QCD in strong magnetic fieldM. N. Chernodub

CNRS, University of Tours, France

1. Link to experiment: Relevance to heavy-ion collisions [hot quark-gluon plasma and cold vacuum exposed to magnetic field]

2. Phase diagram of QCD in strong magnetic field [an unsolved (so far) mystery]

3. Nondissipative transport phenomena in QCD plasma [exotic interplay of QCD topology and strong magnetic field]

Graz, 25-26 November 2013

Plan maximum:

Magnetic field in heavy-ion collisions

Noncentral heavy-ion collisions should produce magnetic field.

The magnetic field is directed out of the collision plane. The duration of the field's pulse is very short however (typically, 1 fm/c).

B

Electromagnetism at work:

The generated magnetic field is very strong!

What is «very strong» field? Typical values:

● Thinking — human brain: 10-12 Tesla

● Earth's magnetic field: 10-5 Tesla

● Refrigerator magnet: 10-3 Tesla

● Loudspeaker magnet: 1 Tesla

● Levitating frogs: 10 Tesla

● Strongest field in Lab: 103 Tesla

● Typical neutron star: 106 Tesla

● Magnetar: 107...10 Tesla

● Heavy-ion collisions: 1015...16 Tesla

● Early Universe: even (much) higher

IgNobel 2000 by A.Geim (got Nobel 2010 for graphene)

Destructive explosion

How strong field is created?

Estimations: eBmax

≃ (2...3)·1016 Tesla

mp ≃ 0.02 GeV ≃ 3·1014 Tesla = 3·1018 Gauss 2

W. T. Deng and X. G. Huang, Phys.Rev. C85 (2012) 044907

A. Bzdak and V. Skokov, Phys.Lett. B710 (2012) 171+ Vladimir Skokov, private communication.

LHC

LHC

RHIC

Ultraperipheral (b < 2R)

2

Conversion of units:

Physical environment

Hot quark-gluon plasma T ~ 200 MeV or 2·1012 K

in strong magnetic fieldeB ~ (0.1...1) GeV2 or B ~ 1015..16 T

B

A natural question:What is the effect of the magnetic field on quark-gluon plasma?

A “simpler” question:What is the phase diagram of QCD in a background ofstrong magnetic field?

Phase diagram of QCD with B = 0

1) Hot quark-gluon plasma phase and cold hadron phase constitute, basically, one single phase because they are separated by a nonsingular transition (“crossover”). 2) The color superconductingphases at high baryon chemical potential m were extensively studied theoretically [they are out of reach of both lattice simulations and Earth-based experiments]

3) The LHC and RHIC experiments probe low baryon density physics. One can safely take m = 0 in further discussions.

From a BNL webpage

Finite-temperature structure of QCD (B = 0 & m = 0)

Two most important physical phenomena in low-T phase: 1) Quark confinement: No quarks and gluons in the physical spectrum; The physical degrees of freedom are hadrons (mesons and baryons)

2) Chiral symmetry breaking: The source of hadron masses (mesons and baryons are massive)

QCD Lagrangian:

Quark confinement (I): order parameter

The relevant order parameter is the Polyakov loop:

1) defined at finite temperature T in the Euclidean space-time (after a Wick rotation, t → t = - it = x4)

2) related to the free energy Fq of a single quark:

3) order parameter:

Quark confinement (II): lattice simulations

The expectation value of the Polyakov loop vs. temperature

Smooth transition (crossover)

Reminder:B = 0 & m = 0

Deconfinement[quark-gluon plasma]

Confinement[hadron phase]

[adapted from Borsanyi et al, JHEP 1009 (2010) 073, ArXiv:1005.3508]

Chiral symmetry breaking (I): The symmetry

Left/right quarks, for each flavor f :

If the quark masses are zero, M = 0, then the global internal continuous symmetries of QCD are as follows:

;projectors

with

The global symmetry group:

Lagrangian:

Chiral symmetry breaking (II): order parameter

The order parameter of the chiral symmetry is the chiral condensate:

In the hadronic phase of QCD the chiral condensate is nonzero and

the chiral symmetry subgroup

is broken spontaneously:

so that the allowed transformations are as follows:

and with

Baryon symmetry (unbroken)

Axial symmetry(broken by an anomaly)

Chiral symmetry breaking (III): lattice results

Chiral condensate vs. temperature (crossover transition):

[adapted from Borsanyi et al, JHEP 1009 (2010) 073, ArXiv:1005.3508]

Chiral symmetry is restored [QGP]

Chiral symmetry is broken [hadron phase]

Physical picture at B = 0

T0Restoration of

the chiral symmetryDeconfinement

transition

What happens with this picture in strong magnetic field?

Flavor symmetry breaking by magnetic field

It is an immediate effect due to the magnetic field background.

Consider QCD with two lightest quarks (Nf =2):

with Electromagnetic gauge field (fixed)

Gluon field

Electric charge operator (matrix)

Explicit flavor breaking:

Magnetic catalysis at zero temperature (I)

In strong magnetic field quarks and antiquarks pair more effectively!

S.P. Klevansky and R. H. Lemmer ('89); H. Suganuma and T. Tatsumi ('91) - effective modelsV. P. Gusynin, V. A. Miransky and I. A. Shovkovy ('94, '95, '96,...) → real QCDxQED

Enhancement of the chiral symmetry breaking at strong B

1) Dimensional reduction (3+1)D → (1+1)D: In a very strong magnetic field the dynamics of electrically charged particles (quarks, in our case) becomes effectively one-dimensional, because the particles tend to move along the magnetic field only.

2) Quarks interact stronger in one spatial dimension: In (1+1)D an arbitrarily weakest interaction between two objects leads to pair formation. This fact: (i) follows from Quantum Mechanics; (ii) is known as a “Cooper theorem” in solid state physics.

Why? Two reasons:

Dimensional reduction (I): energy spectrum

Energy of free relativistic fermion in strong magnetic field:

momentum alongthe magnetic field axis

nonnegative integer number

projection of spin onthe magnetic field axis

Gaps between the levels are

The infrared dynamics of a fermion in strong magnetic field is governed by the lowest Landau Level (LLL) with n = 0 and sz = sgn(q) ½:

The theory in the infrared region becomes effectively one dimensional!

Dimensional reduction (II): phase space

Integral over momentum is split into the sum over Landau levels:

For the LLL:

Dimensional reduction in the phase space:

(3+1)D phase element: (1+1)D phase element: Volume:

Degeneracy of the LLLs:

Dimensional reduction (III): visual illustration

d

negatively charged r meson, charge=-e,

spin=+1

positively charged r meson,

charge=+e,spin=+1

u

d

u

d

direction of the magnetic moments of the r mesons

gluons

ma

gne

tic fi

eld

spins of quarks andantiquarks

du

u

mesonsspin=0charge=0

due to the flavorbreaking thesemesons are different!

Dimensional reduction (IV): lattice visualization

Typical structure of fermionic modes on the lattice

[from studies by Buividovich et al, Phys.Lett. B682 (2010) 484]

Coming back: Magnetic catalysis at T = 0 (II)

1) The quark-antiquark pairing due to gluon exchange is enhanced due to the dimensional reduction.2) The pairing is more effective for u-quarks compared to d-quarks

[picture from Frasca, Ruggieri, Phys.Rev. D83 (2011) 094024]

Attractive channel: spin-0 flavor-diagonal states

1) enhances chiral symmetry breaking

2) breaks flavorsymmetry

eB (GeV2)

Magnetic field:

Magnetic catalysis + flavor breaking at T = 0

u-quarks should become heavier than d-quarks

Dynamical quark mass:

[Miransky, Shovkovy, Phys. Rev. D 66, 045006 (2002)]

Based on truncation of Schwinger-Dyson (gap) equations in QCD in strong-field limit at large number of colors.

with

Magnetic catalysis (III)

Change of the chiral condensate due to strong magnetic field

[G. Bali et al, Phys.Rev. D86 (2012) 071502]

Theoretical estimationsNumerical simulations of lattice QCD

Notations:

Chiral models:

Nambu-Jona-Lasinio model:

[Shushpanov, Smilga, Phys.Lett. B402 (1997) 351]

[Klevansky, Lemmer, Phys.Rev. D39 (1989) 3478]

Summary: basic chiral properties for T = 0

1) Explicit breaking of flavor (u ≠ d)

2) Dimensional reduction (3+1)D → (1+1)D

3) Enhancement of chiral symmetry breaking

(“magnetic catalysis”)

Quark (de)confinement: energy arguments (I)

Free energy arguments. 1) Consider lightest excitations: pions and quarks.2) Physical degrees of freedom: Confinement (cold) phase – pions; Deconfinement (hot) phase – quarks.3) Compare corresponding free energies and find which phase is better.

[the first attempt was done by Agasian and Fedorov, Phys.Lett. B663 (2008) 445]

Pion vacuum is diamagnetic. It does not like the magnetic field because the free energy of pions gets increased in magnetic field.

Quark vacuum is paramagnetic. It likes the magnetic field because the free energy of quarks gets decreased in magnetic field.

Warning 1: here we ignore the dynamics of gluons – which are not electrically charged anywayWarning 2: this is the first serious attempt to get the phase diagram. To be critically revised ...

Qualitative conclusion: magnetic field should lead to the deconfinement!

Quark (de)confinement: energy arguments (II)

Energy of free relativistic particle in strong magnetic field:

momentum alongthe magnetic field axis

nonnegative integer number

projection of spin onthe magnetic field axis

Diamagnetism of the bosonic gas:

Paramagnetism of the fermionic gas:

[Pauli paramagnetism, spin polarization]

Quark (de)confinement: energy arguments (III)

Phase diagram, based on energy arguments[Agasian and Fedorov, Phys.Lett. B663 (2008) 445]

First order line

Second order endpoint

Crossover line

Warning 3: The confinement phase was recently (2013) shown to be paramagnetic.Warning 4: The phase diagram was recently (2012) shown to be different.

Chiral restoration: magnetic catalysis arguments

1) T=0 experience: magnetic field enhances the chiral condensate

2) B=0 experience: Thermal fluctuations destroy the chiral condensate

3) Immediate conclusion: The critical temperature of the chiral phase transition is an increasing function of the magnetic field.

Naive phase diagram of QCD in magnetic field

Additional arguments were used:

1) At B = 0 the chiral and deconfinement transitions almost coincide;

2) Both these transitions are crossovers

3) Magnetic field may enhance the strength of the (chiral) transition

Warning 5: the arguments based on the magnetic catalysis contradict the recent (2012) lattice data; this phase diagram is not correct.

Enhancement of the chiral transition: [Mizher, Fraga, Phys.Rev. D78 (2008) 025016]

s-model coupled to quarks and to Polyakov loop (I)

Full Lagrangian:

Quarks: Mesons:

Polyakov loop (confining properties)

Chiral + Confining + Electromagnetic properties

s-model coupled to quarks and to Polyakov loop (II)

Polyakov-loop Lagrangian:

Potential comes from the phenomenology (describes well a finite-temperature Polyakov potential and thermodynamics in pure Yang-Mills theory without quarks):

Polyakov loop:

- critical temperature in pure SU(3) gauge theory

s-model coupled to quarks and to Polyakov loop (III)

Confining properties (no magnetic field)

Confinement Deconfinement

In the deconfinement phase the center symmetry is spontaneously broken:

s-model coupled to quarks and to Polyakov loop (IV)

Mean-field approximation. Free energy of the system:

Meson fields Quark contributionPolyakov loop

Hadronphase

QGP phase

Crossover

Explicit breaking of the center symmetry due to magnetic field

Confinement Deconfinement

Compare with

[A.Mizher, E.Fraga, M.Ch., Phys.Rev. D82 (2010) 105016]

with

s-model coupled to quarks and to Polyakov loop (V)

The effective potential for the s field

An issue with the vacuum corrections: effect of magnetic field on the T = 0 ground state (Euler-Heisenberg energy):

with the constituent quark mass and

without vacuum corrections with vacuum corrections

The phase diagram(s):

s-model coupled to quarks and to Polyakov loop (VI)

1) at small B: a crossover transition (at a very small region)2) first order transitions elsewhere3) chiral and deconfinement split4) both critical Tc raise with B

1) at small B: a crossover transition (at a very small region)2) first order transitions elsewhere3) chiral and deconf. Coincide4) both critical critical Tc decrease with B

Warning 6: the diagram without corrections is qualitatively correct, but there is a mismatch in the transitions' order.

