to
Region 1 :
Region 2 :
(i) Region 1
1
(ii) Region 2
2
1
As a sum, the conserved quantities are evaluated as
Sawing condition for the two regions :
consistency of the eom and the division requires
This reproduces the quark-string model,
Related issues on the string model
- The quark mass is “constituent quark mass”
- High-spin meson decay can be computed, and
it reproduces Casher-Neuberger-Nussinov model (‘79)
[Peeters, Sonnenschein, Zamaklar 0511044]
- Schwinger effect
- OZI rule
- Tension of the flux string :
Regge behavior in thermal holographic QCD
We consider similar spinning string solution in this
thermal background.
(1) Low T
The spinning string is the same as that of
zero temperature.
Spectrum is independent of the temperature
(2-1) Intermediate T, we will see that
(a) Mesons become lighter for higher T
(b) Meson “melts” at critical temperature for given spin
(c) Melting temperature is lower for higher spin
(2-2) High T
No meson state
[Peeters, Sonnenschein, Zamaklar 0606195]
The computation of the conserved quantities for spinning
strings is quite similar, and we list here the results:
For finite temperature, the energy and the angular
momentum of the string solution reach maximum values
Mesons with spins larger than this maximum melt
Temperature dependence of the conserved quantities
lower T
higher T
(Upper lines should be unstable mesons)
Consistency with exp.
For fixed spin J, the
energy is lower for
higher T, which agrees
with experiments.
Prediction
Melting temperature is higher
for lower spin. Mesons start
melting before the chiral sym.
restoration.
3-8
Quark mass in Sakai Sugimoto model
[T.Hirayama, F.L.Lin, H.U.Yee and KH, 0803.4192]
Causes of the massless quark problem
(a) There is no room to make the D8s separate from D4
(b) L and R quarks live on different locations in 10dim.
L
R
Quark mass term :
Our Solution to the problem
(1) Worldsheet instanton
Ex) Yukawa interactions
in string phenomenology
Dp
Dp
Dp
(2) Extended Technicolor
Techni-sector
QCD sector
“W-boson”
techni-quarks
Our results
We derive analytically
Gell-Mann Oakes Renner relation
Chiral condensate
Chiral lagrangian
Low energy sector of the SS model is drastically refined!
Importance (future directions)
Strange physics
Finite temperature / finite density QCD
Extended technicolor Æ quark mass
Technicolor sector :
QCD
Technicolor
Mediator
Quark mass
Techni-quark
W-induced
4-Fermi
Condensation
Brane configuration of the Technicolor
ref. [Hirayama, Yoshioka (07)]
Technicolor sector
||
Additional D4s
Strongly coupled
Technicolor sector
||
Additional throat
Worldsheet instanton
Double-line
=
Worldsheet instanton
Quark mass
Worldsheet instanton
W-boson mass :
Quark mass :
Pion mass
Gravity dual has 2 throats
Î Non-trivial 1-cycle
on the D8-brane
Worldsheet instanton
Standard chiral lagrangian
Boundary coupling
to D8 gauge field b.g.
GOR relation
Relation to previous attempts
Tachyon condensation
(Higgs mechanism)
Quark mass
[Casero, Kiritsis, Peredes (07)]
[Dhar, Nag (07,08)]
[Bergman, Seki, Sonnenschein (07)]
D8-D8 string : “tachyon”
Gauge invariant term
Difficulty : Tachyon action?
… No consistent truncation?
“Tachyon” picture and our solution
=
=
Tachyon condensation = partial annihilation of D8D8
(non-normalizable mode of “bulk tachyon”Ù quark mass)
The same worldsheet instanton gives the quark mass
Higgs mechanism and technicolor
are s-t channel dual to each other!
4. Discussions
Computed quantities in Sakai-Sugimoto model
Meson
Spectrum
Baryon
Spectrum
[Sakai-Sugimoto(04,05)]
[Hata et al.(07)]
Glueball
Spectrum
[Ooguri et al.]
[Brower et al. (03)]
Glueball decay
Charge radius,
form factors
[S.Terashima, C.I.Tan
and KH, 0709.2208]
[T.Sakai, S.Sugimoto
and KH, 0806.3122]
Computed quantities in Sakai-Sugimoto model
Glueball
Meson
Baryon
Nuclear force
[T.Sakai, S.Sugimoto
and KH, in progress]
Computed quantities in Sakai-Sugimoto model
Glueball
Exotics
Meson
Baryon
Finite temperature,
Finite density,
Chemical potential
More : hidden local symmetry, vector meson dominance,
Jet quenching parameters, viscosity of plasma,
Quark drag force, meson melting and decay width,
Phase transition temperature, …
Problems of the Sakai-Sugimoto model
- Massless quark problem
Æ Solved by worldsheet instanton
- Cut-off problem
Above
, unwanted excitations appear.
Glueball sector : “Susy” exciataions, S4 excitations
Meson sector : S4 excitations
Cause : Size of S4 = size of S1 = QCD scale
Æ Need of a scaling limit ?
- Asymptotic freedom?
Road map : Comparison with lattice QCD
Lattice QCD
Determine
what you
like to
compute
Mass spectra of bound states
Wilson loop, quark antiquark potential
Finite temperature, phase transition,
finite density, critical phenomena
Put gluons
Put QCD
on it
Holographic QCD
Put quarks
Quark mass
Put SM?
Choose consistent
D-brane configurations
String phenomenology
Lattice QCD
Invention of
computation
methods
Monte Carlo
Metropolis
Nuclear force
Finite T
Improvement
of
computation
Feedback
Continuum limit
Improved
programs
Holographic QCD
Gauge/gravity duality
Analogy to QCD string
Blackhole T
Double-Scaling limit?
Large N corrections?
SUGRA computations?
Better backgrounds?
Lattice SUSY
Check of duality
Cond-mat /
Cosmology
New standpoint of
string theory
Concluding words
Holographic QCD is a powerful tool
for analyzing low energy QCD
It is not just a reproduction of known physics…
it gives a new paradigm for understanding QCD.
Further surprises will wait for us…