Jan M. Pawlowski Universität Heidelberg & ExtreMe Matter Institute Trento, November 14th 2012 Quantum fluctuations & magnetic catalysis 1
Jan M. Pawlowski Universität Heidelberg & ExtreMe Matter Institute
Trento, November 14th 2012
Quantum fluctuations & magnetic catalysis
1
Heavy ion collisions
UrQMD Frankfurt/MStrickland
Simulation of a heavy ion collision
2
Functional Methods for QCD
Chiral symmetry breaking
Magnetic catalysis
Summary & Outlook
Outline
3
Functional Methods for QCD
FunMethods: FRG-DSE-2PI-...
Quarks Gluons
4
RG-scale k: t = ln k
Functional Methods for QCD
free energy
JMP, AIP Conf.Proc. 1343 (2011)
gluequantum fluctuations
hadronic quantum fluctuations
quark quantum fluctuations
Yang-Mills:
Rk2R 2k2
k2p20
1
∂tΓk[A, c, c] =1
2Tr
1
Γ(2)[A, c, c] +Rk∂tRk
− ∂tCk
full propagator regulator
∂t = k ∂k
∂tΓk[φ] =
by L. Fister
dynamical
5
RG-scale k: t = ln k
Functional Methods for QCD
Fermions are straightforward though ‘physically’ complicated
no sign problem
chiral fermions
Gluons have cost us decades
Complementary to lattice!
free energy
JMP, AIP Conf.Proc. 1343 (2011)
∂tΓk[φ] =
gluequantum fluctuations
hadronic quantum fluctuations
quark quantum fluctuations
bound states via dynamical hadronisation
dynamical
6
Naturally encorporates PQM/PNJL models as specific low order trunations
pure glue flow + + ...
flow of gluon propagator
Functional Methods for QCD
free energy
∂tΓk[φ] =
gluequantum fluctuations
hadronic quantum fluctuations
quark quantum fluctuations
JMP, AIP Conf.Proc. 1343 (2011)
7
Yang-Mills
!t = 2 + + +2
Functional Methods for QCD
Matter
- + 1
2
!t!1
= +
!t!1
= !!1/2
+
2PI-resummation
∂t = − 3 +6 +3 − 6
− 1
2+
Aconst0
Veff [σ,π;A0]
λψ
h[σ,π]
+matter-contributions
DSE
8
Chiral symmetry breaking
in strong magnetic fields
Quarks
9
EOM(σ)
Chiral symmetry breaking
Perturbative four-fermi coupling
Nf = 2 : τ = (σ1,σ2,σ3)
λψ
2
(qq)2 + (iqγ5τq)
2
λψ ∝ α2s
σ = 0Bosonisation (Hubbard-Stratonovich)
q
q
λψ
2
(ψψ)2 + (iψγ5τψ)
2=
m2σ
2
x
σ2 + π2
+ i h
xψ(σ + iγ5τπ)ψ
∝ + + +λψ =
∝ ∝+ + +
bosonisation
10
ΓΛ[ψ, ψ] = ψ ∂/ψ +1
2ψaαi ψbα
j ΓΛabcdijlm ψcβ
l ψdβm
τ : - generators SU(Nf ) ∝ +λψ = + +
Low energy effective action at high scales
Simplest approximation: NJL-model
+...
Chiral symmetry breakingLow energy effective models
ΓΛ[ψ, ψ] = ψ i∂/ψ − λΛ
2
(ψψ)2 + (iψγ5τψ)
2
11
Mean field free energy at one loop
βΩ/V −NcNfΛ4
4π2
1 +
1− 2π2
NcNf λΛΛ2
M2
Λ2
Chiral phase transition
Chiral symmetry breakingLow energy effective models
λΛΛ2 >
2π2
NcNf
M2 = −λΛψψ
12
Mean field free energy at finite temperature
βΩ/V −NcNfΛ4
4π2
1 +
1− 2π2
NcNf λΛΛ2
M2
Λ2
Chiral phase transition
Chiral symmetry breakingLow energy effective models
Tc =
3Λ2
π2− 6
NcNfλΛ
λΛΛ2 >
2π2
NcNf
1− π2
3
Λ2
T 2
critical temperature
M2 = −λΛψψ
T Λ
13
M2 = −λΛψψ
Mean field free energy at finite temperature & magnetic field
Full sum
Chiral symmetry breakingLow energy effective models
βΩ/V = −Nc
f=u,d
|qfB|2π
dpz2π
ω0 + 2T ln
1 + e−βω0
+
M2
2λΛ
LLLA
2
p2≤Λ2
d3p
(2π)3f(ω) −→ 2
Λ,B
d3p
(2π)3f(ωn) ≡
|qB|2π
NΛ,B
n=0
αn
p2≤Λ2
pz2π
f
p2z + 2|qB|n+M2
14
M2 = −λΛψψ
Mean field free energy at finite temperature & magnetic field
Chiral symmetry breakingLow energy effective models
critical temperature
βΩ/V = −Nc
f=u,d
|qfB|2π
dpz2π
ω0 + 2T ln
1 + e−βω0
+
M2
2λΛ
LLLA
Tc =2eγ
πΛ exp
− 2π2
Nc λΛ
f |qfB|
Chiral phase transition
λΛΛ2 > 0
T Λ
15
λψ
Chiral symmetry breaking
Flow for four-fermion coupling with infrared scale λψ = λψk2 k
+ + + ...
