QCD Phase Diagram from
Finite Energy Sum Rules
Alejandro Ayala
Instituto de Ciencias Nucleares, UNAM
(In collaboration with A. Bashir,
C. Domínguez, E. Gutiérrez, M. Loewe, and A. Raya)
arXiv:1106.5155 [hep-ph]
Outline
• Deconfinement and chiral symmetry
restoration
• Resonance threshold energy as
phenomenological tool to study
deconfinement
• QCD sum rules at finite temperature/chemical
potential
• Results
Deconfinement and chiral
symmetry restoration
Driven by same effect:
• With increasing density, confining interaction gets screened and
eventually becomes less effective (Deconfinement)
• Inside a hadron, quark mass generated by confining
interaction. When deconfinement occurres, generated
mass is lost (chiral transition)
Critical end point?
Lattice quark condensate
and Polyakov loop
A. Bazavov et al., Phys. Rev. D 90, 014504 (2009)
Status of phase diagram
• =0: Physical quark masses, deconfinement and
chiral symmetry restoration coincide. Smooth
crossover for 170 MeV < Tc < 200 MeV
• Analysis tools:
– Lattice (not applicable at finite )
– Models (Polyakov loop, quark condesate)
• Lattice vs. Models:
– Lattices gives:
smaller/larger
chemical potential/temperature values for endpoint than
models
• Critical end point might not even exist!
Alternative signature:
Melting of resonances
Im 
pole
s0
s
For increasing T and/or B the energy threshold for the continuum goes to 0
Correlator of axial currents
Quark – hadron duality
Finite energy sum rules
Operator product expansion
Non-pert part: dispersion relations
Pert part: imaginary parts
at finite T and 
Two contributions:
1) Annihilation channel (available also at T==0)
2) Dispersion channel (Landau damping)
Imaginary parts at finite T and 
Annihilation term
Dispersion term
Pion pole
Threshold s0 at finite T and 
N=1, C2<O2> = 0
2
GMOR
Need quark condensate
at finite T and 
quark condensate T,   0
Poisson summation formula
quark condensate
A. Bazavov et al., Phys. Rev. D 90, 014504 (2009)
Lose of Lorentz covariance means that
Parametrize S-D solution in terms of “free-like” propagators
Parameters fixed by requiring S-D conditions and description of lattice data
Representation makes it easy to
carry out integration
8
_
2
Susceptibilities
QCD Phase Diagram
Summary and conclusions
• QCD phase diagram rich in structure: critical end
point?
• Polyakov loop, quark condensate analysis can be
supplemented with other signals: look at threshold s0
as function of T and 
• Finite energy QCD sum rules provide ideal
framework. Need calculation of quark condesnate.
Use S-D quark propagator parametrized with “freelike” structures.
• Transition temperatures coincide, method not
accurate enough to find critical point, stay tuned.