Mesonic correlation functions at finite temperature in the NJL model
29
Mesonic correlation functions at finite temperature in the NJL model Ricardo Farias a Advisor: G. Krein IFT - UNESP - SP FB18 August 25th, 2006 a G. Dallabona, G.Krein and O.A. Battistel
Transcript of Mesonic correlation functions at finite temperature in the NJL model
Mesonic correlation functions atfinite temperature in the NJL
modelRicardo Farias
a
Advisor: G. Krein
IFT - UNESP - SP
FB18
August 25th, 2006
aG. Dallabona, G.Krein and O.A. Battistel
OutlineMotivation
Properties of dense matter
Lattice QCD
Four-fermion models
NJL model
Scalar, pseudoscalar, vector and axial-vector couplings
Real-time finite temperature
Hadronic correlation functions
Implicit regularization scheme
Numerical Results
Comparison with lattice QCD results
Conclusions
Future applications
Mesonic correlation functions at finite temperature in the NJL model – p.1/28
Motivation
Properties of dense matter −→ diquark condensation and color superconductivity
Inapplicability of weak coupling techniques −→ densities/temperatures ofphenomenological interest
Nonperturbative lattice techniques are not yet sufficiently developed to deal withsuch problems
Phenomenological models −→ to make progress in the field.
Nonrenormalizable four-fermion interactions −→ aspects of dynamical chiralsymmetry breaking (DχSB) and high density/temperature quark matter.
Mesonic correlation functions at finite temperature in the NJL model – p.2/28
NJL model
The lagrangian of the NJL model which we will work is
L = ψ (x) (iγµ∂µ −m0)ψ (x) +GS
»
“
ψ (x)ψ (x)”2
+“
ψ (x) iγ5−→τ ψ (x)
”2–
− GV
»
“
ψ (x) γµ−→τ ψ (x)
”2+“
ψ (x) γ5γµ−→τ ψ (x)
”2–
At one loop approximation, the gap equation is given by
M = 48GMˆ
iIquad(M) − I(T )˜
where Iquad(M2) is the quadratically divergent integral
Iquad(M2) =
Z
d4k
(2π)41
k2 −M2
and I(T, µ) is the finite integral
I(T, µ) =
Z
d3k
(2π)3n(k)
E(k)and n(k) being the Fermi-Dirac distribution
Mesonic correlation functions at finite temperature in the NJL model – p.3/28
NJL model
At this one-loop approximation, besides the quadratically divergent integralIquad(M2) there appears also a logarithmically divergent integral
Ilog(M2) =
Z
d4k
(2π)41
(k2 −M2)2
The traditional approach −→ three- or four-momentum cutoff Λ
The finite integral I(T ) containing the Fermi-Dirac distributions is also cutoff
In this way leaving out the high-momentum components
Mesonic correlation functions at finite temperature in the NJL model – p.4/28
Real-time finite temperature
The self-energy satisfies the following relationsa
Re Π = Re Π11
Im Π = sech (2θp) Im Π11
where Π11 means that the polarization functions are evaluated using the (1,1)component of free-thermal fermionic propagator
iS (6k) =i
6k −m+ iε− 2π (6k +m) δ
`
k2 −m2´
n (k0)
and
sech(2θp) =1 − e−β|p0|
1 + e−β|p0|
aKobes and Semenoff, Nucl. Phys. B260, 747
Mesonic correlation functions at finite temperature in the NJL model – p.5/28
Hadronic correlation functions
The correlation functions, in Minkowski space, are defined as
Cij = i
Z
d4x e−ipx 〈T (Ji (x) Jj (0))〉β
with the currents
Ji (x) = ψα1(x) Γα1β1λiψβ1
(x)
where Γ = (1, iγ5, γµ, γµγ5, σµν) are the Dirac matrices. Here Cij
`
p2, T´
represents the full (meson ) polarization propagator in the random-phaseapproximation.
