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Magnetic aspects of QCD and compact stars
T. Tatsumi Department of Physics, Kyoto University
I. Introduction and motivationII. Chiral symmetry and spin density wave (SDW)III. Ferromagnetism (FM) and magnetic susceptibilityIV. Screening effects for gluonsV. Magnetic properties at T=0VI. Finite temperature effects and Non-Fermi-liquid behaviorVII. Summary and concluding remarks
T.T., Proc. of EXOCT07 (arXiv:0711.3348)T.T. and K. Sato., Phys. Lett., B663 (2008) 322. K. Sato and T.T. , Prog. Theor. Phys. Suppl. 174 (2008) 177.
T.T. and K. Sato, Phys. Lett. B672(2009) 132.K. Sato and T.T., Nucl. Phys. A826 (2009) 74.T.T., Proc. of CSQCDII (2010) in press.
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Meson (Pion) condensation (PIC) Antiferromagnetism (AFM)Deconfinement Ferromagnetism (FM) Chiral restoration Spin density wave (SDW)Color superconductivity . .
CSC
Chiral restoration
Magnetic phase diagram of QCD
Critical end-point
I. Introduction and motivation
Deconfinement
(I) Phase diagram of QCD
?
Spin degrees of freedom
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Origin:(i) Fossil field(ii) Dynamo scenario (crust)(iii)Microscopic origin (core)
(II) Strong magnetic field in compact stars
Magnetars
Its origin is a long-standing problem since the first discovery of pulsars.
Recent discovery of magnetars seems to revive this issue
curveP P- &
P
P
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Ferromagnetism or spin polarization
For recent references,I.Bombaci et al, PLB 632(2006)638G.H. Bordbar and M. Bigdeli, PRC76 (2007)035803
Spontaneous magnetization of quark matter or ferromagnetism in quark matter
Nuclear matter calculations have shown negative results
15For 10 , it corresponds to
The energy scale of the strong interaction is also
(10MeV).
(MeV,GeV).
O
O
B G
It would be rather natural to attribute its origin to strong interaction
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Some ideas in QCD
E.J. Ferrer, de la Incera, PRL 97(2006) 122301; PRD 76(2007) 045011; PRD 76(2007)114012. arXiv:0810.3886
Bloch mechanism
Cf. Other mechanism in CSC:
・ Gluon condensate
D.T. Son, M.A. Stephanov, PRD 77(2008)014021.
・ Axial anomaly in CFL
X
B
8cos sin A A G
8 8sin cos G A G
B
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A typical example of the condensed phase:0
† 03 3 sin( )ck z
Liquid crystal with antiferromagnetic order
II. Chiral symmetry and Spin density wave (SDW)
T.Takatsuka et al., Prog.Theor.Phys.59(1978) 1933.T.Suzuki et al.,arXiv:nucl-th/9901097
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ref. T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006.
A density-wave instability before/around chiral-symmetry restorationor another restoration path due to pseudo-scalar density
A magnetic phase in the intermediate densities
Chiral symmetry restoration and SDW
2 ,M G
qq
= D
D =
vacD
q= ×q r
Chiral-restoration path
cBm
cBm
SDW
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Remarks:(i) Nesting (Overhauser, Peiels) is the key mechanism for generating SDW
Level crossing of the energy spectrum near the Fermi surface
( )
cos
2
i i
F
ie e
k
e
V
® +
=
kx kx k ±q x
qx
q
Model indep.
SDW,CDW
kFq
A.W. Overhauser, PRL 4(1960) 462.R.E. Peierls, Quntum Theory of Solids (1955)
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(ii) Similar idea
2 (Ising) vs (4)Z O
(iii) Similarity to LOFF in superconductor
(uniform) 0
(r) (non-uniform)
D «
D
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Spin density wave ( SDW )
3
1 3 2 21
2 cos(2
,
Average spin:
Magnetic moment
( )(2 )
:
)z
d p mM n
m pM
zΣ = 0
q r p
(c.f. Overhauser)
DCDW
DCDW
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(iii) Phason and spin-wave as NG modes in SDW
12 co ( ,s( ))M tq rr
G.Gruener, Rev.Mod.Phys. 66(1994) 1.
