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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 Conclusion Motivations Motivations I Magnetic field is relevant in QCD if strong enough: |eB| & Λ2QCD ≈ 1019 Gauss · e. I Some neutron stars, called magnetars, have magnetic fields at the surface, B ∼ 1012−15 G (Magnetar SGR 1900+14): 2/31 Introduction QCD under B field Conclusion Motivations Motivations I Magnetic field is relevant in QCD if strong enough: |eB| & Λ2QCD ≈ 1019 Gauss · e. I Some neutron stars, called magnetars, have magnetic fields at the surface, B ∼ 1012−15 G (Magnetar SGR 1900+14): 2/31 Introduction QCD under B field Conclusion Motivations Motivations I In the peripheral collisions of relativistic heavy ions huge magnetic fields are produced at the center: 3/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector condensation I The energy spectrum of (elementary) charged particle under ~ = B ẑ): the magnetic field (B q E (~ p ) = ± pz2 + m2 + n|qB|, where n = 2nr + |mL | + 1 − sign(qB) (mL + 2sz ). I At the lowest Landau level the spin of the rho meson is along the B field direction and n = −1. If elementary, mρ2 (B) = mρ2 − |eB| . 4/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector condensation I The energy spectrum of (elementary) charged particle under ~ = B ẑ): the magnetic field (B q E (~ p ) = ± pz2 + m2 + n|qB|, where n = 2nr + |mL | + 1 − sign(qB) (mL + 2sz ). I At the lowest Landau level the spin of the rho meson is along the B field direction and n = −1. If elementary, mρ2 (B) = mρ2 − |eB| . 4/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector condensation I Vector meson condensation: Vector order parameter develops under strong magnetic field (Chernodub 2011): hūγ1 di = −i hūγ2 di = ρ(x⊥ ) . 5/31 Introduction QCD under B field Conclusion Vector condensation Neutron star 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: 144, mqa=0.01 1644, mqa=0.01 164, mqa=0.02 18 , mqa=0.02 a=0.1383 fm: 1444, mqa=0.01 16 , mqa=0.02 1.5 mρ(s=0), GeV I 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 eB, GeV2 6/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector condensation I Lattice calculation shows vector meson condensation at B > Bc = 0.93GeV2 /e (Barguta et al 1104.3767): 7/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Effective Lagrangian (DKH98 & 2014): I Quarks under strong B field occupy Landau levels: q E = ± pz2 + m2 + 2n|qB|, (n = 0, 1, · · · ) E . . . . n=3 n=2 n=1 ⇤L n=0 8/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Effective Lagrangian (DKH98 & 2014): I Quark propagator under B field is given as Z ∞ X 2 n SF (x) = (−1) e −ik·x e −k⊥ /|qB| Sn (qB, k) n=0 Sn (qB, k) = k Dn (qB, k) [(1 + i)k0 ]2 − kz2 − 2|qB|n 2 2 2 2k⊥ 2k⊥ 2k⊥ Dn = 26k̃q P− Ln − P+ Ln−1 +46k⊥ L1n−1 . |qB| |qB| |qB| 9/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Matching with QCD at ΛL : I At low energy E < ΛL we integrate out the modes in the higher Landau levels (n 6= 0). I A new quark-gluon coupling: L2 = c2 igs2 Q̄0 A 6 γ̃µ · ∂q A 6 γ̃µ Q0 . |qB| 10/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Matching with QCD at ΛL : I At low energy E < ΛL we integrate out the modes in the higher Landau levels (n 6= 0). I A new quark-gluon coupling: L2 = c2 igs2 Q̄0 A 6 γ̃µ · ∂q A 6 γ̃µ Q0 . |qB| 10/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Effective Lagrangian (DKH98): I Four-Fermi couplings for LLL quarks: L1eff 3 I 2 2 i g1s h Q̄0 Q0 + Q̄0 iγ5 Q0 . 