qcd with a magnetic background field -...
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QCD with a Magnetic Background Field
Andreas Schafer for SFB/TRR-55, in particularGergely Endrodi (now Frankfurt)
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Key question of relativistic heavy ion physics:Does one reach thermal equilibrium fast enough to really probethe quark gluon plasma ?
In HICs one produces the strongest magnetic fields in theuniverse
Does this influence thermalization ?Does this lead to novel effects like the CME ?
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Time and space dependence of these magnetic fields
0.5 1.0 1.5 2.0t @fmD
10-510-40.001
0.01
0.1
1
eBy @fm-2D
[Deng et al ’12] [Gursoy et al ’13]
The time profile depends crucially on the electric conductivity.Gursoy et al. use the lattice QCD values for equilibrium.
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relevance of QCD & magnetic fields for astrophysics
arXiv:0801.4387 gravitational wave signal from neutron starmerger
0 5 10 15 20 25time (ms)
-4.0
-2.0
0.0
2.0
4.0
r h +
(m
)
magnetizednon magnetized
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Lee, Fukushima, Kharzeev et al.: The QCD chiral anomalyGµνGµν ∼ Ecolor ⋅ Bcolor can induce an electromagnetic Eparallel or antiparallel to B. STAR 1404.1433 :
0
10
20
30
40
020406080
11.5 GeV
0
10
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020406080
7.7 GeV
0 20 40 60 80
0
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UrQMDopp charge
same charge
27 GeV
0 20 40 60 80
0
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30
19.6 GeV
MEVSIM
5
0
5
10 62.4 GeV
opposite charge
same charge
5
0
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10 39 GeV
Collision centrality (% Most Central)
]⟩)R
Pψ
22
φ+ 1
φc
os
(⟨
≡ γ
[× 4
10
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Lattice QCD cannot help directly , because the CME ishighly dynamical. Also, µB, µiso ≠ 0.While the CME effect should exist in principle, its sizecould well be unmeasurable small, see B. Muller and AS,1009.1053.see below: magnetic fields influence the flow pattern(paramagnetic squeezing).Lattice calculations with magnetic fields allow to checkeffective dynamical descriptions needed for comparisonwith experiment.
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Magnetic field on the LatticeQCD is contained in the generating functional:
Z [Jaµ, η
i , ηi] = ∫ D[Aaµ, ψi , ψi
]
exp(i∫ d4x [LQCD − JaµAa
µ − ψiηi
− ηiψi])
Discretized space time⇒ e.g. the Wilson action
U(l1) = exp(−igAb(l1)
λb
2a)
W◻ = Tr{U(l1)U(l2)U(l3)U(l4)}
∑◻
2g2 (3 −Re W◻) =
14 ∫
d4x (F aµνF a
µν + O(a2))
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Magnetic field on the torus
torus T 2 with surfacearea LxLy
picture from [D’Elia et al ’11]
● phase factor for a charged particle transported along pathC: exp(iq ∮C dxµAµ)
● Stokes theorem: ∮C dxµAµ = ∫∫A dσB = B ⋅Abut also = − ∫∫T 2−A dσB = −B ⋅ (LxLy −A)
● equality of phase factors gives quantization condition[Hashimi, Wiese ’09]
exp(iqBLxLy) = 1 → qBLxLy = 2π ⋅Nb, Nb ∈ Z
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How to discretize Aµ on the lattice?
simplest choice uy = exp(iaqAy) = exp(iφnx), with the flux unitφ = a2qB plus local U(1) gauge transformationψ(Nx ,ny)→ ψ(Nx ,ny) ⋅V ny with V = exp(iφNx)
This restores periodicity in the x-direction.
remark: det( /D(B) +mlatf ) > 0 so no sign problem
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Observables sensitive to the QCD transition
● chiral condensate→ chiral symmetry breaking
ψfψf =∂ logZ∂mf
● chiral susceptibility→ chiral symmetry breaking
χf =∂2 logZ∂m2
f
● Polyakov loop→ deconfinement
P =1V∑x
Tr∏x4
U4(x,x4)
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C. Gattringer arXiv:1004.2200, pure gauge theoryphase of the Polyakov loop: Z3 symmetry for SU(3)
-0.2 0.0 0.2 0.4-0.4
-0.2
0.0
0.2
-0.2 0.0 0.2 0.4-0.4
-0.2
0.0
0.2
T = 0.63 Tc
T = 1.32 Tc
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Transition characteristics
chiral susceptibility(∼ specific heat)
χ =∂2 logZ∂m2
transition temperature: peak maximumorder of transition: volume-dependence of heighth(V)∝ Vα
1st (α = 1), 2nd (0 < α < 1) or crossover (α = 0)bubble nucleation versus smooth transition
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Transition characteristics at B = 0
● simulations with physical mlatf ,
continuum extrapolation● no singular behavior as V →∞
⇒ transition is analyticcrossover[Aoki, Endrodi, Fodor, Katz, Szabo ’06]there is no unique transition temperature
0
0.2
0.4
0.6
0.8
1
120 140 160 180 200
T [MeV]
∆l,s
fK scale
asqtad: Nτ=8
Nτ=12
HISQ/tree: Nτ=6
Nτ=8
Nτ=12
Nτ=8, ml=0.037ms
stout cont.! !!!!!
