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II Russian-Spanish Congress “Particle and Nuclear Physics at all scales and Cosmology”,
Saint Petersburg, Oct. 4, 2013
RECENT ADVANCES IN THE BOTTOM-UP
HOLOGRAPHIC APPROACH TO QCD
Sergey Afonin
Saint Petersburg State University
A brief introduction
AdS/CFT correspondence – the conjectured equivalence between a string theory
defined on one space and a CFT without gravity defined on conformal boundary of
this space.
Maldacena example (1997):
Type IIB string theory on AdS5  S 5
in low-energy (i.e. supergravity)
approximation

4
YM theory on AdS boundary
in the limit
gYM N
1
String theory
Up down
Bottom up
AdS/QCD correspondence – a program to implement such a
duality for QCD following the principles of AdS/CFT
correspondence
We will
discuss
QCD
Basic property: Algebra of SO(4,2) group and that of isometries of AdS5 coincide
SO(4, 2) : Equivalence of energy scales
 The 5-th coordinate – (inverse) energy scale
[Witten; Gubser, Polyakov, Klebanov (1998)]
Essence of the holographic method
Operators in a 4D gauge theory

Classical fields in 5D dual theory
In the sence that the corresponding sources
 Boundary values
One postulates:
generating functional
effective action
The correlation functions are given by
The output of the holographic models: Correlators
Mass spectrum: Poles of the two-point correlator
Alternative way for finding the mass spectrum is to solve e.o.m.
The holographic correspondence dictates the relation
Main assumption of AdS/QCD: There is an approximate 5D holographic dual for QCD
An important example of dual fields for the QCD operators (R=1):
Here
A typical model (Erlich et al., PRL (2005); Da Rold and Pomarol, NPB (2005))
For
Hard wall model:
At
one imposes certain gauge invariant boundary conditions on the fields.
Equation of motion for the scalar field
Solution independent of usual 4 space-time coordinates
quark condensate
bare quark mass
As the holographic
dictionary prescribes
here
Denoting
the equation of motion for the vector fields are (in the axial gauge)
due to chiral symmetry breaking
where
The spectrum of normalizable modes is given by zeros of Bessel function, thus the
asymptotic behavior is
mn
that is not Regge like
mn2
n
n
Soft wall model (Karch et al., PRD (2005))
The IR boundary condition is that the action is finite at
To have the Regge like spectrum:
To have AdS space in UV asymptotics:
The mesons of arbitrary spin J can be considered, the spectrum has the form
But! No natural chiral symmetry breaking!
Self-consistent extension to the arbitrary intercept: Afonin, PLB (2013)
Some applications
 Meson, baryon and glueball spectra
Low-energy strong interactions (chiral dynamics)
Hadronic formfactors
Thermodynamic effects (QCD phase diagram)
Condensed matter (high temperature superconductivity etc.)
...
Deep relations with other approaches
Light-front QCD
Soft wall models: QCD sum rules in the large-Nc limit
Hard wall models: Chiral perturbation theory supplemented by infinite number of vector
and axial-vector mesons
Holographic description of thermal and finite density effects
- corresponds to
Basic ansatz
One uses the Reissner-Nordstrom AdS black hole solution
where
is the charge of the gauge field.
The hadron temperature is identified with the Hawking one:
The chemical potential is defined by the condition
The critical temperature and density (deconfinement) can be found from the condition of
complete dissociation of meson peaks in the correlators. The typical critical temperature at zero
chemical potential for the light flavors lies about 200 MeV, for heavy ones does near 550 MeV.
Some examples of phase diagrams
He et al., JHEP (2013)
Colangelo et al.,
EPJC (2013)
Hadronic formfactors
Definition for mesons:
Electromagnetic formfactor:
In the holographic models for QCD:
Brodsky, de Teramond, PRD (2008)
Linear spectrum and quark masses
The dependence of A and B on the quark masses?
Afonin, Pusenkov, PLB (2013)
Basic construction: The no-wall holographic model (Afonin, PLB (2009))
The result:
From the ω-meson trajectory:
From the holography:
Charmonim:
In the heavy-quark limit:
Bottomonium:
The binding energy grows linearly
with the quark mass!
Interpretation: When a non-relativistic quark is created and moves with the velocity v in the c.m.
frame, should compensate its kinetic energy
In the limit
This coincides with a prediction of the Lovelace-Shapiro dual amplitude!
Thank you!