∂tλψ
k∂kλψ 2λψ A
T
k
λ2ψ B
T
k
λψ αs C
T
k
α2s+ + + + ... =
Chiral symmetry breaking directly sensitive to size of αs
Chiral symmetry breaking in QCD within the FRG
16
σ, π, ...q
qλψ
Chiral symmetry breaking
Flow for four-fermion coupling with infrared scale λψ = λψk2 k
k∂kλψ 2λψ A
T
k
λ2ψ B
T
k
λψ αs C
T
k
α2s+ + + + ... =
+ + + ...
∂tλψ
Dynamical hadronisation
Gies, Wetterich ’01 JMP ’05 Flörchinger, Wetterich ’09
dynamical
Dynamical hadronisation...and now for something completely different...
17
∂th2
m2λψ
∂tλψ
Flow for four-fermion coupling with infrared scale λψ = λψk2 k
k∂kλψ 2λψ +
+ ...
= + +
++
+ - terms
Chiral symmetry breakingDynamical hadronisation
18
∂th2
m2
= 0
k∂kλψ 2λψ +
+ ...
= + +
++
+ - terms
Full bosonisation λψ = 0
Chiral symmetry breaking
Meson potential
σπ
∂th2
m2
+ ...
+ - terms
Dynamical hadronisation
19
h(k)
k[GeV]0.1 0.5 1.0 5.0 10.0 50.0 100.0
10
20
15 UV2 GeVUV5 GeVUV10 GeVUV90 GeV
initial scale
Full bosonisation λψ = 0
h(k)
k[GeV]
0.01 0.1 1 10 100
10
15
UV90 GeV, Ε100, ΛΣ0.001, h0.001UV90 GeV, Ε10, ΛΣ0.001, h0.001UV90 GeV, Ε4.89, ΛΣ0.001, h0.01UV90 GeV, Ε4.89, ΛΣ0.001, h0.1UV90 GeV, Ε4.89, ΛΣ0.001, h1UV90 GeV, Ε4.89, ΛΣ0.001, h0.001
initial conditions
Braun, Fister, Haas, JMP, in prep
Chiral symmetry breakingDynamical hadronisation
20
h(k)
k[GeV]0.1 0.5 1.0 5.0 10.0 50.0 100.0
10
20
15 UV2 GeVUV5 GeVUV10 GeVUV90 GeV
initial scale
Full bosonisation λψ = 0
Low energy models
Braun, Fister, Haas, JMP, in prep
Chiral symmetry breakingStability of low energy models
21
h(k)
k[GeV]0.1 0.5 1.0 5.0 10.0 50.0 100.0
10
20
15 UV2 GeVUV5 GeVUV10 GeVUV90 GeV
initial scale
Full bosonisation λψ = 0
Low energy models
FRG: (
com
plet
ely)
fixe
d from
QCD
Braun, Fister, Haas, JMP, in prep
Chiral symmetry breakingStability of low energy models
21
λψ
Chiral symmetry breaking
Flow for four-fermion coupling with infrared scale λψ = λψk2 k
+ + + ...
∂tλψ
k∂kλψ 2λψ A
T
k
λ2ψ B
T
k
λψ αs C
T
k
α2s+ + + + ... =
Chiral symmetry breaking directly sensitive to size of αs
αs,eff → 0
Nc → ∞
h(k)
k[GeV]0.1 0.5 1.0 5.0 10.0 50.0 100.0
10
20
15 UV2 GeVUV5 GeVUV10 GeVUV90 GeV
initial scale
Λ
IR-stability
k < Λ
Chiral symmetry breaking in QCD within the FRG
22
λψ
Chiral symmetry breaking
Flow for four-fermion coupling with infrared scale λψ = λψk2 k
+ + + ...