4 |qB| Below ΛL the quark-loop does not contribute to the beta-function of αs : At one-loop 1 1 11 µ = + ln . αs (µ) αs (ΛL ) 2π ΛL 11/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Effective Lagrangian (DKH98): I Four-Fermi couplings for LLL quarks: L1eff 3 I 2 2 i g1s h Q̄0 Q0 + Q̄0 iγ5 Q0 . 4 |qB| Below ΛL the quark-loop does not contribute to the beta-function of αs : At one-loop 1 1 11 µ = + ln . αs (µ) αs (ΛL ) 2π ΛL 11/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Effective Lagrangian (DKH98): I One-loop RGE for the four-quark interaction: µ I d s 40 g = − αs2 (ln 2)2 dµ 1 9 Solving RGE to get g1s (µ) = 1.1424 (αs (µ) − αs (ΛL )) + g1s (ΛL ) . I If B ≥ 1020 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 ≥ 1020 G. 12/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Effective Lagrangian (DKH98): I One-loop RGE for the four-quark interaction: µ I d s 40 g = − αs2 (ln 2)2 dµ 1 9 Solving RGE to get g1s (µ) = 1.1424 (αs (µ) − αs (ΛL )) + g1s (ΛL ) . I If B ≥ 1020 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 ≥ 1020 G. 12/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Effective Lagrangian (DKH98): I One-loop RGE for the four-quark interaction: µ I d s 40 g = − αs2 (ln 2)2 dµ 1 9 Solving RGE to get g1s (µ) = 1.1424 (αs (µ) − αs (ΛL )) + g1s (ΛL ) . I If B ≥ 1020 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 ≥ 1020 G. 12/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector mesons in the Effective Lagrangian: I Running coupling under strong B field: ↵s (µ) 1 ⇤QCD ⇤QCD (B) ⇤L µ 13/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector mesons in the Effective Lagrangian: I We need a stronger B field (B > mρ2 /e) to condense vector mesons: ΛQCD (B) 2 eff 2 2 − |eB| . mρ (B) = mρ · ΛQCD I The critical B field occurs at (DKH 2014) eBc = I mρ2 · mρ ΛQCD 4 9 ≈ 0.90 GeV2 . It agrees well with the lattice result by Barguta et al, Bc = 0.93 GeV2 /e. 14/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector mesons in the Effective Lagrangian: I We need a stronger B field (B > mρ2 /e) to condense vector mesons: ΛQCD (B) 2 eff 2 2 − |eB| . mρ (B) = mρ · ΛQCD I The critical B field occurs at (DKH 2014) eBc = I mρ2 · mρ ΛQCD 4 9 ≈ 0.90 GeV2 . It agrees well with the lattice result by Barguta et al, Bc = 0.93 GeV2 /e. 14/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Vector mesons in the Effective Lagrangian: I We need a stronger B field (B > mρ2 /e) to condense vector mesons: ΛQCD (B) 2 eff 2 2 − |eB| . mρ (B) = mρ · ΛQCD I The critical B field occurs at (DKH 2014) eBc = I mρ2 · mρ ΛQCD 4 9 ≈ 0.90 GeV2 . It agrees well with the lattice result by Barguta et al, Bc = 0.93 GeV2 /e. 14/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect QCD Vacuum Energy: I The additional vacuum energy at one-loop is given by Schwinger as Z ∞ ds −Mπ2 s 1 eBs ∆Evac = − e −1 . 16π 2 0 s 3 sinh(eBs) I The chiral condensate becomes by the Gell-Mann-Oakes-Renner relation (Shushpanov+Smilga ’97) |eB| ln 2 B B=0 hq̄qi = hq̄qi 1+ + ··· . 16π 2 Fπ2 15/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect QCD Vacuum Energy: I The additional vacuum energy at one-loop is given by Schwinger as Z ∞ ds −Mπ2 s 1 eBs ∆Evac = − e −1 . 16π 2 0 s 3 sinh(eBs) I The chiral condensate becomes by the Gell-Mann-Oakes-Renner relation (Shushpanov+Smilga ’97) |eB| ln 2 B B=0 hq̄qi = hq̄qi 1+ + ··· . 