k ck
It would be interesting to see that both modes have the linear dispersion relation:
Cf. Spin waves in FM and AFM Counting rule of NG bosons
2
# of NG bosons 1 2 2
Dispersion
FM AFM
(3) (2
S
)
:
DW
ck ck k
SO O
c
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T.T. PLB489(2000)280.T.T.,E. Nakano and K. Nawa, Dark matter,p.39 (Nova Sci. Pub., 2005)
Fock exchange interaction is responsible to ferromagnetism in quark matter (Bloch mechanism)c.f. Ferromagnetism of itinerant electrons (Bloch,1929)
vv
v v
q q’
q’ q
OGE
k
k
q
q
1/ 2 1/ 2 0c abab baN
Is there ferromagnetic instability in QCD?
III. Ferromagnetism (FM) and magnetic susceptibility
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(1 ) / 2n p n
Weakly first order
c.f. A.Niegawa, Prog. Theor. Phys. 113(2005)581, which also concludes ferromagnetism at low density,by the use of the resummation technique.
ferro para
Magnetars as quark stars
15 17(10 )O G
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Relativistic Fermi liquid theory (G.Baym and S.A.Chin, NPA262(1976)527.)
No direct int. Fock exchange int. (tr 0)
: , ':2 '' ,,
,
1= q q
a ba b
s
c k q
m mf
N E EMf kζ q kζk ζζ q qζζ
Color symmetric int.:
; , ' ; , 'ai bj ij a bf fkζ qζ kζ qζNo flavor dep.
In the following we are concerned with only one flavor.
; , '
( ; ) / ( ; )
( ; ) / ( '; )ai bj
ai E n ai
f ai n bj
kζ qζ
kζ kζ
kζ qζ
Ferromagnetism in gauge theories
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4 QED 4i
4
nt
:= / 2Gordon identit
d
d .
y
q
qxL e d
x i
x
A B r
γ
Σ Br
A
:q
2 2/ 2C F FN N Vk k
0L
Fk
0q p - k
Fk
Fk
BFk
Fk
0
spin susceptibilityMagnetization
'','
3
3
)2(
qkk
kkk
kkk
nfqd
N
Bgn
nn
qC
qD
1
exp 1/ 2 1
with the gyromagnetic ratio .
k D q
D
n g B
g
k
2D q
C
gM q q N n n
k k
k
Change of the distribution function
Dirac magneton
Magnetic susceptibility :response to the external magnetic field
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, , 0
M
N T B
M
B
M
MF01
0 M
MF01
0
Free energy
which also measures the curvature of the free energy at the origin
-1 or 0
spontaneous magnetization
2
2
1
2
( ) / 1 ( )2
1 = /
2 3s
D q oM
D q
F F
a
C
gN T N T f
g
N kf f
E
N(T): effective density of states at the Fermi surface
sFCFF
k
FC
fkNk
v
kkNTN
F
12
2
2
2
3
,)(
k
, ' 's af f f k q kq kq
Fermi velocity
f: Landau parameters
Magnetic (spin) susceptibility in the Fermi liquid theory
Infrared (IR) singularities
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2
screen
4
12( ) (
2 2
,
2
)
,
0
( ) ( ) (
( ) / 2
(
) ,
( )
,
)4
, | |
/
T LT L
T L T
OGE
L
fL D f F
f u d s
FF
DF FT
p pD p P D p P D p
p
D p p
u k
p m g k
pu mEp i
p
k q
kq
p
p=k-q
(Debye mass)
(Landau damping)
Gauge choice
( i) Debye screening in the longitudinal (electric) gluons
improve IR behavior(ii) Transverse (magnetic) gluons only gives the dynamical screening, which leads to IR (Log) divergence
Non-Fermi-liquid behavior
HDL resummationIV Screening effects
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V. Magnetic properties at T=0
)()(2
1 002
22
2
2
||||qkqk
Tii
LFC
Ckqk
DMDME
mg
N
Nf
F
Quasiparticle interaction:
1
1 cos kq kqcos1
1
longitudinal transverse
1
Pauli
2
2
2 2
1 (2 )12
1
2l ( 4 2 )
( 0) /
n2
fF
F F
F F
M
C gm E m
E k
E
T
E m m
Susceptibility
Screening effect 4 2( ln )O g g
Pauli : Pauli paramagnetism
2
2
2 F
D
km
log div
Simple OGE
cancellation
k q
kq
p
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s quark only
u,d,s symmetric matter
Ferromagneticphase
Paramagneticphasewithout
screening
cFk
suppression enhancement
●
●
2ln1
2
2
2 F
D
km
iFiFD Ekg
m ,,flavors
2
22
2
NF dependence
2cFN
0( )O r:
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spin free/
free 2Fk
1(fm )Fk
cFk
Ferromagnetism
Paramagnetism
Large fluctuations (or paramagnon)cf. emissivity (talk by S. Reddy)
SDW?