∂tλψ
k∂kλψ 2λψ A
T
k
λ2ψ B
T
k
λψ αs C
T
k
α2s+ + + + ... =
∂tλk = −NcNf λ2kk
2
3π2
∂tλk = 2λk − NcNf
3π2λ2k
Chiral symmetry breaking directly sensitive to size of αs
αs,eff → 0
Nc → ∞
Chiral symmetry breaking in QCD within the FRG
23
Fukushima, JMP ’12
∂tΠk(0) =
2Nc +
34
Nfk2
3π2
1
2− 1
eβk + 1− βk
eβk
(eβk + 1)2
t = + + · · ·+ +
(a) (b) (c) (d)
Πk(p) 2Nc 1 1/4-
Flow of four-point correlation function
Chiral symmetry breaking!t!k["] = 1
2! ! + 1
2
λΛΛ2
3>
2π2
NcNf
λk =λΛ
1 +NcNf λΛ
6π2(k2 − Λ2)
∂tλψ λk = Πk(0)
Chiral symmetry breaking in QCD within the FRG
24
Fukushima, JMP ’12
∂tΠk(0) =
2Nc +
34
Nfk2
3π2
1
2− 1
eβk + 1− βk
eβk
(eβk + 1)2
t = + + · · ·+ +
(a) (b) (c) (d)
Flow of four-point correlation function
Chiral symmetry breaking!t!k["] = 1
2! ! + 1
2
∂tλψ
Tc =
Λ2
π2− 6
NcNf λΛ
critical temperature
λk =λΛ
1 +NcNf λΛ
6π2
k2 − Λ2 + π2T 2
λk = Πk(0)
Chiral symmetry breaking in QCD within the FRG
25
Fukushima, JMP ’12
Flow of four-point correlation function
critical temperature
Integrated flow λk =λΛ
1 +Nc λΛ
2π2
f
|qfB| t
Chiral symmetry breakingStrong magnetic fields with RG in LLLA
!t!k["] = 1
2! ! + 1
2
Dimensional Reduction
λk = Πk(0)
Tc = 0.42Λ exp
− 2π2
NcλΛ
f |qfB|
t = + + · · ·+ +
(a) (b) (c) (d)
∂tλk = − Nc
2π2
f
|qfB|λ2k
26
Fukushima, JMP ’12
Flow of four-point correlation function
Chiral symmetry breaking!t!k["] = 1
2! ! + 1
2
µ = 0
0 0.1
0.2 0.3
0.4 0.5
0 0.2
0.4 0.6
0.8 1
1
10
100
Temperature T [ ]Magnetic Field |eB| [ 2]
*
0 0.1
0.2 0.3
0.4 0.5
0 0.2
0.4 0.6
0.8 1
1
10
100
Temperature T [ ]Magnetic Field |eB| [ 2]
*
λψ∗
µ = 0
0 0.1
0.2 0.3
0.4 0.5
0 0.2
0.4 0.6
0.8 1
1
10
100
Temperature T [ ]Magnetic Field |eB| [ 2]
*
λψ∗
NJL-model
t = + + · · ·+ +
(a) (b) (c) (d)
∂tλk = ∂tΠk(p = 0, B)
Strong magnetic fields with RG beyond LLLA
Skokov ’11PQM+
27
Fukushima, JMP ’12
Flow of four-point correlation function
Chiral symmetry breaking!t!k["] = 1
2! ! + 1
2Strong magnetic fields with RG with full non-localities
λk = Πk(0, B)
t = + + · · ·+ +
(a) (b) (c) (d)
!"##
!$##
!%##
#
# $ & ' ( %#!)*+
,-./01233456-7
/01233456-7)*+
λψ
bubble resummation
s, t & u channel
s-channel
LLL approximation
∂tλk = ∂tΠk(p = 0, B)
Πk(p) =λk(p)
1− λk(p)[Πk(p)−Πk(0)]
28
Summary & Outlook
29
RG point of view of magnetic catalysis
magnetic catalysis via dimensional reduction
fluctuations & magnetic catalysis
momentum dependence
Gluonic fluctuations
B-dependence of strong coupling
dynamical hadronisation
Summary & outlook
Dimensional Reduction
µ = 0
0 0.1
0.2 0.3
0.4 0.5
0 0.2
0.4 0.6
0.8 1
1
10
100
Temperature T [ ]Magnetic Field |eB| [ 2]
*
0 0.1
0.2 0.3
0.4 0.5
0 0.2
0.4 0.6
0.8 1
1
10
100
Temperature T [ ]Magnetic Field |eB| [ 2]
*
λψ∗
!"##
!$##
!%##
#
# $ & ' ( %#!)*+
,-./01233456-7
/01233456-7)*+
λψ
30
Additional material
31
Confinement & Thermodynamics
Strickland
Fister, JMP
T =0T A
− p (T ; A) =
0
Λ
dk