16π 2 Fπ2 15/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Neutron star cooling: I Cooling through axion bremsstrahlung (DKH98) (a) I (b) Energy loss per unit volume per unit time at low temp. (Iwamoto, Ellis, Brinkman+Turner) Q1a ∝ f Mπ 4 2 mn2.5 gan T 6.5 . 16/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Neutron star cooling: I Cooling through axion bremsstrahlung (DKH98) (a) I (b) Energy loss per unit volume per unit time at low temp. (Iwamoto, Ellis, Brinkman+Turner) Q1a ∝ f Mπ 4 2 mn2.5 gan T 6.5 . 16/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Neutron star cooling: I The axion coupling gan ∝ mn /fPQ , mn ∝ hq̄qi, and since fPQ does not get any corrections from the magnetic field, we have |eB| ln 2 gan (B) = gan (0) 1 + + ··· . 16π 2 Fπ2 I The correction then becomes (DKH ’98), using the Goldberg-Treiman relation and GOR relation, Q1a (B) ' Q1a (0) g1s 2 |eB| ln 2 6.5 1+ 2 1+ . 2π 16π 2 Fπ2 17/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Neutron star cooling: I The axion coupling gan ∝ mn /fPQ , mn ∝ hq̄qi, and since fPQ does not get any corrections from the magnetic field, we have |eB| ln 2 gan (B) = gan (0) 1 + + ··· . 16π 2 Fπ2 I The correction then becomes (DKH ’98), using the Goldberg-Treiman relation and GOR relation, Q1a (B) ' Q1a (0) g1s 2 |eB| ln 2 6.5 1+ 2 1+ . 2π 16π 2 Fπ2 17/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Neutron star cooling: I Similarly, the lowest-order energy emission rate per unit volume by the pion-axion conversion π − + p → n + a (Schramm) has corrections (DKH98): − Qπa (B) ' Qπa − (0) I g1s 2 |eB| ln 2 −1 1+ 2 1+ 2π 16π 2 Fπ2 Order of magnitude enhancement for B = 1020 G. Magnetars cool quickly! 18/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Neutron star cooling: I Similarly, the lowest-order energy emission rate per unit volume by the pion-axion conversion π − + p → n + a (Schramm) has corrections (DKH98): − Qπa (B) ' Qπa − (0) I g1s 2 |eB| ln 2 −1 1+ 2 1+ 2π 16π 2 Fπ2 Order of magnitude enhancement for B = 1020 G. Magnetars cool quickly! 18/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Chiral Magnetic Effects (DKH2011) I An instanton number may be created in RHIC: Z gs2 d4 x G a G̃ a = nL − nR . nw = 32π 2 whose conjugate variable is µA . I Chiral magnetic effect (Fukushima+Kharzeev+Warringa): Under a strong magnetic field q2 ~. J~ = 2 µA B 2π 19/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Chiral Magnetic Effects (DKH2011) I An instanton number may be created in RHIC: Z gs2 d4 x G a G̃ a = nL − nR . nw = 32π 2 whose conjugate variable is µA . I Chiral magnetic effect (Fukushima+Kharzeev+Warringa): Under a strong magnetic field q2 ~. J~ = 2 µA B 2π 19/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I Off-center collision produces strong B field: I Such CP-odd effect can be then observable (Fukushima+ Kharzeev+Warringa 2008): 20/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I Off-center collision produces strong B field: I Such CP-odd effect can be then observable (Fukushima+ Kharzeev+Warringa 2008): 20/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I CME seen at RHIC? (STAR prl ’09) I May be (see talk by Bzdak at HIC10) I May be not. (Muller+Schafer arXiv:1009.1053) 21/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I CME seen at