Introduction QCD under B field Conclusion QCD under a strong magnetic field Deog-Ki Hong Pusan National University SCGT14Mini, KMI, March. 7, 2014 (Based on DKH 98, 2011, 2014) 1/31
Introduction QCD under B field Motivations Conclusion Motivations ◮ Magnetic field is relevant in QCD if strong enough: QCD ≈ 10 19 Gauss · e . | eB | � Λ 2 ◮ Some neutron stars, called magnetars, have magnetic fields at the surface, B ∼ 10 12 − 15 G (Magnetar SGR 1900+14): 2/31
Introduction QCD under B field Motivations Conclusion Motivations ◮ Magnetic field is relevant in QCD if strong enough: QCD ≈ 10 19 Gauss · e . | eB | � Λ 2 ◮ Some neutron stars, called magnetars, have magnetic fields at the surface, B ∼ 10 12 − 15 G (Magnetar SGR 1900+14): 2/31
Introduction QCD under B field Motivations Conclusion Motivations ◮ In the peripheral collisions of relativistic heavy ions huge magnetic fields are produced at the center: 3/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector condensation ◮ The energy spectrum of (elementary) charged particle under the magnetic field ( � B = B ˆ z ): � z + m 2 + n | qB | , p 2 E ( � p ) = ± where n = 2 n r + | m L | + 1 − sign ( qB ) ( m L + 2 s z ). ◮ At the lowest Landau level the spin of the rho meson is along the B field direction and n = − 1. If elementary, m 2 ρ ( B ) = m 2 ρ − | eB | . 4/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector condensation ◮ The energy spectrum of (elementary) charged particle under the magnetic field ( � B = B ˆ z ): � z + m 2 + n | qB | , p 2 E ( � p ) = ± where n = 2 n r + | m L | + 1 − sign ( qB ) ( m L + 2 s z ). ◮ At the lowest Landau level the spin of the rho meson is along the B field direction and n = − 1. If elementary, m 2 ρ ( B ) = m 2 ρ − | eB | . 4/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector condensation ◮ Vector meson condensation: Vector order parameter develops under strong magnetic field (Chernodub 2011): � ¯ u γ 1 d � = − i � ¯ u γ 2 d � = ρ ( x ⊥ ) . 5/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector condensation ◮ Lattice calculation shows vector meson becomes lighter under the B field (Luschevskaya and Larina 2012): 2 a=0.0998 fm: 14 4 , m q a=0.01 16 4 , m q a=0.01 16 4 , m q a=0.02 18 4 , m q a=0.02 1.5 a=0.1383 fm: 14 4 , m q a=0.01 16 4 , m q a=0.02 m ρ (s=0), GeV 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 eB, GeV 2 6/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector condensation ◮ Lattice calculation shows vector meson condensation at B > B c = 0 . 93 GeV 2 / e (Barguta et al 1104.3767): 7/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Effective Lagrangian (DKH98 & 2014): ◮ Quarks under strong B field occupy Landau levels: � z + m 2 + 2 n | qB | , p 2 E = ± ( n = 0 , 1 , · · · ) E . . . . n=3 n=2 n=1 Λ L n=0 8/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Effective Lagrangian (DKH98 & 2014): ◮ Quark propagator under B field is given as ∞ � e − ik · x e − k 2 � ( − 1) n ⊥ / | qB | S n ( qB , k ) S F ( x ) = k n =0 D n ( qB , k ) S n ( qB , k ) = [(1 + i ǫ ) k 0 ] 2 − k 2 z − 2 | qB | n � 2 k 2 � 2 k 2 � 2 k 2 � � �� � D n = 2 � ˜ ⊥ ⊥ +4 � k ⊥ L 1 ⊥ − P + L n − 1 . k � P − L n n − 1 | qB | | qB | | qB | 9/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Matching with QCD at Λ L : ◮ At low energy E < Λ L we integrate out the modes in the higher Landau levels ( n � = 0). ◮ A new quark-gluon coupling: ig 2 s ¯ L 2 = c 2 Q 0 � A ˜ γ µ · ∂ � � A ˜ γ µ Q 0 . | qB | 10/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Matching with QCD at Λ L : ◮ At low energy E < Λ L we integrate out the modes in the higher Landau levels ( n � = 0). ◮ A new quark-gluon coupling: ig 2 s ¯ L 2 = c 2 Q 0 � A ˜ γ µ · ∂ � � A ˜ γ µ Q 0 . | qB | 10/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Effective Lagrangian (DKH98): ◮ Four-Fermi couplings for LLL quarks: g s �� ¯ � ¯ � 2 + � 2 � L 1 1 eff ∋ Q 0 Q 0 Q 0 i γ 5 Q 0 . 