k
32
p2A A(p2)
perturbative
2
3
4
1
0 5 64321
!"!! #"$
%&!! #"$
!"!! '"(
'#!! '"(
%&!! '""
p [GeV]
FRG: Fischer, Maas, JMP ’08
lattice: Sternbeck et al. ’06
non-perturbative and phenomenologically
relevant
Propagators
Propagators phenomenologically well described in 1/N expansion
33
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
p [GeV]
Longitudinal Propagator GL
FRG: T = 0
FRG: T = 0.361Tc
FRG: T = 0.903Tc
FRG: T = 1.81Tc
Lattice: T = 0
Lattice: T = 0.361Tc
Lattice: T = 0.903Tc
Lattice: T = 1.81Tc
0.0 0.5 1.0 1.5 2.00
1
2
3
4
5
p [GeV]
Transversal Propagator GT
FRG: T = 0
FRG: T = 0.361Tc
FRG: T = 0.903Tc
FRG: T = 1.81Tc
Lattice: T = 0
Lattice: T = 0.361Tc
Lattice: T = 0.903Tc
Lattice: T = 1.81Tc
Fister, JMP ’11
Lattice: Maas, JMP, Spielmann, von Smekal ’11
!t = 2 + + +2
!t!1
= +
!t!1
= !!1/2
+
+ RG-dressed gluonic vertices
ConfinementThermal gluon propagators
confirmed with the full system, JMP, Fister, in prep
see talk of L.Fister
34
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4Tσ−1/2
0
0.1
0.2
0.3
0.4
0.5
0.6
DL(0
)−1/2
[GeV
]
322×4
642×4
1282×4
2562×4
1282×6
cT/T0 0.5 1 1.5 2
[GeV
]-1
/2(0
)LD
0
0.5
1
Electric screening mass for SU(2)
3d
4d
critical scaling in Landau gauge props on the lattice?
ν ≈ 0.68
ν ≈ 1
Maas, JMP, Spielmann, von Smekal ’11
ConfinementChromo-electric propagator
FRG
DL(0)−1/2 ∝ |T−Tc|ν + · · ·
DL(0)−1/2 ∝ V[A0] + · · ·
DL(0) = AAT(0)
global gauge fixing35
Confinement
Braun, Gies, JMP ‘07
SU(3)
βgA0
2π
-0.5-0.4-0.3-0.2-0.1
0 0.1 0.2 0.3 0.4
0 0.2 0.4 0.6 0.8 1
!4 V(!
<"
0>)
! <A0>/(2#)
0.3 0.5 0.7
276 MeV
295 MeV
286 MeV
280 MeV
276 MeV
271 MeV
β4 VYM[A0]
Tc = 276± 10MeV
Order parameter
Φ[43π
1βg
] = 0
Φ
0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25
0.0
0.2
0.4
0.6
0.8
1.0
SU(3)
T/Tc
Polyakov loop
lattice : Tc/√
σ = 0.646
Φ[A0] =13(1 + 2 cos
12βgA0)
Tc/√
σ = 0.658± 0.023
SU(N), Sp(2), E(7): Braun, Eichhorn, Gies, JMP ’10
SU(2) & critical scaling: Marhauser, JMP ’08
36
pGT=0,k ∂tRk
T =0T A
− p (T ; A) =
0
Λ
dk
k
Confinement & Thermodynamics
FRGBorsanyi et al.
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.2
0.4
0.6
0.8
1.0
TTc
p YMp SB
A0,min
p
GT ,k ∂tRk
pre
lim
inary
Fister, JMP, in prep
1/2 * 2 polarisations
37
Full dynamical QCDPhase structure
0
0.2
0.4
0.6
0.8
1
150 160 170 180 190 200 210 220 230
T [MeV]
f!(T)/f!(0)
Dual density
Polyakov Loop
160 180 200
"L,d
ual
0
0.2
0.4
0.6
0.8
1
1 1.5 2 2.5 3
T/Tc
Lren(T)
HISQ: N =8N =6
stout cont.SU(3)
Polyakov loopChiral condensate
Budapest-Wuppertal ’10 hotQCD ’10
Braun, Haas, Marhauser, JMP ‘09
0
50
100
150
200
0 50 100 150 200 250 300 350
T [M
eV]
µ [MeV]
χ crossoverΦ crossover—Φ crossoverCEPχ first order
Phase diagram of quantised PQM-model
Herbst, JMP, Schaefer ’10
Nf =2 & chiral limit
FRG QCD results at finite density
Braun, Fister, Haas, JMP, in prep
FRG QCD surveyJMP, Aussois ’12
38