RHIC? (STAR prl ’09) I May be (see talk by Bzdak at HIC10) I May be not. (Muller+Schafer arXiv:1009.1053) 21/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I CME seen at RHIC? (STAR prl ’09) I May be (see talk by Bzdak at HIC10) I May be not. (Muller+Schafer arXiv:1009.1053) 21/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Quark matter under strong B field: I Under strong B field quark spectrum takes: q EA = −µ ± kz2 + 2 |qB| n . I Quark propagator under B field is given as Z ∞ X 2 n e −ik·x e −k⊥ /|qB| Sn (qB, k) SF (x) = (−1) n=0 k Dn (qB, k) . [(1 + i)k0 + µ]2 − kz2 − 2|qB|n 2 2 2 2k⊥ 2k⊥ 2k⊥ 1 Dn = 26k̃q P− Ln − P+ Ln−1 +46k⊥ Ln−1 , |qB| |qB| |qB| Sn (qB, k) = 22/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Quark matter under strong B field: I Under strong B field quark spectrum takes: q EA = −µ ± kz2 + 2 |qB| n . I Quark propagator under B field is given as Z ∞ X 2 n e −ik·x e −k⊥ /|qB| Sn (qB, k) SF (x) = (−1) n=0 k Dn (qB, k) . [(1 + i)k0 + µ]2 − kz2 − 2|qB|n 2 2 2 2k⊥ 2k⊥ 2k⊥ 1 Dn = 26k̃q P− Ln − P+ Ln−1 +46k⊥ Ln−1 , |qB| |qB| |qB| Sn (qB, k) = 22/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I Anomalous current for LH fermions at one-loop Z d4 k α α α ∆L (µL ) ≡ ψ̄L γ ψL = − Tr γ S̃(k) L . (2π)4 I Matter-dependent part is finite and explicitly calculable: Z µL ∂ α α α ∆mat ≡ ∆ (µL , B) − ∆ (0, B) = dµ0 0 ∆α (µ0 , B) ∂µ 0 I We change variables: α k α −→ k 0 = k α + u α µ , u α = (1, ~0 ) 23/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I Anomalous current for LH fermions at one-loop Z d4 k α α α ∆L (µL ) ≡ ψ̄L γ ψL = − Tr γ S̃(k) L . (2π)4 I Matter-dependent part is finite and explicitly calculable: Z µL ∂ α α α ∆mat ≡ ∆ (µL , B) − ∆ (0, B) = dµ0 0 ∆α (µ0 , B) ∂µ 0 I We change variables: α k α −→ k 0 = k α + u α µ , u α = (1, ~0 ) 23/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect I Anomalous current for LH fermions at one-loop Z d4 k α α α ∆L (µL ) ≡ ψ̄L γ ψL = − Tr γ S̃(k) L . (2π)4 I Matter-dependent part is finite and explicitly calculable: Z µL ∂ α α α ∆mat ≡ ∆ (µL , B) − ∆ (0, B) = dµ0 0 ∆α (µ0 , B) ∂µ 0 I We change variables: α k α −→ k 0 = k α + u α µ , u α = (1, ~0 ) 23/31 Introduction QCD under B field Conclusion I Vector condensation Neutron star Chiral Magnetic effect Differentiating with respect to the chemical potential, we get ∞ k2 X ∂ − ⊥ S̃ = ie |qB| (−1)n Dn 2πi δ kq2 − Λn · δ(k 0 − µ) ∂µ n=0 I Integrating over ~k⊥ , we get " ∆αmat = |qB| (0) Γαβ L Iβ # X αβ + 2gq (n) Iβ , n=1 αβ12 sign(qB) + g αβ and where Γαβ q L = I (n)β Z µ Z (n) pF β0 β 0 2 0 0 δ . = dµ kq δ kq − Λn · δ(k − µ ) = 4π 2 0 kq 24/31 Introduction QCD under B field Conclusion I Vector condensation Neutron star Chiral Magnetic effect Differentiating with respect to the chemical potential, we get ∞ k2 X ∂ − ⊥ S̃ = ie |qB| (−1)n Dn 2πi δ kq2 − Λn · δ(k 0 − µ) ∂µ n=0 I Integrating over ~k⊥ , we get " ∆αmat = |qB| (0) Γαβ L Iβ # X αβ + 2gq (n) Iβ , n=1 αβ12 sign(qB) + g αβ and where Γαβ q L = I (n)β Z µ Z (n) pF β0 β 0 2 0 0 δ . = dµ kq δ kq − Λn · δ(k − µ ) = 4π 2 0 kq 24/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect The Fermi momentum at the n-th Landau level: (p µ2 − 2|qB|n, if µ > 2|qB|n; (n) pF (µ, B) = 0, otherwise. E . . . ( 2|qB|n ) ( 2|qB| ) 1/2 1/2 P z P F (n) ... P F (1) P (0) F 25/31 Introduction QCD under B field Conclusion I Vector condensation Neutron star Chiral Magnetic effect The density of states (n) |qB| X pF (µL , B) nL = · . 