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.

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?