1. Finite temperature in modified Soft-Wall AdS/QCD Model
Ling-xiao Cui
Institute of Theoretical Physics
Chinese Academy of Science

2. Outline Zero temperature meson sector
Thermodynamics of Soft-Wall AdS/QCD
Effects of finite chemical potential

3. The AdS/CFT Correspondence
Gauge/Gravity Correspondence: Extend to less or non SUSY theory
- d+1 Gravity on the bulk d QFT on the boundary
- the classical gravity theory strongly coupled QFT
GKP-W relation

4. Gauge/Gravity duality in QCD
-What we can do with gauge/gravity duality in QCD?
Transport coefficients (e.g. universal viscosity bound)
Compute observables in strongly coupled QFTs
Thermodynamics/QCD Phase diagrams
Meson/Nucleon spectra(melting)
Model QCD equation of state
Confinement/Deconfinement , Chiral breaking/restoration
…………………

5. Top-Down Vs. Bottom-up Top-Down AdS/QCD: Bottom-Up AdS/QCD:
Start from String Theory
Find brane config. for the gravity dual
Advantage:
Descriptions of theory are
relatively well understood, duality
is exact
Disadvantage:
So far QCD with fundamental flavors
does not have weakly-coupled AdS/
CFT dual, even at large-N
Bottom-Up AdS/QCD:
Do not dynamically generate a background geometry, but assume
one to incorporate a limited number of QCD properties:
Advantage:
Freedom to match model to aspects
of QCD.
Disadvantage:
Some features of model
disagree with QCD

6. Phenomenological AdS/QCD models
Do not dynamically generate a background geometry, but assume one to incorporate a limited number of QCD properties:
chiral symmetry
quark and gluon condensates
a background dilaton field to model a running coupling.(soft-wall)
Hard wall:
Confinement
Soft wall:

7. Soft wall model 5D action: Boundary Bulk
Global SU(3)L x SU(3)R symmetry
SU(3)L x SU(3)R gauge symmetry
5D action:
5D mass:

8. Field components
transform the L and R gauge fields into the vector (V ) and axial-vector (A) fields:
rewrite the action (3.3) in terms of the vector and axial-vector fields:
with
And the covariant derivative now becomes

9. Gauge coupling
In AdS, correlation function of source fields on UV brane
Bulk to boundary propagator solution to the equation of motion with V(0)=1
In QCD, correlation function of vector current is
Large Nc
small coupling

10. Chiral symmetry breaking by X
The field X becomes a complex field to incorporate the scalar S and the
pseudoscalar P fields:
One can find the expectation value of X
defined as the classical solutions of field equations, using UV boundary conditions
M is quark mass matrix (Explicit chiral breaking)
Σ is chiral condensate (Spontaneous chiral breaking)

11. Finding mass eigenvalues
In the hard-wall model, to find the poles of the scalar two-point function. Taking two functional derivatives of the effective action with respect
to the source
In the soft wall, Kaluza-Klein (KK) decomposition is the preferred method for finding mass eigenvalues. In KK decomposition, we break the field into an innite tower of 4D components and purely z-dependent parts:
Using Proca's equation:
The mass eigenvalue equation can be formulated from the equation of motion.

12. Modified Soft-Wall AdS/QCD
Y.Q.Sui, YLW, Z.F.Xie, Y.B.Yang PRD arXiv:  
Hard-wall AdS/QCD models contain the chiral symmetry breaking, the resulting mass spectra for the excited mesons are contrary to the experimental data
Soft-wall AdS/QCD models describe the linear confinement and desired mass spectra for the excited vector mesons, while the chiral symmetry breaking can't consistently be realized.
A quartic interaction in the bulk scalar potential was introduced to incorporate linear confinement and chiral symmetry breaking. While it causes an instability of the scalar potential and a negative mass for the lowest lying scalar meson state.
How to naturally incorporate two important features into a single AdS/QCD model and obtain the consistent mass spectra.

13. Modified Soft-Wall AdS/QCD
The 5D action with the background field of dilaton Φ(z) and a quartic term in the bulk scalar potential
Modified 5D Metric in IR Region
• IR region is modified
• UV region is not changed
Equation for the bulk vacuum:
IR&UV behaviour of dilaton and vacuum:
Solutions for the dilaton field at the UV & IR boundary

14. Three type of models for v(z)
Two IR boundary conditions of the bulk VEV:

15. 参数输入
scales the mass spectra of meson resonances, it can be found out from a global fitting.
The three parameters , and are fixed by the known experimental values:

17. Thermodynamics properties of Soft-Wall AdS/QCD

18. For finite temperature
QCD
gravity theory
deconfinement phase transition
Hawking-Page transition
dual
[ Herzog , Phys.Rev.Lett.98:091601,2007 ]
AdS BH (Deconfinement)
:Horizon
(HP)
Thermal AdS (Confinement)

