Thermodynamic properties of QCD in external magnetic ...Thermodynamic properties of QCD in external magnetic ﬁelds: an update Falk Bruckmann SFB Meeting, Regensburg, April 2013 G

Thermodynamic properties of QCD in externalmagnetic fields: an update

Falk Bruckmann

SFB Meeting, Regensburg, April 2013

G. Bali, M. Constantinou, M. Costa, G. Endrodi, Z. Fodor, F. Gruber,S. Katz, S. Krieg, T. Kovács, H. Panagopoulos, A. Schäfer, K. Szabó

JHEP 1202 044, PRD 86 071502 + 09451, JHEP (accepted + accepted)

Falk Bruckmann Thermodynamic properties of QCD in external magnetic fields 0 / 17

Motivation: Strong Magnetic fields

early universe√

eB ' 2 GeV

RHIC/LHC 0.1..0.5 GeV QCD scale!non-central collisionscharged spectatorsB perp. to reaction plane

neutron stars, magnetars 1 MeV B ' 1014 G

cf. strongest field in lab 105 G(107 G unstable)

refrigerator magnet 100 G

earths magn. field 0.6 G


Technical details

idealized: constant external magn. field B + QCD in equilibrium

lattice: abelian space-dependent phases on the links mimicking AµB quantized and bounded, but no sign problem

simulations similar to transition studies at B = 0 Budapest-Wuppertal

tree-level improved gauge actionstout smeared staggered fermions (rooting trick)2 light quarks + strange quark, charges (qu,qd ,qs) = (2

3 ,−13 ,−

13)e

physical pion massesB does not alter scale setting a(β) and line of constant physicsm(β) more

state-of-the-art:√

eB = 0.1 . . . 1 GeV


Main result: QCD phase diagram with magn. field

(pseudo-)Tc as a function of magnetic field:

strange numbersusceptibilitylight quark condensateboth renormalized

⇒ Tc decreases by O(10) MeV for eB . 0.5 GeV2

⇒ transition not becoming stronger nature


(light) condensate and Polyakov loop as a function of T at fixed B’s:

T = 0: Magnetic Catalysis

Model ConjectureMonte Carlo X D’Elia et al. 10

Inverse Magnetic Catalysis!? Tc ↘

Important: Masses of Current quarksInvestigate Monte Carlo configurations!⇒ sea and valence effects and P-loop

& MC and IMC in the gluon sector& anisotropies


Magnetic catalysis Müller, Schramm2 92; Gusynin, Miransky, Shovkovy 96

change of condensate[〈ψψu,d〉(B)− 〈ψψu,d〉(0)

]mu=d at T = 0:

with χPT, NJL Cohen, McGady, Werbos 07, Andersen 12; Gatto, Ruggieri 10

⇒ well approximated unless eB > 0.1,0.3 GeV2

note that : ∆... = ...(B)− ...(0) removes additive divergences (as T )m · ... removes multiplicative divergences


Magnetic catalysis

change of condensate and gluonic action:

very similar shape

⇒ gluons inherit magnetic catalysis from quarks via strong coupling

magnitude O(100) larger for gluons, but B = 0 scale (= gluoncondensate) already O(200) larger: relative effect larger on quarks


Intermezzo: Trace anomaly

I = ε− 3p . . . interaction measure, since free gas: ε = 3p

lattice= − T

Vd log Zd log a

. . . scale anomaly

= − TV

(∂ log Z∂β

∂β

∂ log a+

∂ log Z∂ log am

∂ log am∂ log a

)β =

6g2

= −(〈sg〉

−∂β∂ log a

+ m 〈ψψ〉 ∂ log am∂ log a

)∆I = −

(∆〈sg〉

−∂β∂ log a

+ m ∆〈ψψ〉 ∂ log am∂ log a

)⇒ change of gluonic action density and condensate go together

with beta and gamma function [LCP]!?⇒ similarity in B-dependence

(used to improve convergence of gluonic contribution)Falk Bruckmann Thermodynamic properties of QCD in external magnetic fields 6 / 17

Inverse magnetic catalysisagain change of condensate and gluonic action, now finite T :

non-monotonic behaviour, again similar shape for quarks and gluons⇒magnetic catalysis turns into inverse magnetic catalysis around Tc

physical quark masses essential check

higher masses in other simulations D’Elia et al. 10, Ilgenfritz et al. 12

missed in most non-lattice approachesFalk Bruckmann Thermodynamic properties of QCD in external magnetic fields 7 / 17