cf in electron gas FM(Bloch) SDW(Overhauser) low density high-density
3 ln (Pethick and Carneiro)VC T TD :
Note that this SDW has nothing to do with chiral symmetry,but nesting is also important
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( i) Debye screening in the longitudinal (electric) gluons improve IR behavior
(ii) Transverse (magnetic) gluons only gives the dynamical screening, which leads to IR (Log) divergence
(iv) Results are independent of the gauge choice
(iii) Divergences cancel each other to give a finite
Non-Fermi-liquid behavior
...)ln()()(1 242
spin
Pauli ggOgO
Screening effect
Some features:
To summarize:
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VI. Finite temperature effects and Non-Fermi-liquid behavior
2 2
1
( )
2 1 ( )
12 ( )
FD q
a al t
Fq
MD
a atl
g N T
N T f f f f
g
N T
3
; ,3
spin dependent FL interaction s.t, :
( )2 / ( )
(2 )
.
s
a al t
a aki c i k k
k
f f
nd kf N f N T
・ We consider the low T case, T/but the usual low-T expansion cannot be applied.
( )sk
3
3
1( ) 2 ,
(2 ) 1 exp( ( ))k
C kk k
nd kN T N n
○ Density of state:
・ Quasiparticle energy exhibits an anomalous behavior near the Fermi surface
○
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Quark self-energy
D
SSchwinger-Dyson
2
2reg
2
F
( ) ln1
or
Re ( ) Re (2 | |
) ( ),
1
2c
fc
f FC g u
NC
N
32
3( ) ( ) ( )
(2 )n
d qp e T S p q D q
Anomalous term
(C. Manuel, PRD 62(2000) 076009)
One loop result:
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Role of transverse gluons=relevant interactions in RG
Non-Fermi liquid behavior
・ Specific heat
・ Gap equation
How about susceptibility?
Peculiar temperature dep. of susceptibility
Curie temperature
lnVC T T
2 2exp ( 4)( 1) /16 exp( 3 / 2 )C gN (D.T. Son, PRD 59(1999)094019)
(A. Ipp et al., PRD 69(2004)011901)
Effective coupling is 2eff FC g v= Infrared free
(Schaefer, K. Schwenzer, PRD 70(2004) 054007)
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T-indep. term
T2-term Non Fermi-liquid effect
Magnetic susceptibility at T>0
Cf. paramagnon effect
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Paramag.
FM
Magnetic phase diagram of QCD
Curie (critical) temperature should be order of several tens of MeV.
Non-Fermi-liquid effect
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VII. Summary and concluding remarks
・ We have considered magnetic susceptibility q=0) of QCD within Fermi-liquid theory Roles of static and dynamic screening are figured out:
StaticDynamic
24 ln ggTT ln2 Novel non-Fermi liquid effect!
・ Since the order parameter is color singlet, FM and SDW survive even in the large Nc limit.
・ Observational signatures of magnetic phasesThermal evolution as well as magnetic evolution
Novel mechanism of neutrino emissivity!?Specific heat and thermal conductivity? ・ Spin wave or magnons in FM・ SDW and phasons
( T.T., arxiv:07113349 )
・ Possibilities and properties of SDW and FM have been discussed