Nambu-Jona-Lasinio model

The phase diagram(s):

Warning 8: both diagrams do not agree with the results of lattice simulations.

2-point + 4-point terms 2-point + 4-point + 8-point terms

[Gatto, Ruggieri, Phys. Rev. D 83, 034016 (2011)]

Lattice QCD: inverse magnetic catalysis

Surprise 1: At finite temperature the chiral condensate decreases with magnetic field (“inverse magnetic catalysis”)!

[G. S. Bali et al., JHEP 1202, 044 (2012)][G. S. Bali et al., JHEP 1202, 044 (2012)]

Lattice QCD: finite-T phase diagram

[G. S. Bali et al., JHEP 1202, 044 (2012)]

Surprise 2: Transitions converge, not split.Surprise 3: No enhancement of the transition (weak crossover remains weak crossover)

Note: shown maximal field strengths Bmax are too modest according to new estimations (2013).

Magnetic susceptibility

Magnetization:

Magnetic susceptibility*:

F is the free energy density.

Paramagnetic: - likes magnetic field (energy is lowered)

Diamagnetic: - dislikes magnetic field (energy is increased)

*) assuming linear behavior – surely valid for a weak magnetic fieldbut may also be valid for stronger fields if magnetization is linear in B

Magnetic susceptibility of quark-gluon plasma (QGP) and hot vacuum for weak magnetic fields

At weak magnetic fields both hot vacuum and QGP are paramagnetic!

[Bonati, D’Elia, Mariti, Negro, Sanfilippo, Phys. Rev. Lett. 111, 182001 (2013)]

[more detailed study by the same Authors in arXiv:1310.8656]

May lead to paramagnetic squeezing of the QGP fireball [Bali, Bruckmann, Endrodi, Schafer, arXiv:1311.2559]

Strong paramagnet: 10 times more paramagnetic compared to ordinary materials.

Magnetic susceptibility of cold vacuum at strong magnetic field

At strong field the vacuum becomes paramagnetic as well:

[Bali, Bruckmann, Gruber, Endrodi, Schafer, JHEP 1304 (2013) 130]

Do we understand the paramagnetic behavior?

1) Hot vacuum and QGP.YES: the paramagnetism of emerging quarks dominates over thediamagnetism of pions.

2) Cold vacuum at strongmagnetic fields: ? - next(the results at weak fields agree withHadron Resonance Gas model:[Endrodi, JHEP 1304 (2013) 023])

Phase diagram – continue

(T, m) phase diagram

(T, B) phase diagram

paramagnetic region

Possible superconducting phase at strong field

This claim seemingly contradicts textbooks which state that: 1. Superconductor is a material (= a form of matter, not an empty space)

2. Weak magnetic fields are suppressed by superconductivity 3. Strong magnetic fields destroy superconductivity

In a background of strong enough magnetic field the vacuum may become a superconductor.

The superconductivity emerges in empty space.Literally, “nothing becomes a superconductor”.

[M. Ch., PRD82 (2010) 085011; PRL 106 (2011) 142003]

General features of superconducting state

1. spontaneously emerges above the critical magnetic field

2. usual Meissner effect does not exist

3. perfect conductor (= zero DC resistance) in one spatial dimension (along the axis of the magnetic field).

4. No superconductivity in other (perpendicular) directions

5. Hyperbolic metamaterial (Smolyaninov, 2011 ): has a negative refraction index (“perfect lens”).

6. Strong paramagnet (contrary to a perfect diamagnetism of ordinary superconductors).

Bc ≃ 1016 Tesla = 1020 Gauss

eBc≃ mr ≃ 31 mp ≃ 0.6 GeV2or 2 2

Too strong critical magnetic field?