4π π n I As ∆α (0, B) = 0, the anomalous electric (axial) vector currents become JVα ≡ q (∆αL + ∆αR ) = δ α3 JAα ≡ q (∆αL − ∆αR ) = δ α3 q2B µA + δ α0 q n , 2π 2 q2B µ + δ α0 q nA . 2π 2 26/31 Introduction QCD under B field Conclusion I Vector condensation Neutron star Chiral Magnetic effect The density of states (n) |qB| X pF (µL , B) nL = · . 4π π n I As ∆α (0, B) = 0, the anomalous electric (axial) vector currents become JVα ≡ q (∆αL + ∆αR ) = δ α3 JAα ≡ q (∆αL − ∆αR ) = δ α3 q2B µA + δ α0 q n , 2π 2 q2B µ + δ α0 q nA . 2π 2 26/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect No more corrections I Full contributions to the anomalous current: δΓmat (A, G ; µ) α . hJ i = δAα A=0=G I The full effective action is obtained by two steps: + 27/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect No more corrections I Full contributions to the anomalous current: δΓmat (A, G ; µ) α . hJ i = δAα A=0=G I The full effective action is obtained by two steps: + 27/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Anomalous Currents I Matter contribution to the anomalous current: Z µ δΓmat (A) δ ∂ α hJ imat = = dµ0 Γ(A ; µ0 ) , δAα δAα 0 ∂µ0 I When derivative acts on the loop, we get vertex correction: ∂ Tr[6AS1 · · · Sn ] = Tr[6AS1 · · · k6 lq · · · Sn ] 2πi δ(klq2 )δ(kl0 − µ) . ∂µ _d dµ = q q 28/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Anomalous Currents I Matter contribution to the anomalous current: Z µ δΓmat (A) δ ∂ α hJ imat = = dµ0 Γ(A ; µ0 ) , δAα δAα 0 ∂µ0 I When derivative acts on the loop, we get vertex correction: ∂ Tr[6AS1 · · · Sn ] = Tr[6AS1 · · · k6 lq · · · Sn ] 2πi δ(klq2 )δ(kl0 − µ) . ∂µ _d dµ = q q 28/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Anomalous Currents I Because the electric charge is not renormalized (Ademollo-Gatto theorem), the vertex correction should vanish. I Furthermore the density of states is also not subject to corrections due to interaction. (Luttinger theorem) I The one-loop result is exact. 29/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Anomalous Currents I Because the electric charge is not renormalized (Ademollo-Gatto theorem), the vertex correction should vanish. I Furthermore the density of states is also not subject to corrections due to interaction. (Luttinger theorem) I The one-loop result is exact. 29/31 Introduction QCD under B field Conclusion Vector condensation Neutron star Chiral Magnetic effect Anomalous Currents I Because the electric charge is not renormalized (Ademollo-Gatto theorem), the vertex correction should vanish. I Furthermore the density of states is also not subject to corrections due to interaction. (Luttinger theorem) I The one-loop result is exact. 29/31 Introduction QCD under B field Conclusion Conclusion Conclusion I Magnetic field is relevant in QCD if B ≥ 1019 G. I We derive an effective theory for LLL QCD, which has a new marginal four-quark interactions. I Scale separation between chiral symmetry breaking and confinement. I LLL quarks are one dimensional and does not contribute to running QCD coupling. I Condensation along the vector channel occurs, when mρ2 · B> e mρ ΛQCD 4 9 . 