4 | qB | ◮ Below Λ L the quark-loop does not contribute to the beta-function of α s : At one-loop � µ 1 α s (Λ L ) + 11 1 � α s ( µ ) = 2 π ln . Λ L 11/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Effective Lagrangian (DKH98): ◮ Four-Fermi couplings for LLL quarks: g s �� ¯ � ¯ � 2 + � 2 � L 1 1 eff ∋ Q 0 Q 0 Q 0 i γ 5 Q 0 . 4 | qB | ◮ Below Λ L the quark-loop does not contribute to the beta-function of α s : At one-loop � µ 1 α s (Λ L ) + 11 1 � α s ( µ ) = 2 π ln . Λ L 11/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Effective Lagrangian (DKH98): ◮ One-loop RGE for the four-quark interaction: µ d 1 = − 40 s (ln 2) 2 d µ g s 9 α 2 ◮ Solving RGE to get g s 1 ( µ ) = 1 . 1424 ( α s ( µ ) − α s (Λ L )) + g s 1 (Λ L ) . ◮ If B ≥ 10 20 G , the four-quark interaction is stronger than gluon interaction. Therefore the chiral symmetry should break at a scale higher than the confinement scale for B ≥ 10 20 G . 12/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Effective Lagrangian (DKH98): ◮ One-loop RGE for the four-quark interaction: µ d 1 = − 40 s (ln 2) 2 d µ g s 9 α 2 ◮ Solving RGE to get g s 1 ( µ ) = 1 . 1424 ( α s ( µ ) − α s (Λ L )) + g s 1 (Λ L ) . ◮ If B ≥ 10 20 G , the four-quark interaction is stronger than gluon interaction. Therefore the chiral symmetry should break at a scale higher than the confinement scale for B ≥ 10 20 G . 12/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Effective Lagrangian (DKH98): ◮ One-loop RGE for the four-quark interaction: µ d 1 = − 40 s (ln 2) 2 d µ g s 9 α 2 ◮ Solving RGE to get g s 1 ( µ ) = 1 . 1424 ( α s ( µ ) − α s (Λ L )) + g s 1 (Λ L ) . ◮ If B ≥ 10 20 G , the four-quark interaction is stronger than gluon interaction. Therefore the chiral symmetry should break at a scale higher than the confinement scale for B ≥ 10 20 G . 12/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector mesons in the Effective Lagrangian: ◮ Running coupling under strong B field: α s ( µ ) 1 µ Λ QCD Λ QCD ( B ) Λ L 13/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector mesons in the Effective Lagrangian: ◮ We need a stronger B field ( B > m 2 ρ / e ) to condense vector mesons: � 2 � Λ QCD ( B ) 2 ( B ) = m 2 m eff ρ · − | eB | . ρ Λ QCD ◮ The critical B field occurs at (DKH 2014) � m ρ � 4 9 ≈ 0 . 90 GeV 2 . eB c = m 2 ρ · Λ QCD ◮ It agrees well with the lattice result by Barguta et al, B c = 0 . 93 GeV 2 / e . 14/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector mesons in the Effective Lagrangian: ◮ We need a stronger B field ( B > m 2 ρ / e ) to condense vector mesons: � 2 � Λ QCD ( B ) 2 ( B ) = m 2 m eff ρ · − | eB | . ρ Λ QCD ◮ The critical B field occurs at (DKH 2014) � m ρ � 4 9 ≈ 0 . 90 GeV 2 . eB c = m 2 ρ · Λ QCD ◮ It agrees well with the lattice result by Barguta et al, B c = 0 . 93 GeV 2 / e . 14/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect Vector mesons in the Effective Lagrangian: ◮ We need a stronger B field ( B > m 2 ρ / e ) to condense vector mesons: � 2 � Λ QCD ( B ) 2 ( B ) = m 2 m eff ρ · − | eB | . ρ Λ QCD ◮ The critical B field occurs at (DKH 2014) � m ρ � 4 9 ≈ 0 . 90 GeV 2 . eB c = m 2 ρ · Λ QCD ◮ It agrees well with the lattice result by Barguta et al, B c = 0 . 93 GeV 2 / e . 14/31
Introduction Vector condensation QCD under B field Neutron star Conclusion Chiral Magnetic effect QCD Vacuum Energy: ◮ The additional vacuum energy at one-loop is given by Schwinger as � ∞ 1 � � ds eBs s 3 e − M 2 π s ∆ E vac = − sinh( eBs ) − 1 . 16 π 2 0 ◮ The chiral condensate becomes by the Gell-Mann-Oakes-Renner relation (Shushpanov+Smilga ’97) � 1 + | eB | ln 2 � qq � B = � ¯ qq � B =0 � ¯ + · · · . 16 π 2 F 2 π 15/31
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