Over-critical magnetic fields (of the strength B ~ 2...3 Bc) may be generated in ultraperipheral heavy-ion collisions (duration is short, however – detailed calculations are required)

eBc≃ mr ≃ 31 mp ≃ 0.6 GeV2 2 2

W. T. Deng and X. G. Huang, Phys.Rev. C85 (2012) 044907

A. Bzdak and V. Skokov, Phys.Lett. B710 (2012) 171+ Vladimir Skokov, private communication.

LHC

LHC

RHIC

eBc

eBc

ultraperipheral

Conventional BCS superconductivity

1) The Cooper pair is the relevant degree of freedom!

2) The electrons are bounded into the Cooper pairs by the (attractive) phonon exchange.

Three basic ingredients:

The vacuum (T = 0) in strong magnetic field

Ingredients needed for possible superconductivity:

A. Presence of electric charges? Yes, we have them: there are virtual particles which may potentially become “real” (= pop up from the vacuum) and make the vacuum (super)conducting.

B. Reduction to 1+1 dimensions? Yes, we have this phenomenon: in a very strong magnetic field the dynamics of electrically charged particles (quarks, in our case) becomes effectively one-dimensional, because the particles tend to move along the magnetic field only.

C. Attractive interaction between the like-charged particles? Yes, we have it: the gluons provide attractive interaction between

the quarks and antiquarks (qu=+2 e/3 and qd=+e/3)

Pairing of quarks in strong magnetic field

Similar to the magnetic catalysis at T = 0

B0

B0 Bc

attractive channel: spin-0 flavor-diagonal states

attractive channel: spin-1 flavor-offdiagonal states (quantum numbers of r± mesons)

This talk:

enhances chiral symmetry breaking

electrically chargedcondensates: lead to electromagneticsuperconductivity

Naïve qualitative picture of quark pairing in the electrically charged vector channel: r mesons

- Energy of a relativistic particle in the external magnetic field Bext:

momentum alongthe magnetic field axis nonnegative integer number

projection of spin onthe magnetic field axis

(the external magnetic field is directed along the z-axis)

- Masses of ρ mesons and pions in the external magnetic field

- Masses of ρ mesons and pions:

becomes heavier

becomes lighter

Electrodynamics of ρ mesons = NSSM**NSSM - “naive simplest solvable model”

- Lagrangian (based on vector dominance models):

[D. Djukanovic, M. R. Schindler, J. Gegelia, S. Scherer, PRL (2005)]

- Tensor quantities - Covariant derivative

- Kawarabayashi-Suzuki- Riadzuddin-Fayyazuddin relation

- Gauge invariance

- Tensor quantities

Nonminimal couplingleads to g=2

Condensation of ρ mesons (mean-field)

masses in the external magnetic fieldKinematical impossibility of dominant decay modes

stops at certain valueof the magnetic field

- The decay

The pion becomes heavier while the r meson becomes lighter

Energy of the condensed state:Ginzburg-Landau potential for ordinary superconductivity:

Mean-field: the ρ± mesons become condense at certain Bc

Symmetries

- The condensate “locks” rotations around field axis and gauge transformations:

In terms of quarks, the state implies

Abelian gauge symmetry

Rotations around B-axis

(the same structure of the condensates in the Nambu-Jona-Lasinio model)

(similar to “color-flavor locking” in color superconductors at high quark density)

No light goldstone boson: it is “eaten” by the electromagnetic gauge field!

+ Discussion in the literature [Y.Hidaka and A. Yamamoto, arXiv:1209.0007; Chuan Li and Qing Wang, PLB, arXiv:1301.7009; M. Ch., PRD, arXiv:1209.3587]

Condensates of ρ mesons, solutionsSuperconducting condensate (charged ρ condensate):

Superfluid condensate (neutral ρ condensate)

Condensates of ρ mesons, solutionsSuperconducting condensate Superfluid condensate (charged rho mesons) (neutral rho mesons)

New objects, topological vortices, made of the rho-condensates

B = 1.01 Bc

similar results in holographic approaches by M. Ammon, Y. Bu, J. Erdmenger, P. Kerner, J. Shock, M. Strydom (2012)

The phases of the rho-meson fields wind around vortex

centers, at which the condensates vanish.