!!!!!!!!!!!!!!!!!!!!!!
""""""""""""""""""
!!!!!!!!!!!!!!!!!!
# # ####### ##### #
##!!""!!
Nt"16Nt"12Nt"10Nt"8
Continuum
100 150 200 250 300 3500.0
0.2
0.4
0.6
0.8
1.0
T !MeV"RenormalizedPolyakovloop
● T ψψc ≈ 150 MeV, T P
c ≈ 175 MeV [BWc ’06,’09,’10]
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Condensate at B > 0: ‘magnetic catalysis’
● what happens to ψψ (⟨+q ↑,−q ↓⟩) in magnetic field?⇒ magnetic moments parallel, energetically favored state
● dimensional reduction 3 + 1→ 1 + 1 for LLL
E0(cont .) =√
p2z +m2 + (2n + 2sz + 1)mωc +
aeeBm
E0(LLL) =√
p2z +m2, #0 =
∣qB∣ ⋅ LxLy
2π● chiral condensate↔ spectral density near zero
ψψ ∝ ρ(0) [Banks, Casher ’80]
● in the chiral limit, to maintain ψψ > 0(NJL [Gusynin, Shovkovy et al ’96])
B = 0 ρ(p)dp ∼ p2dp “we need a strong interaction”B ≫ m2 ρ(p)dp ∼ qBdp “the weakest interaction suffices”
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● magnetic catalysis at zero temperature is a robust concept:χPT, NJL model, AdS-CFT, linear σ model, . . .
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● magnetic catalysis at zero temperature is a robust concept:χPT, NJL model, AdS-CFT, linear σ model, . . .
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Inverse magnetic catalysis
● lattice QCD, physical mπ, continuum limit [Bali et al ’11, ’12]
● at T ≈ 150 MeV the condensate is reduced by Bdubbed ‘inverse magnetic catalysis’
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Phase diagram● inflection point of ψψ(T ) defines Tc● significant difference whether IMC is exhibited or not:
lattice QCD, physical mπ, continuum limit [Bali et al ’11]
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● Suggestion: Two competing mechanisms at finite B[D’Elia et al ’11, Bruckmann, Endrodi, Kovacs 1303.3972]
direct (valence) effect B ↔ qfindirect (sea) effect B ↔ qf ↔ g
⟨ψψ(B)⟩∝ ∫ DU e−Sg det( /D(B,U) +m)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
sea
Tr [( /D(B,U) +m)−1]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶valence
BBBB
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● Suggestion: Two competing mechanisms at finite B[D’Elia et al ’11, Bruckmann, Endrodi, Kovacs 1303.3972]
direct (valence) effect B ↔ qfindirect (sea) effect B ↔ qf ↔ g
⟨ψψ(B)⟩∝ ∫ DU e−Sg det( /D(B,U) +m)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
sea
Tr [( /D(B,U) +m)−1]
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶valence
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Matter in magnetic fields (linear response)●paramagnets: attracted by magnetic field●diamagnets: repel magnetic field
paramagnet: liquid oxygen diamagnet: frog
is thermal QCD as a medium para- or diamagnetic?
