Download QCD in strong magnetic field

QCD in strong magnetic field
M. N. Chernodub
CNRS, University of Tours, France
Plan maximum:
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
Magnetic field in heavy-ion collisions
Noncentral heavy-ion collisions
should produce magnetic field.
B
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).
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:
●
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
1 Tesla
IgNobel 2000 by
A.Geim (got Nobel
2010 for graphene)
Destructive explosion
How strong field is created?
Estimations: eBmax ≃ (2...3)·1016 Tesla
LHC
LHC
RHIC
Ultraperipheral (b < 2R)
W. T. Deng and X. G. Huang,
Phys.Rev. C85 (2012) 044907
Conversion of units:
2
2
A. Bzdak and V. Skokov,
Phys.Lett. B710 (2012) 171
+ Vladimir Skokov,
private communication.
mp ≃ 0.02 GeV ≃ 3·1014 Tesla = 3·1018 Gauss
Physical environment
Hot quark-gluon plasma
B
T ~ 200 MeV or 2·1012 K
in strong magnetic field
eB ~ (0.1...1) GeV2 or B ~ 1015..16 T
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 of
strong 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 superconducting
phases 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
From a BNL webpage
baryon density physics. One can safely take m = 0 in further discussions.
Finite-temperature structure of QCD (B = 0 & m = 0)
QCD Lagrangian:
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)
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)
Confinement
[hadron phase]
Deconfinement
[quark-gluon plasma]
Reminder:
B=0&m=0
[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 :
;
projectors
Lagrangian:
If the quark masses are zero, M = 0, then the global internal
continuous symmetries of QCD are as follows:
with
The global symmetry group:
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
Baryon symmetry
(unbroken)
the chiral symmetry subgroup
is broken spontaneously:
Axial symmetry
(broken by an anomaly)
so that the allowed transformations are as follows:
and
with
Chiral symmetry breaking (III): lattice results
Chiral condensate vs. temperature (crossover transition):
Chiral symmetry is
restored [QGP]
Chiral symmetry
is broken [hadron phase]
[adapted from Borsanyi et al, JHEP 1009 (2010) 073, ArXiv:1005.3508]
Physical picture at B = 0
0
T
Restoration of
the chiral symmetry
Deconfinement
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)
Enhancement of the chiral symmetry breaking at strong B
S.P. Klevansky and R. H. Lemmer ('89); H. Suganuma and T. Tatsumi ('91) - effective models
V. P. Gusynin, V. A. Miransky and I. A. Shovkovy ('94, '95, '96,...) → real QCDxQED
In strong magnetic field quarks and antiquarks pair more effectively!
Why? Two reasons:
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.
Dimensional reduction (I): energy spectrum
Energy of free relativistic fermion in strong magnetic field:
momentum along
the magnetic field axis
nonnegative
integer number
projection of spin on
the 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:
Degeneracy of the LLLs:
Dimensional reduction in the phase space:
(3+1)D phase element:
(1+1)D phase element:
Volume:
Dimensional reduction (III): visual illustration
mesons
spin=0
charge=0
negatively charged
r meson,
charge=-e,
spin=+1
u
magnetic field
due to the flavor
breaking these
mesons are
different!
u
d
direction of
the magnetic
moments of
the r mesons
d
d
u
spins of
quarks and
antiquarks
d
positively charged
r meson,
charge=+e,
spin=+1
u
gluons
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
Attractive channel: spin-0
flavor-diagonal states
Magnetic field:
1) enhances chiral
symmetry breaking
2) breaks flavor
symmetry
eB (GeV2)
[picture from Frasca, Ruggieri, Phys.Rev. D83 (2011) 094024]
Magnetic catalysis + flavor breaking at T = 0
u-quarks should become heavier than d-quarks
Dynamical quark mass:
with
Based on truncation of Schwinger-Dyson (gap) equations in QCD
in strong-field limit at large number of colors.
[Miransky, Shovkovy, Phys. Rev. D 66, 045006 (2002)]
Magnetic catalysis (III)
Numerical simulations of lattice QCD
Change of the chiral condensate
due to strong magnetic field
Theoretical estimations
Chiral models:
[Shushpanov, Smilga, Phys.Lett. B402 (1997) 351]
Nambu-Jona-Lasinio model:
[Klevansky, Lemmer, Phys.Rev. D39 (1989) 3478]
Notations:
[G. Bali et al, Phys.Rev. D86 (2012) 071502]
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.
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.
Qualitative conclusion: magnetic field should lead to the deconfinement!
[the first attempt was done by Agasian and Fedorov, Phys.Lett. B663 (2008) 445]
Warning 1: here we ignore the dynamics of gluons – which are not electrically charged anyway
Warning 2: this is the first serious attempt to get the phase diagram. To be critically revised ...