Anisotropic superconductivity(via an analogue of the London equations)

- Apply a weak electric field E to an ordinary superconductor

- Then one gets accelerating electric current along the electric field:

[London equation]

- In the QCDxQED vacuum, we get an accelerating electric current along the magnetic field B:

Written for an electric current averaged over one elementary (unit) rho-vortex cell

( )

Acoustic phonon spectrum

Transverse phonons Longitudinal phonons

Low-energy phonon spectrum:!!!

Normalized energy of the ρ meson condensate in the transverse plane.

Check x-y slice at fixed time t and distance z

Instead of a regular lattice structure we see an irregular vortex pattern (liquid/glass?) The vortices move as we move the slice.

V.Braguta et al, 2013

Condensate in the transverse plane: plane by plane

Normalized energy of the ρ meson condensate along the magnetic field.

Check x-z slice at fixed time t and coordinate y

Vortices are not straight and static: they are curvy moving one-dimensional (in 3d) structures.

Phase diagram – current status

Effects of magnetic field: (temporary) conclusions

Bad news: No solid/full picture so far …Good news: No solid/full picture so far … a very active (surging) area of research!

More or less understood:

1) Explicit breaking of flavor (u ≠ d)2) Dimensional reduction (3+1)D → (1+1)D3) Enhancement of chiral condensate (“magnetic catalysis”) at T = 04) Explicit breaking of the center group (“induced deconfinement”);

Not well understood:

5) Inverse magnetic catalysis at T > 0;6) Converging chiral and deconfinement transitions;7) No enhancement of finite-T transition strength;8) Strong paramagnetism both in weak magnetic fields (warm/hot plasma) and strong magnetic fields (cold vacuum)9) Electromagnetic superconductivity of cold vacuum?

The Chiral Magnetic Effect (CME)

Electric current is induced by applied magnetic field:

Spatial inversion (x → - x) symmetry (P-parity):● Electric current is a vector (parity-even quantity);● Magnetic field is a pseudovector (parity-odd quantity).

Thus, the CME medium should be parity-odd!

In other words, the spectrum of the medium which supports the CME should not be invariant under the spatial inversion transformation.

An example of a parity-odd system?

Consider a massless fermion:

Right handed Left handed

P-parity

The chirality (left/right) is a conserved number.

The CME in heavy-ion collisions (II)

Red: momentumBlue: spin

Electric charges:u-quark: q=+2e/3d-quark: q= - e/3

Role of topology:u

L → u

R

dL → d

R

Visual picture:

P-invariantplasma

of quarks(and gluons)

Sphaleron(instanton)

= topological

charge

P-oddplasma

of quarks(and gluons)

The CME in heavy-ion collisions (II)

Red: momentumBlue: spin

Electric charges:u-quark: q=+2e/3d-quark: q= - e/3

Role of topology:u

L → u

R

dL → d

R

Visual picture:

P-invariantplasma

of quarks(and gluons)

Sphaleron(instanton)

= topological

charge

P-oddplasma

of quarks(and gluons)

Why the CME is astonishing?

1) Because it gives an equilibrium dissipationless current! (similar to the electric current in superconductivity)

2) Because it exists in an interacting system: interactions cannot destroy the CME if they do not wash out the chiral imbalance.

3) Because the CME does not need a presence of a condensate. A) Thus, thermal fluctuations cannot destroy the CME. B) Thus, the dissipationless electric current may exist at high temperatures. BTW, the temperature of the quark-gluon plasma is of the order of 1012 Kelvin (200 MeV and more)

4) … now, an analogue of the CME is under very active search in condensed matter physics (for example, in suggested Weyl semi-metals). Why? Because the CME is a very good alternative to the room-temperature superconductivity.