30/31 Introduction QCD under B field Conclusion Conclusion Conclusion I Magnetic field is relevant in QCD if B ≥ 1019 G. I We derive an effective theory for LLL QCD, which has a new marginal four-quark interactions. I Scale separation between chiral symmetry breaking and confinement. I LLL quarks are one dimensional and does not contribute to running QCD coupling. I Condensation along the vector channel occurs, when mρ2 B> · e mρ ΛQCD 4 9 . 30/31 Introduction QCD under B field Conclusion Conclusion Conclusion I Magnetic field is relevant in QCD if B ≥ 1019 G. I We derive an effective theory for LLL QCD, which has a new marginal four-quark interactions. I Scale separation between chiral symmetry breaking and confinement. I LLL quarks are one dimensional and does not contribute to running QCD coupling. I Condensation along the vector channel occurs, when mρ2 B> · e mρ ΛQCD 4 9 . 30/31 Introduction QCD under B field Conclusion Conclusion Conclusion I Magnetic field is relevant in QCD if B ≥ 1019 G. I We derive an effective theory for LLL QCD, which has a new marginal four-quark interactions. I Scale separation between chiral symmetry breaking and confinement. I LLL quarks are one dimensional and does not contribute to running QCD coupling. I Condensation along the vector channel occurs, when mρ2 B> · e mρ ΛQCD 4 9 . 30/31 Introduction QCD under B field Conclusion Conclusion Conclusion I Magnetic field is relevant in QCD if B ≥ 1019 G. I We derive an effective theory for LLL QCD, which has a new marginal four-quark interactions. I Scale separation between chiral symmetry breaking and confinement. I LLL quarks are one dimensional and does not contribute to running QCD coupling. I Condensation along the vector channel occurs, when mρ2 B> · e mρ ΛQCD 4 9 . 30/31 Introduction QCD under B field Conclusion Conclusion Conclusion I The vector condensation is being calculated in the effective theory (DKH2014). I We show that magnetars cool too quickly by emitting axions if B > 1020 G. I We calculate the spontaneous generation of anomalous current of dense quark matter under the magnetic field. I The one-loop is shown to be exact: JVα = δ α3 JAα = δ α3 q2B µA + δ α0 q n , 2π 2 q2B µ + δ α0 q nA . 2π 2 31/31 Introduction QCD under B field Conclusion Conclusion Conclusion I The vector condensation is being calculated in the effective theory (DKH2014). I We show that magnetars cool too quickly by emitting axions if B > 1020 G. I We calculate the spontaneous generation of anomalous current of dense quark matter under the magnetic field. I The one-loop is shown to be exact: JVα = δ α3 JAα = δ α3 q2B µA + δ α0 q n , 2π 2 q2B µ + δ α0 q nA . 2π 2 31/31 Introduction QCD under B field Conclusion Conclusion Conclusion I The vector condensation is being calculated in the effective theory (DKH2014). I We show that magnetars cool too quickly by emitting axions if B > 1020 G. I We calculate the spontaneous generation of anomalous current of dense quark matter under the magnetic field. I The one-loop is shown to be exact: JVα = δ α3 JAα = δ α3 q2B µA + δ α0 q n , 2π 2 q2B µ + δ α0 q nA . 2π 2 31/31 Introduction QCD under B field Conclusion Conclusion Conclusion I The vector condensation is being calculated in the effective theory (DKH2014). I We show that magnetars cool too quickly by emitting axions if B > 1020 G. I We calculate the spontaneous generation of anomalous current of dense quark matter under the magnetic field. I The one-loop is shown to be exact: JVα = δ α3 JAα = δ α3 q2B µA + δ α0 q n , 2π 2 q2B µ + δ α0 q nA . 2π 2 31/31