Inverse magnetic catalysis: mechanism

〈ψψ〉full =

∫e−Sg det( /D[B] + m) tr( /D[B] + m)−1∫e−Sg det( /D[B] + m) tr( /D[B] + m)−1

〈ψψ〉val =

∫e−Sg det( /D[0] + m) tr( /D[B] + m)−1∫e−Sg det( /D[0] + m) tr( /D[B] + m)−1

B in observable

〈ψψ〉sea =

∫e−Sg det( /D[B] + m) tr( /D[0] + m)−1∫e−Sg det( /D[B] + m) tr( /D[0] + m)−1

B in config. generation

to lowest order approx. : 〈ψψ〉full ' 〈ψψ〉val + 〈ψψ〉sea D’Elia, Negro 11

∆〈ψψ〉 at low T and around Tc : FB, Endrodi, Kovács 13


Inverse magnetic catalysis: mechanism

〈ψψ〉full =

∫e−Sg det( /D[B] + m) tr( /D[B] + m)−1∫e−Sg det( /D[B] + m) tr( /D[B] + m)−1

〈ψψ〉val =

∫e−Sg det( /D[0] + m) tr( /D[B] + m)−1∫e−Sg det( /D[0] + m) tr( /D[B] + m)−1

B in observable

〈ψψ〉sea =

∫e−Sg det( /D[B] + m) tr( /D[0] + m)−1∫e−Sg det( /D[B] + m) tr( /D[0] + m)−1

B in config. generation

to lowest order approx. : 〈ψψ〉full ' 〈ψψ〉val + 〈ψψ〉sea D’Elia, Negro 11

∆〈ψψ〉 at low T and around Tc : FB, Endrodi, Kovács 13


/D has more small eigenvalues with Bin valence trace ⇓generates condensate= statement about thechange of the spectrum(even quenched)

⇓ in sea determinantleads to a B-dep. probability= statement about the typicalgauge field= feedback of quarks

sea effect is particularly effective near Tc ! why increasing at low T ?washed out for heavy quarksincreasing Polyakov loop indicates change of typical gauge field


Anisotropy I: Magnetic susceptibility

tensor polarization: Ioffe, Smilga 84⟨ψσµνψ

⟩∝ Fµν

⟨ψσ12ψ

⟩≡ qB

⟨ψψ⟩· χ︸︷︷︸

τ

+O((qB)3)

for each quark flavorχ relevant for radiative Ds meson transitions, anomalous magn.

moment of the muon, chiral-odd photon distribution amplitudesτ like order parameter, physical values numbers

in free energy:

F ∝ −τ(qB)2 + angular momentum + O((qB)4)

negative τ as we find⇒ free energy increases with B ⇒diamagnetic contribution from spinbut: total O(B2) at T = 0 completely fixed by charge renormalization


Anisotropy II: Field strength componentsplaquette expectation values in various planes:

〈tr E2‖ 〉 < 〈tr E

2⊥〉

T =0' 〈trB2⊥〉 < 〈trB2

‖〉

A(E) ≡ 〈tr E2⊥〉 − 〈tr E2

‖ 〉 > 0 A(B) ≡ 〈trB2⊥〉 − 〈trB2

‖〉 < 0

same for coarse and heavy Nf = 2 simulations Ilgenfritz et al. 12

in line with perturbative Euler-Heisenberg eff. Lagrangian:

Seff =− log det( /D[B] + m)∼ (qB)2

m4

[52〈tr E2

‖ 〉 − 〈trB2⊥〉 − 〈tr E2

⊥〉 − 3〈trB2‖〉]


Anisotropy III: Quark sectorquark action:

Af ≡ 〈ψf /D⊥ψf 〉 − 〈ψf /D‖ψf 〉 for all flavors

⇒ negative⇒ bigger than gluonic anisotropy⇒ roughly independent of temperature


Anisotropy IV: Topological chargetwo-point correlator in different directions:

〈q(0)q(r)〉~r ‖~B vs. 〈q(0)q(r)〉~r ⊥~B vs. 〈q(0)q(r)〉

⇒ no stat. significant anisotropyalthough topology related to quark zero modes (index theorem)and those should become elongated along B (Landau levels) . . .


(An)isotropic pressure and magnetizationpressure p is change of free energy under compression of volume,now sensitive to direction:

pi = −Li

VdFdLi

specify whether magn. field eB or magn. flux Φ = eB · LxLy constantphys. question, e.g. Φ-scheme: perfect conductors, field lines frozen in

homogeneous system: free energy extensive with density f

F = LxLyLz · f (eB) = LxLyLz · f (Φ

LxLy)

additional Lx ,y = L⊥-dependence⇒

p(B)⊥ = p(B)

‖︸︷︷︸isotropic pressure

= p(Φ)‖ 6= p(Φ)

⊥︸︷︷︸anisotropic pressure

not very clear in the literature (!) on the lattice fortunately . . .



pi = −Li

VdFdLi




LxLy)


p(B)⊥ = p(B)


= p(Φ)‖ 6= p(Φ)


not very clear in the literature (!) on the lattice fortunately . . .



pi = −Li

VdFdLi




LxLy)


p(B)⊥ = p(B)


= p(Φ)‖ 6= p(Φ)


not very clear in the literature (!) on the lattice fortunately . . .Falk Bruckmann Thermodynamic properties of QCD in external magnetic fields 14 / 17

. . . we’ve got a quantized magn. flux and thus the Φ-scheme, moreover:

p(Φ)⊥ − p(Φ)