Lattice simulations of the CME: a task (II)

4) Then, impose an external magnetic field.

5) According to the CME, the chirally-imbalanced quark ensembles will lead to the generation of electric current. [Depending on the sign of the topological charge, these currents may be positive or negative].

6) Check the presence of the electric current numerically!

Checked: [Buividovich et al, '09]

[QCD was studied in the quenched limit (= “backreaction of fermions on gluons was neglected”) because this feature (the presence of the vacuum quark loops) is not important for the CME]

Lattice simulations of the CME: currents (I)

eB = 0 eB = (500 MeV)2

eB = (1.1 GeV)2eB = (780 MeV)2

Typical density of the electric current

Lattice simulations of the CME: currents (II)

Typical density of the electric current in gradually increasing magnetic field. Each step in magnetic field is (350 MeV)2.

Lattice simulations of the CME: instanton

An illustration:

1) take an instanton-like gluon configuration (= “numerically obtained configuration of the non-Abelian field of a unit topological charge”)

2) Apply the external magnetic field.

3) Check for the existence of the electric current.

The induced current along the magnetic-field axis

The (absence of induced)current in the transverse (perpendicular) directions

A list of anomalous non-dissipative effects (I)

1) Chiral Magnetic Effect – the electric current is induced in the direction of the magnetic field due to the chiral imbalance:

2) Chiral Vortical Effect – the axial current is induced in the rotating quark-gluon plasma along the axis of rotation:

Note: The axial current = the difference in currents of right-handed and left-handed quarks.

[all formulae are written for one flavor of fermions]

How to understand the Chiral Vortical Effect.

Imagine massless fermions ...

Right handed Left handed

… in a rotating frame.

A list of anomalous non-dissipative effects (II)

3) Chiral Separation Effect – the axial current is induced in the direction of the magnetic field:

Notes: A) This effect is realized even if the plasma is chirally-trivial. B) is the the quark chemical potential (the plasma is dense).

4) Axial Magnetic Effect – the energy flow is induced in the direction of the axial (=chiral) magnetic field:

All these effects will become more complicated (richer) in a rotating, chirally-imbalanced dense plasma subjected to both the usual and axial magnetic fields.

The Axial Magnetic Effect

For one quark's flavor:

The energy flow is given by the off-diagonal component of the energy-momentum tensor

The axial magnetic field is the magnetic field which acts on left-handed and right-handed quarks in the opposite way.It can be described by the axial field in the Lagrangian:

The Axial Magnetic Effect vs. the Chiral Vortical Effect

For a many-flavor system:

The Axial Magnetic Effect:

The Chiral Vortical Effect:

Their anomalous conductivities are the same:

They have the same origin! (It is, BTW, quite impressive: the generation of the axial current in a rotating system is tightly related to the induction of the energy flow in a static system).

The Axial Magnetic Effect on the lattice

Task: simulate lattice QCD in the background of the axial magnetic field and study the energy flow of fermions

both in the direction of the field and in transverse direction .

[QCD was studied in the quenched limit (= “backreaction of fermions on gluon files was neglected”) because the vacuum fermion loops are not important for the AME]

Results [Braguta et al., '13]:a) the effect was not observed in low-temperature phase, where the plasma is absent. This is explained by the quark confinement as the individual quarks do not exist in this phase.b) the effect is negligible at the deconfinement phase transition.

The Axial Magnetic Effect on the lattice

c) the Axial Magnetic Effect is clearly seen in the high-temperature (deconfinement) phase:

However, the linear coefficient is one order of magnitude smaller than the one predicted by the theory (a renormalisation of the effect?)

Summary

The quark-gluon plasma exhibits various dissipationless transport phenomena in thermodynamic equilibrium:

A) Chiral Magnetic Effect: generation of the electric current by magnetic field

B) Axial Magnetic Effect: generation of the energy flow by axial magnetic field

C) Chiral Vortical Effect: generation of the axial current in rotating plasma

D) Chiral Separation Effect: generation of the axial current by magnetic field