19. Difficulties with Minkowski thermal correlators
The Minkowski version of the AdS/CFT
In the Euclidean version, the classical solution φ is uniquely determined by its value φ0 at the boundary z = 0 and the requirement of regularity at the horizon z = zH. Correspondingly, the Euclidean correlator obtained by using the correspondence is unique. However, in the Minkowski space, the requirement of regularity at the horizon is insufficient; to select a solution one needs a more refined boundary condition there. This is thought to reflect the multitude of real-time Green’s functions (Feynman, retarded,
advanced) in finite-temperature field theory.
incoming-wave boundary condition
retarded Green’s function
outgoing-wave boundary condition
advanced Green’s function

20. Difficulties still remain
Consider fluctuations of a scalar on AdS BH
The equation of motion:
The solution with fixed boundary condition at zB
and satisfy the incoming-wave condition at z=zH

21. The action on shell reduces to the surface terms:
where
Through GKP-W relation, retarded Green’s function:
due to
The imaginary part at the horizon completely cancels out the imaginary part at the boundary
D.T.Son’s prescription:

22. Phase Transition On-shell action density Free energy tAdS: AdSBH:
C.P.Herzog,PRL (2007)
On-shell action density Free energy
On-shell
tAdS:
AdSBH:
Difference between V1 and V2
When △V is positive (negative), tAdS (AdSBH) is stable.
Critical Temperature (1st order)
Tc,Hard = 122 and Tc,soft = 191 MeV.

23. mesons dissociation in soft-wall model
Fukushima, et al
(arXiv: )
Colangelo,et al
(arXiv: )
One way to increase the temperature is to set up a different mass scale in soft-wall.

24. Extend to finite temperature
L.X.Cui, Shingo Takeuchi, Y.L.Wu JHEP04(2012)144
5D action:
Modified AdS Black Hole:
with
• IR region is improved
• If , it become an AdS BH
: Horizon
Hawking temperature is not changed:

25. The EOM for Vector and Axial-Vector
Action at the quadratic order:
Vector:
Axial-Vector:

26. Taking in-falling boundary condition
From equation of V(z):
We assume:
divergent term can be read:
finite temperature part of vacuum:

27. Retarded Green’s function
2 independent solutions near the boundary
2 independent solutions near the horizon
: Out-coming from Black Hole
Advanced Green's function
: In-falling into Black Hole
Retarded Green's function
[D.T.Son and A.O.Starinets (2002)]

28. Spectral Function Linear combination of the 2 solution
In-falling solution
boundary condition:
Retarded Green's function
The surface term of vector part :
following D.T.Son’s prescription:

29. The retarded Geen’s function:
Spectral function:

30. Numerical Results Low temperature results
At low temperature, sharp peaks stand in accord with t=0 spectrum.

31. Numerical Results
Higher temperature results
Properties
1. The lowest-lying state melts gradually as T increases.
2. The peak moves to a smaller mass with increasing T.
3. The excited states melt much earlier and shift more.
4. The lowest-lying state survies until T~200MeV

32. comparison with other models
Our results: : :
Fukushima, et al
(arXiv: )
Colangelo,et al
(arXiv: )

33. Pseudo-scalar mesons
The pseudoscalar sector is the most intriguing part:
We adopt a complementary way to focus only on the finite temperature and density effects:
g5 → 0

34. Scalar and pseudo-scalar mesons
Finite temperature effects:

35. Scalar and pseudo-scalar mesons
Mass shift:
fitting the spectral function by following Breit-Wigner form:
with

36. Quark Number Susceptibility in modified soft wall model
The QNS is known to be the response of quark number density to infinitesimal change of the quark chemical potential
The lattice QCD studies suggest that the behavior of on the transition
between the hadronic phase and the Quark-Gluon-Plasma phase at zero quark chemical potential is a continuous but rapid in a narrow temperature interval around T ≃ 170 MeV
夸克数密度响应率

37. Quark Number Susceptibility in modified soft wall model
Using the fluctuation-dissipation theorem, it can be expressed as ：
The 5D action:
modified AdS BH:

38. Equations of motion :
with
Assuming the solution takes the form of in-falling condition at horizon:
due to
the function X can be expanded as:
equations becomes:

39. solution for At
with:
the on shell action reduced to the surface term:
then the Green’s function:

42. Effects Of Finite Chemical Potential

43. Modified Reissner-Nordstrom AdS BH
The gravity action describe the interaction of the U(1) gauge field with the AdS space is given by:
Under the ansatz:
The most general solution known as the Reissner-Nordstrom AdS black hole:
To characterize the low energy dynamic of QCD, IR region is modified:

44. Hawking temperature and chemical potential
According to AdS/CFT correspondence, the quark chemical potential, which
is the coefficient of the quark number operator, corresponds to the time-component of U (1) gauge field in the bulk.

45. Scalar and pseudo-scalar mesons
Finite density effects:

46. T-μ plane:
Lattice Result:
JHEP 0203 (2002) 014

47. Thanks you!