Solving the Bethe-Salpeter equation (for reviews seea)
Cσ (q) =Πσ (q)
1 − 2GSΠσ (q)
aU. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195 (1991), S.P. Klevansky, Rev. Mod. Phys. 64,
649 (1992), M.K. Volkov, Part. Nucl. B 24, 1 (1993) Mesonic correlation functions at finite temperature in the NJL model – p.6/28
Hadronic correlation functions
Cπ =Ππ (q) [1 + 2GV ΠA (q)] − 2q2GV Π2
πA (q)
[1 + 2GV ΠA (q)] [1 − 2GSΠπ (q)] + 4q2GSGV Π2πA (q)
Cρ =ΠB (q)
1 − 2GV ΠB (q)
„
−gµν +qµqν
q2
«
defining
CA (p) = CµµA (p)
CA (p) = −4`
Π(B) − Π(A)´
1 −GV
ˆ
Π(B) − Π(A)˜
+CP (p)
n
[1 −GSΠπ ] Π(B) + p2GS (ΠπA)2o
n
Ππˆ
1 +GV Π(A)˜
− p2GV (ΠπA)2o
˘
1 +GV
ˆ
Π(A) − Π(B)˜¯
Mesonic correlation functions at finite temperature in the NJL model – p.7/28
Hadronic correlation functions
where
ΠπA =Π(A)
2M
Axial polarization:
ΠµνA = gµνΠ(A) +
„
−gµν +pµpν
p2
«
Π(B)
and
Π(A) = Π(A)00 + Π
(A)0β + Π
(A)β0 + Π
(A)ββ
Π(B) = Π(B)00 + Π
(B)0β + Π
(B)β0 + Π
(B)ββ
Mesonic correlation functions at finite temperature in the NJL model – p.8/28
Scalar polarization function
In the case of the scalar mesons
Cσ =Πσ
1 − 2GS (T )Πσ
The imaginary part of Cσ can be written as
ImCσ =Im Πσ
(1 − 2GSRe Πσ)2 + (2GS Im Πσ)2
where
Π11σ = −i
Z
d4k
(2π)4Tr [iS (6k) iS (6k−6p)]
The free thermal propagator is
iS (6k) = iS0 (6k) + iSβ (6k)
Mesonic correlation functions at finite temperature in the NJL model – p.9/28
Scalar polarization function
where
iS0 (6k) =i
6k −m+ iε
iSβ (6k) = −2π (6k +m) δ`
k2 −m2´
n (k0)
and
n (k0) =1
eβ|k0| + 1
Then
Π11σ = −i
Z
d4k
(2π)4Tr˘ˆ
iS0 (6k) + iSβ (6k)˜ ˆ
iS0 (6k−6p) + iSβ (6k−6p)˜¯
With these definitions we may write
Π11σ = Π00 + Π0β + Πβ0 + Πββ
Mesonic correlation functions at finite temperature in the NJL model – p.10/28
Scalar polarization function
Performing trace we rewrite Π00 as
Π00 = iNcNf [Tσ (p)]
where
Tσ (p) = 4
ˆ
Iquad
`
m2´˜
+1
2
`
4m2 − p2´ ˆ
Ilog
`
m2´˜
−1
2
`
4m2 − p2´
„
i
16π2
«
ˆ
Z0
`
p2,m2´˜
ff
and
Z0
`
p2,m2´
= −2 +
s
1 − 4m2
p2ln
0
B
@
1 +q
1 − 4m2
p2
1 −q
1 − 4m2
p2
1
C
A
− iπ
s
1 − 4m2
p2Mesonic correlation functions at finite temperature in the NJL model – p.11/28
Scalar polarization function
The imaginary part of Π00 is
Im Π00 =NcNf
8πp2„
1 − 4m2
p2
«3
2
and the real part is
Re Π00 = 4NcNf iˆ
Iquad
`
m2´˜
+ 2NcNf
`
4m2 − p2´
iˆ
Ilog
`
m2´˜
− NcNf
8π2
`
4m2 − p2´
2
6
42 −
s
1 − 4m2
p2ln
0
B
@
q
1 − 4m2
p2+ 1
1 −q
1 − 4m2
p2
1
C
A
3
7
5
Mesonic correlation functions at finite temperature in the NJL model – p.12/28
Scalar polarization function
For the Π0β contribution
Im Π0β = −NcNf
8πp2„
1 − 4m2
p2
«3
2
n“p
2
”
ReΠ0β =1
πPZ ∞
4M2
dq2ImΠ0β
`
q2´
q2 − p2
For the Πββ contribution
Im Πββ =NcNf
4πp2„
1 − 4m2
p2
«3
2 h
n“p
2
”i2
Mesonic correlation functions at finite temperature in the NJL model – p.13/28
Scalar polarization function
for |p| = 0 and p0 > 0
Im Πσ =NcNf
8πp20
„
1 − 4M2
p20
«3
2 n
1 − 2n“p0
2
”o
and
Re Πσ = Re Π00
`
p2´
+ 2Re Π0β
`
p2, T´
where
Re Π00 = 4NcNf iˆ
Iquad
`
m2´˜
+ 2NcNf
`
4m2 − p20´
iˆ
Ilog
`
m2´˜
− NcNf
