1. Spin Polarization due to Tensor Fields in NJL Model
Tomoyuki Maruyama, BRS, Nihon Univ.
Collaborators
T. Tatsumi, Dep. of Phys., Kyoto Univ.
T. Tatsumi, Phys. Lett. B489 (2000) 280.

2. Consider Spin-Polarization (SP) for Quark Matter
§1 Introductio
Neutron Stars (NSs) Magnetic Field
Magnetor ~ 1015 Gauss, Normal NS ~ 1012 Gauss
Non-Central Heavy-Ion Collisions ~ 1015 Gauss
D. E. Kharzeev, et al., NPA 803, 227 (08)
How to keep the Strong Magnetic Field
One Possible Candidate Spin-Polarization Matter
T. Tatsumi, Phys. Lett. B489 (2000) 280.
Spin-Spin Interaction for Quark in NJL-Type
Vector-Vector (One-Gluon Exchange)
Fock Term ⇒ － σ·σ ⇒ spin-alignment
Consider Spin-Polarization (SP) for Quark Matter
NM in RMFT, T.M & T.Tatsumi, Nucl. Phys. A693, 710

3. Two Type Spin-Forces NJL Type (Zero-Range)
Energy Density
In Quark Matter
Axial Vector Density
Tensor Density
Dirac Eq.
ℎ 𝑠𝑝 𝑢 𝐩,𝑠 = 𝜶∙𝒑+𝑚+ 𝛴 𝑧 𝐴+𝛽 𝛴 𝑧 𝑈 𝑢 𝒑,𝑠 =e 𝒑,𝑠 𝑢 𝐩,𝑠
Mean-Fields
𝐴= 𝐺 𝐴 𝜌 𝐴 , 𝑈= 𝐺 𝑇 𝜌 𝑇
When GT (GA) < 0, Non-Zero U (A) Spin-Polarization
It is Very Difficult to calculate SP- Phase with Both AV and T
Only A (AV-Type) or Only U (T-Type) SP

4. Two Type Spin Polarization, AV-type & T-type
Single Particle Energy
When U = 0,
When A = 0,
Momentum Distribution at e = 3m
𝐴=0, 𝑈=3𝑚
𝐴=3𝑚, 𝑈=0
T-Type
AV-Type
e = 3m

5. Phase-Diagram between Paramag. and Ferromag.
: a condition of Spontaneous Spin-Polarization
Large Mass
AV > T
Small Mass
T > AV
Mass Zero Limit
No AV-Type SP
T-Type survives

6. One-Gluon Exchange Force ⇒ AV-Type Spin-Spin interaction
T. Tastumi, PL B489, 280 (2000)
Spin-Polarization Phase below Chiral Restoration Density
NJL with AV-Int E. Nakano D-Thesis, S.Maedan PTP118, 729 (2008)
Spin-P phase appears in small density region
below the chiral retsroring
AV –Int does not work in m = 0 limit
T-int works even if m = 0
Fixed Mass : T. Tatsumi, E. Nakano and K. Nawa,
Dark Matter, p.39 (Nova Science Pub., NY, (06).
Zero Mass Y. Tsue, J. de. Providencia,
C. Providencia and M. Yamamura
Prog. Theor. Phys. (2012).

7. §2 Formalizm Lagrangian Dirac Eq. Mean Field Approximation NJL + T-int
SU(2)　Chiral Symmetry
Dirac Eq.
Mean Field Approximation

8. Scalar Density
Medium Part
Vacuum Part
Proper Time Appr.
Tensor Density
Strong Cut-Off and Regularization Dependence
Proper-Time & Energy Cut-Off unfavor to Spin-Polarization (SP)
Momentum Cut-Off & Effective Potential Method favor to SP
Ignoring Vacuum Parts of T-Density in this work

9. §3 Results Parameters Zero Current Mass m = 0 Hartree Equation
E. Nakano and T. Tatsumi, PR D71, (05).
Hartree
Equation
Spontaneous SP
Condition

10. Results of PM1

11. Results of PM2 Two Kinds of SP phase
Chiral Breaking SP Phase ( M ≠ 0 )
Chiral Restoring SP Phase ( M = 0 )
1st Order Transition

12. Critical Density of SP phase
Chiral Breaking SP
PM2
PM1
Chiral Restoring SP
PM1
PM2
Critical Density of
Chiral Phase Transition
in Spin-Saturated Phase

13. §5 Summary Hybrid SP phase with AV and T interaction
NJL model with Scalar and Tensor　Interaction
　　　　Spontaneous Spin-Polarization and Chiral Restoration
T-Type Spin Polarization Phases
Chiral Breaking SP Phase ( M ≠ 0 ) when the coupling －GT is large
Chiral Restoring SP Phase ( M ＝ 0 )　with any GT
GT < 0 : Iso-Scalar SP Same Spin-Directionss for u an d,
GT > 0 : Iso-Vector SP Opposite Spin-Directions (Stronger Mag Fld)
Interaction Breaking of U(1)A ChS makes Spin-Polarization
Mass, Magnetic Field, Tensor Interaction
Hybrid SP phase with AV and T interaction
m = 0 case T.M., E.Nakano, K.Yanase, N.Yoshinaga, PRD 97, (18)
T-Type SP → U(1)A ChS Breaking → AV-T Mixed Type
(Larger SP)