‖ = − ∂f∂eB

· L⊥∂eB∂L⊥

∣∣∣∣Φ=const.: eB∝L⊥

= −M · eB

⇒measure the magnetization M

via anisotropic lattices, schematically:

a‖ = ξa⊥Nµ const.−→ L‖ = ξL⊥

S = ξg,0 · plaq‖ + ξ−1g,0 · plaq⊥ + ξf ,0 · ψf /D‖ψf + ψf /D⊥ψf + ψf mfψf

ξ is the physical anisotropy, ξ.,0(β) are bare anisotropies = parametersto be tuned in the continuum limit along a line of fixed ξ⇒ (bare) magnetization through gluonic and quark anisotropies:

−M · eB = −ζg[A(E)− A(B)]− ζf

∑f

Af


. . . we’ve got a quantized magn. flux and thus the Φ-scheme, moreover:

p(Φ)⊥ − p(Φ)

‖ = − ∂f∂eB

· L⊥∂eB∂L⊥

∣∣∣∣Φ=const.: eB∝L⊥

= −M · eB

⇒measure the magnetization Mvia anisotropic lattices, schematically:

a‖ = ξa⊥Nµ const.−→ L‖ = ξL⊥

S = ξg,0 · plaq‖ + ξ−1g,0 · plaq⊥ + ξf ,0 · ψf /D‖ψf + ψf /D⊥ψf + ψf mfψf

ξ is the physical anisotropy, ξ.,0(β) are bare anisotropies = parametersto be tuned in the continuum limit along a line of fixed ξ⇒ (bare) magnetization through gluonic and quark anisotropies:

−M · eB = −ζg[A(E)− A(B)]− ζf

∑f

Af


with coefficients: Karsch 82

ζ. ∼∂ξ.,0∂ξ

∣∣∣∣ξ=1

pert.−→ 1 +O(g2)

first approx. ζ = 1 and subtraction of O(B2) term (charge renorm.):

including result from Hadron Resonance Gas Endrodi 13

⇒QCD vacuum is paramagneticFalk Bruckmann Thermodynamic properties of QCD in external magnetic fields 16 / 17

Summaryphase diagram: Tc(B) decreases by O(10) MeV, still crossoverMagnetic Catalysis: 〈ψψ〉(B)↗ at zero temperaturem also for gluons, cf. trace anomaly

Inverse Magnetic Catalysis: 〈ψψ〉(B)↘ near transitionquark back reaction: sea effect dominates, only at phys. massesPolyakov loop↗

anisotropies:magn. susceptibility: 〈ψσµνψ〉 ∝ Bfield strength: 〈tr E2

‖ 〉 < 〈tr E2⊥〉, 〈trB2

⊥〉 < 〈trB2‖〉

cf. perturbative Euler-Heisenbergquark action: 〈ψf /D‖ψf 〉 > 〈ψf /D⊥ψf 〉, dominatestopology: no anisotropy in correlator

(an)isotropic pressure:depends on scheme: fixed field vs. fluxallows to compute on the lattice magnetization from anisotropies:QCD vacuum paramagnetic (here renorm. still to perform)


Backup: more simulation details

tree-level improved gauge actionstout smeared staggered fermions (rooting trick)2 light quarks + strange quark, charges (qu,qd ,qs) = (2

3 ,−13 ,−

13)e

lattice spacing set at T = 0,B = 0physical pion massesset by fK , fK/mπ and fK/mK

T = 0: 243 × 32, 323 × 48 and 403 × 48 latticesT > 0: Nt = 6, 8, 10 meaning a = 0.2, 0.15, 0.12 fm

Ns = 16, 24, 32 for finite volumescondensates from stochastic estimator method with 40 vectors

magn. flux quanta: NB ≤ 70 < Nx Ny4 = 144


Backup: Nature of the transition

volume dependence of light susceptibility:

no volume scaling⇒ remains a crossover up to√

eB ' 1 GeV


Backup: Mass sensitivity

what if we put (mlight,mlight,mstrange)→ (mstrange,mstrange,mstrange)?

T -dep. of u-susceptibility (top) and change of u-condensate (bottom)⇒ effects of decreasing Tc & inverse magn. catalysis disappearlight quark masses are important


Backup: Physical valuesat T = 0 and MS scheme at 2 GeV:

τup = −(40.7± 1.3) MeVτdown = −(39.4± 1.4) MeV

τstrange = −(53.0± 7.2) MeV

χup = −(2.08± 0.08) GeV−2

χdown = −(2.02± 0.09) GeV−2

quenched unrenorm.: τup/down = −52 MeV Braguta, Buividovich et al. 10

QCD sum rules: χlight = −(2.11± 0.23) GeV−2 Ball, Braun, Kivel 03

vector dominance: χlight = − 2m2

ρ≈ −3.3 GeV−2

at finite T : |τlight| decreases like an order parameter

inflection point: Tc = 162(3)(3) MeV (compatible with T 〈ψψ〉c at B = 0)


Documents

Thermodynamic properties of QCD in external magnetic ...Thermodynamic properties of QCD in external magnetic ﬁelds: an update Falk Bruckmann SFB Meeting, Regensburg, April 2013 G