Basic Ingredients of the NJL Model

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Basic Ingredients of the NJL Model
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Ref. T.Hatsuda and T.K., Phys. Rep.247(1994),221
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QCD???????????????NJL??
a. massive gluon exchange Fiertz ??

b. Instanton-induced Interaction
for low energy phenomena, local det-interaction
c. Strong coupling Lattice QCD
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One-flavor NJL model
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Mean Field Approximation
(Dirac equation!)
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Solution
helicity
Let us try to solve the Dirac equation with the
initial condition,
massless field
Solution
chiralityhelicity
others0.
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Non-diagonal! due to the mass term
Heisenberg eq.
I.C.
Bogoliubov transformation
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a superposition of states of various
chiralities
Spontaneous Symmetry Breaking!
C.f.
destroy 1 chirality
create 1 chirality
1 helicity
destroy 1 helicity
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Determination of the vacuum gap equation
Self-consistency condition
Gap equation!
For small
,
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The vacuum Energy
(Effective Potential)
Stationary cond
Gap equation
When
,i.e.,
,
the vacuum with Mgt0 is realized, and hence
chiral symmetry is spontaneously (dynamically)
broken!
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With current quark mass
(
)
The gap equation
current-quark mass dependence of the constituent
mass
graphical representation of the gap equation
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The effective potential
MeV
Chiral limit
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(No Transcript)
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SCC for static case
Excited states
Scalar mode
1-loop
Pionic mode
1-loop
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Cutoff scheme
The imaginary parts are finite without cutoff
The coupling constant may have the energy cutoff
From the dispersion relation, the real part is
defined as,
A change of variable
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For the scalar channel
The consistency in the cutoffs in the condensate
and the loops
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The dispersion relations of the pion and sigma
meson
0
Gap eq.!
0
Thus,
Nambu-Goldstone boson!
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Soft mode associated with chiral transition
In the normal phase.Putting M0,
Analytic continuation to the second Rieman sheet
dispersion relation(pole position in the Rieman
sheet)
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Strength function (Spectral function)
RPA
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Coupling dependence of the Spectral
function Precursory soft mode
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Two-flavor case
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The meson dispersion relations
Meson-quark coupling
Pion decay constant
Goldberger-Treiman relation
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With the G-T relation,
Gell-Mann-Oakes-Renner relation