Cold, dense quark matter
NJL type models and the need to go beyond
THOMAS KLÄHN
Collaborators: D. Zablocki, J.Jankowski, C.D.Roberts, R.Lastowiecki, D.B.Blaschke
2013/09/B/ST2/01560
Neutron Stars are born in Supernovae
SN in A.D.1054 was visible from Earth
Neutron Stars are born in Supernovae
SN in A.D.1054 was visible from Earth
with naked eyes
Neutron Stars are born in Supernovae
SN in A.D.1054 was visible from Earth
with naked eyes for 23 days
Compact Stars are born in Supernova
SN in A.D.1054 was visible
with naked eyes for 23 days
AT DAYLIGHT
Neutron Stars are born in Supernova
SN in A.D.1054 was visible
with naked eyes for 23 days
Records found in:
China
Neutron Stars are born in Supernova
SN in A.D.1054 was visible
with naked eyes for 23 days
Records found in:
China
Middle East
?
Neutron Stars are born in Supernova
SN in A.D.1054 was visible
with naked eyes for 23 days
Records found in:
China
Middle East
America
?
Europe ... probably missed it
East-West schism the very same year
Supernova remnant in Crab Nebula:
Hubble ST
detectable frequencies:
- optical
Courtesy of http://chandra.harvard.edu/photo/2009/crab
Supernova remnant in Crab Nebula:
detectable frequencies:
- optical
- infrared
Spitzer ST
Courtesy of http://chandra.harvard.edu/photo/2009/crab
Supernova remnant in Crab Nebula:
detectable frequencies:
- optical
- infrared
- x-ray
Chandra ST
Courtesy of http://chandra.harvard.edu/photo/2009/crab
Supernova remnant in Crab Nebula:
detectable frequencies:
- optical
- infrared
- x-ray
obviously a
complex &
structured
system
Hubble ST
Spitzer ST
Chandra ST
Courtesy of http://chandra.harvard.edu/photo/2009/crab
Supernova remnant in Crab Nebula:
Hubble ST
detectable frequencies:
- optical
- infrared
- x-ray
Spitzer ST
obviously a
complex &
structured
system
Chandra ST
CRAB PULSAR
Courtesy of http://chandra.harvard.edu/photo/2009/crab
Supernova remnant in Crab Nebula:
Hubble ST
detectable frequencies:
- optical
- infrared
- x-ray
Spitzer ST
obviously a
complex &
structured
system
Chandra ST
CRAB PULSAR
Age
958 yrs
Rotation
29.6 /s
Mass
?
Radius
?
Luminosity
?
B-field
?
Courtesy of http://chandra.harvard.edu/photo/2009/crab
CRAB PULSAR
Age
958 yrs
Rotation
29.6 /s
Mass
?
Radius
?
Luminosity
?
B-field
?
Courtesy of http://www.ast.cam.ac.uk/~optics/Lucky_Web_site
Neutron Stars

Variety of scenarios regarding inner structure: with or without QM

Question whether/how QCD phase transition occurs is not settled

Most honest approach: take both (and more) scenarios into
account and compare to available data
Neutron Stars = Quark Cores?

Variety of scenarios regarding inner structure: with or without QM

Question whether/how QCD phase transition occurs is not settled

Most honest approach: take both (and more) scenarios into
account and compare to available data
Neutron Stars = Quark Cores?

Variety of scenarios regarding inner structure: with or without QM

Question whether/how QCD phase transition occurs is not settled

Most honest approach: take both (and more) scenarios into
account and compare to available data
Neutron Stars = Quark Cores?

Variety of scenarios regarding inner structure: with or without QM

Question whether/how QCD phase transition occurs is not settled

Most honest approach: take both (and more) scenarios into
account and compare to available data
Neutron Stars = Quark Cores?

Variety of scenarios regarding inner structure: with or without QM

Question whether/how QCD phase transition occurs is not settled

Most honest approach: take both (and more) scenarios into
account and compare to available data
QCD Phase Diagram

dense hadronic matter
HIC in collider experiments
Won’t cover the whole diagram
Hot and ‘rather’ symmetric
NS as a 2nd accessible option
Cold and ‘rather’ asymmetric
Problem is more complex than
It looks at first gaze
www.gsi.de
QCD Phase Diagram

dense hadronic matter
pQCD?
HIC in collider experiments
Won’t cover the whole diagram
Hot and ‘rather’ symmetric
NS as a 2nd accessible option
Cold and ‘rather’ asymmetric
Problem is more complex than
It looks at first gaze
www.gsi.de
Neutron Star Data

Data situation in general terms is good (masses, temperatures, ages, frequencies)

Ability to explain the data with different models in general is good, too.
... sounds good, but becomes tiresome if everybody explains everything …

