Ferromagnetic properties of quark matter a microscopic origin of magnetic field in compact stars T. Tatsumi Department of Physics, Kyoto University I.Introduction and motivation II.Relativistic Fermi-liquid theory and magnetic susceptibility III.Screening effects for gluons IV.Magnetic properties at T=0 V.Finite temperature effects and Non-Fermi-liquid behavior VI.Summary and concluding remarks T.T., Proc. of EXOCT07 (arXiv: ) 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. A (2009) in press. （ arXiv: ） 30 th anniversary of March 5, 1979 Gamma-ray burst event

Origin: (i)Fossil field (ii)Dynamo scenario (crust) (iii)Microscopic origin (core) (II) Strong magnetic field in compact stars I. Introduction and motivation (I) Phase diagram of QCD In density-temperature plane Hadron Quark-Gluon Plasma Ferromagnetism ? Phase diagram of QCD Compact stars ~150MeV CSC Low temperature and moderate densities Magnetars

No possibility for nuclear matter For recent references, I.Bombaci et al, PLB 632(2006)638 G.H. Bordbar and M. Bigdeli, PRC76 (2007) Spontaneous magnetization of quark matter Is there ferromagnetic instability in QCD? It is a long-standing problem since the first discovery of pulsars. There have been done many calculations of nuclear matter. （ ungapped quark matter ） E.Nakano, T. Maruyama, T.T., PRD 68(2003)

Spontaneous spin polarization in quark matter 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) v v vv qq’ q OGE k k q q Ferromagnetism in gauge theories

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. Magnetars as quark stars (Pauli-Lubanski vector)

II. Relativistic Fermi liquid theory: Quarks as Quasi-particles: (G.Baym and S.A.Chin, NPA262(1976)527.) No direct int. Fock exchange int. Color symmetric int.: No flavor dep. In the following we are concerned with only one flavor.

0 spin susceptibility Magnetization Change of the distribution function Dirac magneton Magnetic susceptibility :response to the external magnetic field

０ ０ Free energy which also measures the curvature of the free energy at the origin spontaneous magnetization N(T): effective density of states at the Fermi surface Fermi velocity f: Landau parameters Magnetic (spin) susceptibility in the Fermi liquid theory Infrared (IR) singularities

k q k q  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 resummation III Screening effects

IV. Magnetic properties at T=0 Quasiparticle interaction: longitudinal transverse Susceptibility Screening effect log div Simple OGE cancellation k q k q  p 

s quark only u,d,s symmetric matter Ferromagnetic phase Paramagnetic phase without screening suppressionenhancement ● ● N F dependence

（ 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 Screening effect Some features: To summarize:

V. Finite temperature effects and Non-Fermi-liquid behavior ・ We consider the low T case, T/  but the usual low-T expansion cannot be applied. ○ Density of state: ・ Quasiparticle energy exhibits an anomalous behavior near the Fermi surface ○

Quark self-energy Schwinger-Dyson Anomalous term (C. Manuel, PRD 62(2000) ) 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 (D.T. Son, PRD 59(1999)094019) (A. Ipp et al., PRD 69(2004)011901)

T-indep. term T 2 -term Non Fermi-liquid effect Magnetic susceptibility at T>0

Paramag. FM Magnetic phase diagram of QCD Curie (critical) temperature should be order of several tens of MeV. Non-Fermi-liquid effect

VI. Summary and concluding remarks ・ We have considered magnetic susceptibility of QCD within Fermi-liquid theory Roles of static and dynamic screening are figured out: Static Dynamic Novel non-Fermi liquid effect! ・ Non-perturbative effects, Large Nc, Nf etc. other than OGE ・ How to distinguish various ideas about the origin Thermal evolution as well as magnetic evolution No decay of B, suggested by statistical analysis of X-ray pulsars Novel mechanism of emissivity or superconductivity Spin wave or magnons （ T.T., arxiv: ）