Download QCD under a strong magnetic field

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):
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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