For our purpose only a few observables are of real interest

Most promising: High Massive NS with 2 solar masses (Demorest et al., Nature 467, 1081-1083 (2010))
NS masses and the (QM) Equation of State

NS mass is sensitive
mainly to the sym. EoS
(In particular true for
heavy NS)

Folcloric:
QM is soft, hence no
NS with QM core

Fact:
QM is softer, but able
to support QM core in NS

Problem:
(transition from NM to)
QM is barely understood
NS masses and the (QM) Equation of State

NS mass is sensitive
mainly to the sym. EoS
(In particular true for
heavy NS)

Folcloric:
QM is soft, hence no
NS with QM core

Fact:
QM is softer, but able
to support QM core in NS

Problem:
(transition from NM to)
QM is barely understood
NS masses and the (QM) Equation of State

NS mass is sensitive
mainly to the sym. EoS
(In particular true for
heavy NS)

Folcloric:
QM is soft, hence no
NS with QM core

Fact:
QM is softer, but able
to support QM core in NS

Problem:
(transition from NM to)
QM is barely understood
NS masses and the (QM) Equation of State

NS mass is sensitive
mainly to the sym. EoS
(In particular true for
heavy NS)

Folcloric:
QM is soft, hence no
NS with QM core

Fact:
QM is softer, but able
to support QM core in NS

Problem:
(transition from NM to)
QM is barely understood
NS masses and the (QM) Equation of State

NS mass is sensitive
hadrons
mainly to the sym. EoS
Dense Nuclear Matter in terms of Quark DoF is barely understood
Problem is attacked in vacuum Faddeev Equations
(In particular true for
heavy NS)
Folcloric:
QM is soft, hence no
NS with QM core

QCD

Bethe Salpeter Equations
Fact:
QM is softer, but able
to support QM core in NS

Problem:
(transition from NM to)
QM is barely understood
Baryons as composites of confined quarks and diquarks
quarks
NS masses and the (QM) Equation of State
(In particular true for
heavy NS)

Folcloric:
QM is soft, hence no
NS with QM core

Fact:
QM is softer, but able
to support QM core in NS

Problem:
(transition from NM to)
QM is barely understood
confinement
mainly to the sym. EoS
hadrons
QCD
NS mass is sensitive
dynamical breaking of chiral symmetry

quarks
Dense Nuclear Matter in terms of Quark DoF is barely understood
Problem is attacked in vacuum Faddeev Equations
Baryons as composites of confined quarks and diquarks
Bethe Salpeter Equations
QCD in dense matter

LQCD fails in dense (like DENSE) matter (Fermion-sign problem)

Perturbative QCD fails in non-perturbative domain
DCSB is explicitly not covered by perturbative approach:

Solution: ‘some’ non-perturbative approach ‘as close as possible’ to QCD
some = solvable; as close as possible = if possible DCSB, if possible confinement

State of the art: Nambu-Jona-Lasinio model(s) (+t.d. bag models, +hybrids)
NJL type models: brief reminder
Partition function in path integral representation
Current-current interaction
Hubbard Stratonovich transformation
mean field approximation w.r. to bosonic fields -> shifts in mass, chem. Pot., 1p-energy -> quasi particles
NJL type models: brief reminder
Effectively an ideal (SC) gas with medium dependent masses and chemical potentials
2Nc
2Nc
NJL type models

S: DCSB

V: renormalizes μ

D: diquarks → 2SC, CFL

TD Potential minimized
in mean-field approximation

Effective model by its nature;
can be motivated (1g-exchange)
doesn’t have to though and can
be extended (KMT, PNJL)

possible to describe hadrons
NJL model study for NS
(TK, R.Łastowiecki, D.Blaschke, PRD 88, 085001 (2013))
Set A
Set B
Conclusion: NS may or may not support a significant QM core.
additional interaction channels won’t change this if coupling strengths are not precisely known.
DSE ↔ NJL

NJL model can be understood as an approximate solution of Dyson-Schwinger equations
quark
gluon
q-g-vertex
DSE ↔ NJL

NJL model can be understood as an approximate solution of Dyson-Schwinger equations
quark
gluon
q-g-vertex
single particle: quark self energy
Inverse (Single-)Quark Propagator:

S ( p ;  )  1  Z 2 ( i  p  i  4 ( p 4  i  )  m bm )   ( p ;  )
i p
revokes Poincaré covariance
Renormalised Self Energy:

 ( p ;  )  Z 1  g (  ) D  ( p  q ;  )
2
q

a
2
  S ( q ;  )  ( q , p ;  )
a
Loss of Poincaré covariance increases complexity
→ technically and numerically more challenging → no surprise, though
General Solution:
Vacuum:   0
S ( p 2 ) 1  i  p A ( p 2 )  B ( p 2 )

