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Ferromagnetism in quark matter and origin of magnetic field in compact stars
Toshitaka Tatsumi (Kyoto U.)
(for a recent review, hep-ph/0506002)I. Introduction
Magnetic phase diagram of QCD in T-plane.
Pion condensationFerromagnetismSpin density wave
RHIC and Compact stars
Discovery of magnetars
CSC
curve
Cyclotron absorpsion lines
P P
CSC
QCD at finite density and/or temperature
Chiral symmetry restoration
B
New class of compact stars with high B
A naïve working hypothesis, conservation of magnetic flux during stellar evolution, looks to break down for magnetars.
2B R
curveP P
P
P
Three groups of compact stars
Microscopic origin of the magnetic field
Long standing problem for the origin of B
II. Relativistic ferromagnetism in quark matter
ref. T.T., PLB489(2000) 280.
T. Maruyama and T.T., NPA 693 (2001) 710.
Perturbative treatment in analogy with
Bloch’s mechanism (Pauli principle)
Fock exchange int. favors spin alignment
5/3 4/3
4/3ex
( ) or ( )K q q
q
n NR n
n
UR
for the Coulomb (gauge) interaction.
Notes (QCD):
Spin couples with motion
Spin configurati
in relativistic theories
in mon omentum space becomes nontrivial
Pauli-Lubanski vector with ( , ) : spin 4-vectora k ζ
[ (1 ) / 2]qn p n
Kinetic energy increases, while the
Weakly first order
meta-stable
OGE
“Spin” is not a good quantum number
5 3AV U
Then
int 3A AH U σ U
1det 0 givesAG
2 2 2 2 2 2( )
, 1
A A A
n
m m
p U U p U
ζ = -1
ζ = +1
AU
Deformation of the Fermi seas
2
exchange splitting ( ), as NR limit.2 A
pm U
m
Self-consistent (H-F) approach
(Axial-vector mean-field)
AG
D
1( ) ( ),AG p p m V p V
2/ 2
2 2
4, e
: Euler's cons
x ,
t
p3
EF FC C C
F
E
C
em N N G m
An effective model with zero-range approx. for OGE (contact interaction)
2 5 5int , (NJL type)H G
(i) Is the genuine Fermi sea effect(ii) Vacuum pol. works against alignment.(iii) Mass effect:
FC
: regularization
scale parameter
Critical density
0, or FC Cm
Careful treatment of the interaction range!
Second order
A
A z UU Self-consistent eq.
( / 1)m
FM
Qualitative change may happen.
(spin mean-value over both Fermi seas 、 and Dirac sea)
III. Color magnetic superconductivity ref. E. Nakano, T. Maruyama and T.T.,PRD68 (2003) 105001.
Coexistence of FM and CSC
OGE interaction
2 2(c.f. UGe , ZrZn in condensed matter physics)
(Nambu-Gorkov)
FM
CSC
One may also expect CSC in high-density quark matter.
Fock self-energy
FM:
CSC:
3 32c.f. phase of He or nucleon superfluidityA P
s = -1s = +1
0s
0s
Diquark pairing structure
IV. Chiral-symmetry restoration and Spin density wave
ref. T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006.
( )3p( )s
Chiral manifold 3S
Restoration paths
Density-wave instability before chiral-symmetry restorationor another restoration path due to pseudo-scalar density
q r
NJL model for simplicity.
2 20 5( ) ( ) ( )NJLL i m G i τ
MFA and the Weinberg transformation,
5 3exp[ / 2] ,W i q rgives
25 31/ 2MF W WqL i m G
with a2 (0, ).nd m G q qAxial-vector MF
Spin density wave
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
V. Summary and Concluding remarks
A magnetic phase diagramof QCD
Ferromagnetism (FM) and Spin density wave (DCDW=SDW)
Resemblance of DCDW to pion condensation (PIC) in the hadron phase
Hadron-quark continuity?
CSC
We have discussed the magnetic aspect of quark matter.
Magnetized strangelets (small “magnets”) Relativistic heavy-ion collisions
Primordial magnetic field as a relic of hadron-quark phase transition in the early universe.
CSC
Implications:
Hierarchy of magnetic field observed in compact stars
NG excitations (spin wave, phason) in the magnetic phase.
Magnetic domain formation
9 12 1510 10 10G G G
recycledpulsars
radiopulsars
magnetars
Magnetic field:
Period: 2 0 110 sec 10 sec 10 sec
9 6 310 yr 10 yr 10 yr Age:
15 17(10 )O G
As a direct implication
Magnetars=Quark stars
Field decay?