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.

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

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

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

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

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

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

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)

(ii) Similar idea

2 (Ising) vs (4)Z O

(iii) Similarity to LOFF in superconductor

(uniform) 0

(r) (non-uniform)

D «

D

0 p

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

(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

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

(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

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

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

, , 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

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

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

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:

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

（ 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:

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

○

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:

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)

T-indep. term

T2-term Non Fermi-liquid effect

Magnetic susceptibility at T>0

Cf. paramagnon effect

Paramag.

FM

Magnetic phase diagram of QCD

Curie (critical) temperature should be order of several tens of MeV.

Non-Fermi-liquid effect

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