Medium:   0 S ( p 2 , p 4 ;  )  1  i  p A ( p 2 , p 4 ,  )  i  4 ( p 4  i  ) C ( p 2 , p 4 ,  )  B ( p 2 , p 4 ,  )
Similar structured equations in vacuum and medium, but in medium:
1. one more gap
2. gaps are complex valued
3. gaps depend on (4-)momentum, energy and chemical potential
Effective gluon propagator

S ( p ;  )  1  Z 2 ( i  p  i  4 ( p 4  i  )  m bm )   ( p ;  )

 ( p ;  )  Z 1  g (  ) D  ( p  q ;  )
2
q

a
2
  S ( q ;  )  ( q , p ;  )
a
Ansatz for self energy (rainbow approximation, effective gluon propagator(s))
Specify behaviour of
Infrared strength
running coupling for large k
(zero width + finite width contribution)
EoS (finite densities):
1st term (Munczek/Nemirowsky (1983))
2nd term
NJL model:
delta function in momentum space → Klähn et al. (2010)
→ Chen et al.(2008,2011)
delta function in configuration space = const. In mom. space
NJL model within DS framework
gap solutions are
momentum independent.
Simple: A=1
Renormalization of
chem. pot. due to
vector interaction
mass gap equation
This is a 1 to 1 reproduction of the (basic) NJL model
NJL model within DS framework
Renormalization of
chem. pot. due to
vector interaction
mass gap equation
This is a 1 to 1 reproduction of the (basic) NJL model
NJL model within DS framework
Steepest descent approximation
1 to 1 NJL (regularization issue ignored)
Renormalization of
chem. pot. due to
vector interaction
mass gap equation
This is a 1 to 1 reproduction of the (basic) NJL model
NJL model within DS framework
Steepest descent approximation
1 to 1 NJL (regularization issue ignored)
Renormalization of
chem. pot. due to
vector interaction
mass gap equation
This is a 1 to 1 reproduction of the (basic) NJL model
Regularization
Regularization completely ignored so far. Two sources for infinities:
- zero point energy (usually resolved by vacuum subtraction)
- scalar density (quadratically divergent, no vacuum subtraction if mass is not const/medium dep.)
‘Standard’ NJL: Cutoff (hard, soft (e.g. Lorentz, Gauss), 4-momentum)
- finite quark density for any T > 0 at any μ > 0 (an ideal gas is an ideal gas is an…)
- Have no ‘strict’ definition of confinement, BUT
- Intuitively confinement implies nq=0
- problem for description of truly confined bound states in medium
beyond a ‘chemical’ picture
Recently exploited to investigate meson properties (Gutierrez-Guerrero et al. 2010, …)
proper time regularization scheme (Ebert et al. 1996)
UV: removes UV divergence
+IR: removes poles in complex plane
Proper Time Regularization
neat scheme in vacuum:
more complicated at finite chem. potential:
and then a bit more at finite chem. potential and temperature:
(leading term in θ4 =1 )
NJL
*
const while μ*<B
No surprises
*
Preliminary
Munczek/Nemirowsky -> NJL‘s complement
MN antithetic to NJL
NJL:contact interaction in x
MN:contact interaction in p
Munczek/Nemirowsky

P (    )  P0 
 d   n (  )
 P0  const  
0
2
2
2
p    2
Wigner Phase
~μ4
~μ 5
.2
.4
2 GeV
2

p
 2  2 2 to obtain
2
f1 ( p  0)  1
model is scale invariant regarding μ/η
P ( )  
well satisfied up to  /   1
5
(  1 . 09 GeV)
‚small‘ chem. Potential:
2
f 1 ( p  0,    )  
←
n (   ) 
T. Klahn, C.D. Roberts, L. Chang, H. Chen, Y.-X. Liu PRC 82, 035801 (2010)
2N cN
2
2
f
d
3


p f1 ( p )  
4
5
DSE – simple effective gluon coupling
Wigner Phase Less extreme, but again, 1particle number density distribution
different from free Fermi gas distribution
Chen et al. (TK) PRD 78 (2008)
Conclusions
NJL model is a powerful tool to explore possible features of dense QCD
It possibly might be a too powerful tool
NJL mf approximation equals gluon mf in momentum space in DSE
NB: Momentum independent gap solutions in their very nature
result in quasi particle picture → essentially an ideal gas, eff. m and μ
momentum dependent gap solutions enrich model space significantly
and provide ability to investigate confinement/deconfinement + DCSB
Conclusions
Thank you!
QCD in medium (near critical line):
-
Task is difficult
Not addressable by LQCD
Not addressable by pQCD
DSE are promising tool to tackle
non-perturbative in-medium QCD
- Qualitatively very different
results depending on
effective gluon coupling