8π2
`
4m2 − p20´
2
6
6
4
2 −s
1 − 4m2
p20ln
0
B
B
@
r
1 − 4m2
p2
0
+ 1
1 −r
1 − 4m2
p2
0
1
C
C
A
3
7
7
5
Mesonic correlation functions at finite temperature in the NJL model – p.14/28
Implicit regularization scheme
aWe assume that each integral is regularized through an unspecified distribution
f(k/Λ), where Λ is a parameter with the dimensions of momentum such that
Iquad(M2) =
Z
d4k
(2π)4f(k/Λ)
k2 −M2
Ilog(M2) =
Z
d4k
(2π)4f(k/Λ)
(k2 −M2)2
One interesting aspect of this regularization scheme is that Iquad and Ilog respectthe scaling relations
iˆ
Iquad(M2)˜
= iˆ
Iquad(M20 )˜
+`
M2 −M20
´
iˆ
Ilog(M20 )˜
− 1
16π2
»
M2 −M20 −M2 ln
„
M2
M20
«–
iˆ
Ilog`
M2´˜
= iˆ
Ilog`
M20
´˜
+1
16π2ln
„
M2
M20
«
aO.A. Battistel
Mesonic correlation functions at finite temperature in the NJL model – p.15/28
Implicit regularization scheme
In the vacuum part we have basic divergent objects, Iquad(M2) and Ilog(M2),which should be treated in order to make a numerical analysis. One convenientapproach is to fit them using the following results of the NJL model
iˆ
Iquad(M20 )˜
= −
D
ψψE
12M0
iˆ
Ilog(M20 )˜
= − f2π
12M20
where M0 = 460 MeV is a vacuum mass obtained for 〈ψψ〉 = −260MeV3 andfπ = 93 MeV .
The gap equation can be written as
M = 48GM
(
−〈ψψ〉012M0
− (M2 −M20 )
f2π
12M20
− 1
(4π)2
»
M2 −M20 −M2 log
„
M2
M20
«–
− I(T )
)
Mesonic correlation functions at finite temperature in the NJL model – p.16/28
Scalar polarization function
Im Πσ =NcNf
8πp20
„
1 − 4M2
p20
«3/2n
1 − 2n“p0
2
”o
Re Πσ = ReΠ00
`
p20´
+ 2Re Π0β
`
p20, T´
Re Π00 = −NcNf
3
D
ψψE
M0− NcNf
3
`
M2 −M20
´ f2π
M20
+NcNf
6p20
„
1 − 4M2
p20
«
f2π
M20
− NcNf
4π2
»
M2 −M20 −M2 ln
„
M2
M20
«–
− NcNf
8π2p20
„
1 − 4M2
p20
«
ln
„
M2
M20
«
+NcNf
8π2p20
„
1 − 4M2
p20
«
2
6
6
4
2 −s
1 − 4M2
p20ln
0
B
B
@
1 +
r
1 − 4M2
p2
0
1 −r
1 − 4M2
p2
0
1
C
C
A
3
7
7
5
Re Π0β = −NcNf
8π2PZ ∞
4M2
dq2q2
q2 − p20
„
1 − 4M2
q2
«3
2
n
p
q2
2
!
Mesonic correlation functions at finite temperature in the NJL model – p.17/28
Pseudoscalar polarization function
Im Ππ =NcNf
8πp20
s
1 − 4M2
p20
n
1 − 2n“p0
2
”o
Re Ππ = ReΠ00
`
p20´
+ 2Re Π0β
`
p20, T´
Re Π00 = −NcNf
3
D
ψψE
M0− NcNf
6
ˆ
2`
M2 −M20
´
− p20˜ f2
π
M20
− NcNf
4π2
»
M2 −M20 +
1
2
`
p20 − 2M2´
ln
„
M2
M20
«–
+NcNf
8π2p20
8
>
>
<
>
>
:
2 −s
1 − 4M2
p20ln
0
B
B
@
1 +
r
1 − 4M2
p2
0
1 −r
1 − 4M2
p2
0
1
C
C
A
9
>
>
=
>
>
;
Re Π0β = −NcNf
8π2PZ ∞
4M2
dq2qp
q2 − 4m2n
„√q2
2
«
q2 − p20
Mesonic correlation functions at finite temperature in the NJL model – p.18/28
Vector polarization function
Im Πρ =NcNf
12π(p20 + 2m2)
s
1 − 4m2
p20
n
1 − 2n“p0
2
”o
Re Πρ = Re Π00
`
p20´
+ 2Re Π0β
`
p20, T´
Re Π00 =NcNf
9p20
f2π
M20
− NcNf
12π2p20 ln
„
M2
M20
«
− NcNf
12π2p20
8
>
>
<
>
>
:
1
3−„
1 +2M2
p20
«
2
6
6
4
2 −s
1 − 4M2
p20ln
0
B
B
@
r
1 − 4M2
p2
0
+ 1
1 −r
1 − 4M2
p2
0
1
C
C
A
3
7
7
5
9
>
>
=
>
>
;
Re Π0β = −NcNf
12π2PZ ∞
4M2
dq2(q2+2M2)
q
p
q2 − 4M2
q2 − p20n“ q
2
”
Mesonic correlation functions at finite temperature in the NJL model – p.19/28
Axial polarization function
Im Π(A) =NcNf