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● free energy density in background magnetic field
f (B) = −TV
logZ(B)
● magnetization
M = −∂f
∂(eB), M∣B=0 = 0
● susceptibility
χ =∂M
∂(eB)∣B=0
= −∂2f
∂(eB)2 ∣B=0
● sign distinguishes betweenparamagnets (χ > 0) drawn into magnetic fielddiamagnets (χ < 0) repelled by magnetic field
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fermions give paramagnetic behaviour, bosons diamagnetic⇒ Expectation for the susceptibility
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Magnetic susceptibility on the lattice
magnetic flux quantization
a2qB =2πNb
NxNy, Nb = 0,1, . . . ,NxNy
χ as derivative is not directly accessible⇒ variousmethods to circumvent this problem: Bali et al 1303.1328,DeTar et al 1309.1142, Bonati et al 1307.8063, Bali et al1406.0269here: calculate f (B) and differentiate it numericallylattice setup: stout smeared staggered quarks + Symanzikgauge action, physical pion mass, continuum estimatebased on Nt = 6,8,10
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with conventional Monte-Carlo techniques, derivatives oflogZ can be calculated, but not logZ ∝ f itselfrewrite logZ as the integral of its derivatives at constant Nb
logZ(∞) − logZ(mphf ) = ∫
∞
mphf
dmf∂ logZ∂mf
take difference ∆ logZ = logZ(Nb) − logZ(0)
∆ logZ(∞)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
0
−∆ logZ(mphf ) = ∫
∞
mphf
dmf∂∆ logZ∂mf
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶∆ψfψf
∆ logZ obtained as integral of condensates
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with conventional Monte-Carlo techniques, derivatives oflogZ can be calculated, but not logZ ∝ f itselfrewrite logZ as the integral of its derivatives at constant Nb
logZ(∞) − logZ(mphf ) = ∫
∞
mphf
dmf∂ logZ∂mf
take difference ∆ logZ = logZ(Nb) − logZ(0)
∆ logZ(∞)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
0
−∆ logZ(mphf ) = ∫
∞
mphf
dmf∂∆ logZ∂mf
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶∆ψfψf
∆ logZ obtained as integral of condensates
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obtain ∆ logZ as an integral for each Binterpolate ∆ logZ as function of Bdifferentiate to obtain χ∝ ∆ logZ ′′
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Susceptibility from the lattice
● comparison to Hadron Resonance Gas model (low T )and to perturbation theory (high T )
● The quark gluon plasma is paramagnetic
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Other results: Gluon anisotropies 1303.1328
A(E) =TV
⟨β
6∑n
(trE2⊥(n) − trE2
∥ (n))⟩
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The Pisa group found an anisotropic heavy quark potential.1403.6094
3 4 5 6 7 8ns
0.3
0.4
0.5
0.6
0.7
0.8
0.9aV
(nsa
)b=0 b=24 XYb=24 XYZb=24 Z
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There is hardly any effect on topological charge density1303.1328
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The correlation between topological charge and electric current
J fν(x) = ψfγνψf (x)
Df (∆) =⟨qtop(x) ⋅ J f
t (x +∆)⟩
√
⟨q2top(x)⟩⟨Σf
xy(x)⟩
lattice results:
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To interpret this result we extended the modelBasar, Dunne and Kharzeev, 1112.0532to arbitrary magnetic field strength and got in our model theprediction for LLL dominance
Df (model) ≈ 1
in contrast toDf (lattice) ≈ 0.1
While you can criticize our model extension, this result fits verywell to what we wrote in 1009.1053
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Are any of these effects phenomenologically relevant ?Example CME: comparing Hirono, Hirano and Kharzeev (HHK),1412.0311 with Muller and Schafer (MS), 1009.1053
∆±=
dN+ − dN−
dN+ + dN− = Cem τBeB∣Q5∣
V
HHK MS
n5 = ∣Q5∣/V (0.35 GeV)3 (0.4 GeV)3
Cem 0.2 GeV−4 0.02 GeV−4
τBeB 4 GeV 0.04 GeV !!
⟨∆±(th)⟩ ≈ ⟨∆±(exp)⟩ ⟨∆±(th)⟩ ≪ ⟨∆±(exp)⟩
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Can experiment decide ?The planned isobar run at RHIC comparing 96Zr with 96Ru(same A different Z).
problem: possible effects of µiso
What can the lattice say ?
Endrodi and Brandt have recently studied the case µiso ≠ 0,µB = 0 1611.06758
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Z = ∫ DU e−βSG (detMlight)1/4
(detMs)1/4
Mlight = /D(τ3µiso) +mlight1 + iλη5τ2
= (/Dµ +mlight λη5−λη5 /D−µ +mlight
)
Ms = /D(0) +ms
/D = γµDµ + γ0 µiso τ3
One has to analyze λ→ 0 to get a well-defined result.
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calculable quantities
⟨π⟩ =T
2V⟨tr
λ
∣ /D(µiso) +mlight∣2 + λ2
⟩
⟨ψψ⟩ =T
2V⟨Re tr
/D(µiso) +mlight
∣ /D(µiso) +mlight∣2 + λ2
⟩
⟨niso⟩ =T
2V⟨Re tr
( /D(µiso) +mlight)† /D(µiso)
′
∣ /D(µiso) +mlight∣2 + λ2
⟩
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The “Silver Blaze” phenomenon
100
110
120
130
140
150
160
170
180
0 0.2 0.4 0.6 0.8 1 1.2 1.4
T[M
eV]
µI/mπ
(TC , µI,C)P(TC , µI,C)C
preliminary
crossover
Pion condensation
243 × 6 lattice; red: phase boundary to the pion condensationphase, (Tc , µI,c)P ; blue: crossover line, (Tc , µI,c)C .crucial question: Is µiso(
96Zr ,96 Ru) close to mπ/2?
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Conclusions
QCD in a magnetic field is a truly fascinating topicmuch insight can be obtained from lattice calculationsSimple models are misleading and have to be substitutedby descriptions which agree with all lattice results.Without tight quantitative control of magnetic field effectsmany aspects of HICs cannot be rigorously interpretedFor the RHIC isobar run it is important to knowµiso(
96Zr ,96 Ru)This is also relevant for astrophysics and cosmology
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