2πM2
s
1 − 4M2
p2
h
1 − 2n“p0
2
”i
Re Π(A) =2
3NcNfM
2 f2π
M20
− NcNf
2π2M2 ln
„
M2
M20
«
+NcNf
2π2M2
2
6
6
4
2 −s
1 − 4M2
p20ln
0
B
B
@
1 +
r
1 − 4M2
p2
0
1 −r
1 − 4M2
p2
0
1
C
C
A
3
7
7
5
− NcNf
π2M2
Z ∞
0ds
1
s+ 4M2 − p20
×
8
<
:
s
1 − 4M2
s+ 4M2n
√s+ 4M2
2
!
−s
1 − 4M2
p20n“p0
2
”
9
=
;
Mesonic correlation functions at finite temperature in the NJL model – p.20/28
Axial polarization function
Im Π(B) =NcNf
12π
„
1 +2m2
p20
«
p20
s
1 − 4m2
p20(1 − 2n)
Re Π(B) =NcNf
9p20
f2π
M20
− NcNfp20
12π2ln
„
M2
M20
«
− NcNf
12π2p20
8
>
>
<
>
>
:
1
3−„
1 +2m2
p20
«
2
6
6
4
2 −s
1 − 4m2
p20ln
0
B
B
@
1 +
r
1 − 4m2
p2
0
1 −r
1 − 4m2
p2
0
1
C
C
A
3
7
7
5
9
>
>
=
>
>
;
+2
πPZ ∞
4M2
dq2ImΠ
(B)0β
`
q2´
q2 − p20
Mesonic correlation functions at finite temperature in the NJL model – p.21/28
T dependence of coupling constants
Using the model proposed By C.M. Skakin et alla
G (T ) = G0
„
1 − 0.3T
Tc
«
0 50 100 150 200 250 300 350 400T (MeV)
0
50
100
150
200
250
300
350
400
450
500
M (
MeV
)
G0
G(T)
aC.M. Shakin et all PRD 67, 114012 (2003) Mesonic correlation functions at finite temperature in the NJL model – p.22/28
Numerical results
Pseudoscalar spectral function
0 5 10 15 20 25ω / Τ
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
σ / ω
2
T/Tc = 1.5
T/Tc = 3.0
0 5 10 15 20 250.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0 5 10 15 20 250.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
ω / T
σ / ω
2
Mesonic correlation functions at finite temperature in the NJL model – p.23/28
Numerical results
Lattice results - Peter Petreczky J. Phys. G: Nucl. Part. Phys 30 (2004)S431
Mesonic correlation functions at finite temperature in the NJL model – p.24/28
Numerical results
Vector spectral function
0 5 10 15 20 25ω / Τ
0,00
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
σ / ω
2
T/Tc = 1.5
T/Tc = 3.0
0 5 10 15 20 250.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
σ / ω
2
ω / T
Mesonic correlation functions at finite temperature in the NJL model – p.25/28
Numerical results
Lattice results - Peter Petreczky J. Phys. G: Nucl. Part. Phys 30 (2004)S431
Mesonic correlation functions at finite temperature in the NJL model – p.26/28
Conclusions
Hadronic correlation functions −→ 0 (absence of asymptotic freedom)
With implicit regularization scheme:
we obtain results which are consistent with all chiral properties
we avoid global and gauge symmetry violations, breaking of causality andunitarity
scaling relations −→ short-distances effects −→ quark mass increases
NJL model gives good results for hadronic correlation functions at low energies
Mesonic correlation functions at finite temperature in the NJL model – p.27/28
Future applications
Momentum dependence of hadronic correlation functions (work in progressa)
consider finite three-momentum in our calculations
compare with recent lattice results
aR.L.S.Farias, G. Dallabona, G.Krein and O.A. Battistel
Mesonic correlation functions at finite temperature in the NJL model – p.28/28