Quark (de)confinement: energy arguments (II)
Energy of free relativistic particle in strong magnetic field:
momentum along
the magnetic field axis
nonnegative
integer number
Diamagnetism of the bosonic gas:
Paramagnetism of the fermionic gas:
[Pauli paramagnetism, spin polarization]
projection of spin on
the magnetic field axis
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
Enhancement of the chiral transition:
[Mizher, Fraga, Phys.Rev. D78 (2008) 025016]
Warning 5: the arguments based on the magnetic catalysis contradict the recent (2012) lattice data;
this phase diagram is not correct.
s-model coupled to quarks and to Polyakov loop (I)
Full Lagrangian:
Quarks:
Polyakov loop
(confining properties)
Mesons:
Chiral + Confining + Electromagnetic properties
s-model coupled to quarks and to Polyakov loop (II)
Polyakov-loop Lagrangian:
Polyakov loop:
Potential comes from the phenomenology (describes well a finite-temperature
Polyakov potential and thermodynamics in pure Yang-Mills theory without quarks):
- 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
Polyakov loop
Quark contribution
Hadron
phase
Crossover
QGP phase
Explicit breaking of the center symmetry due to magnetic field
[A.Mizher, E.Fraga, M.Ch., Phys.Rev. D82 (2010) 105016]
Confinement
Deconfinement
Compare with
with
s-model coupled to quarks and to Polyakov loop (V)
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
The effective potential for the s field
without vacuum corrections
with vacuum corrections
s-model coupled to quarks and to Polyakov loop (VI)
The phase diagram(s):
1) at small B: a crossover transition
(at a very small region)
2) first order transitions elsewhere
3) chiral and deconfinement split
4) both critical Tc raise with B
Warning 6: the diagram without corrections is qualitatively
correct, but there is a mismatch in the transitions' order.
1) at small B: a crossover transition
(at a very small region)
2) first order transitions elsewhere
3) chiral and deconf. Coincide
4) both critical critical Tc decrease with B
Nambu-Jona-Lasinio model
The phase diagram(s):
2-point + 4-point terms
2-point + 4-point + 8-point terms
[Gatto, Ruggieri, Phys. Rev. D 83, 034016 (2011)]
Warning 8: both diagrams do not agree with the results of lattice simulations.
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)]
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 field
but 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]
Strong paramagnet: 10 times more
paramagnetic compared to ordinary materials.
May lead to paramagnetic squeezing
of the QGP fireball
[Bali, Bruckmann, Endrődi, Schäfer, arXiv:1311.2559]
Magnetic susceptibility of cold vacuum
at strong magnetic field
At strong field the vacuum becomes paramagnetic as well:
Do we understand the
paramagnetic behavior?
1) Hot vacuum and QGP.
YES: the paramagnetism
of emerging quarks
dominates over the
diamagnetism of pions.
2) Cold vacuum at strong
magnetic fields: ? - next
(the results at weak fields agree with
Hadron Resonance Gas model:
[Endrődi, JHEP 1304 (2013) 023])
[Bali, Bruckmann, Gruber, Endrődi, Schäfer, JHEP 1304 (2013) 130]
Phase diagram – continue
(T, B) phase diagram
paramagnetic region
(T, m) phase diagram
Possible superconducting phase at strong field
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”.
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
[M. Ch., PRD82 (2010) 085011; PRL 106 (2011) 142003]
General features of superconducting state
1. spontaneously emerges above the critical magnetic field
or
Bc ≃ 1016 Tesla = 1020 Gauss
2
2
2
eBc≃ mr ≃ 31 mp ≃ 0.6 GeV
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).
Too strong critical magnetic field?
eBc≃ mr ≃ 31
2
2
mp ≃ 0.6 GeV
2
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)
LHC
eBc
LHC
eBc
RHIC
ultraperipheral
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.
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
attractive channel: spin-0 flavor-diagonal states
enhances chiral
symmetry breaking
0
B
This talk:
±
attractive channel: spin-1 flavor-offdiagonal states (quantum numbers of r mesons)
electrically charged
condensates: lead
to electromagnetic
superconductivity
0
B
Bc
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 along
the magnetic field axis
nonnegative integer number
projection of spin on
the magnetic field axis
(the external magnetic field is directed along the z-axis)
- Masses of ρ mesons and pions in the external magnetic field
becomes heavier
becomes lighter
- Masses of ρ mesons and pions:
Electrodynamics of ρ mesons = NSSM*
*NSSM - “naive simplest solvable model”
- Lagrangian (based on vector dominance models):
Nonminimal
coupling
leads to g=2
- Tensor quantities
- Covariant derivative
- Kawarabayashi-SuzukiRiadzuddin-Fayyazuddin relation
- Gauge invariance
[D. Djukanovic, M. R. Schindler, J. Gegelia, S. Scherer, PRL (2005)]
Condensation of ρ mesons (mean-field)
Mean-field: the ρ± mesons become condense at certain Bc
masses in the external magnetic field
Kinematical impossibility
of dominant decay modes
The pion becomes heavier while
the r meson becomes lighter
- The decay
stops at certain value
of the magnetic field
Energy of the condensed state:
Ginzburg-Landau potential
for ordinary superconductivity:
Symmetries
In terms of quarks, the state
implies
(the same structure of the condensates
in the Nambu-Jona-Lasinio model)
Abelian gauge symmetry
Rotations around B-axis
- The condensate “locks” rotations around field axis and gauge transformations:
(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, solutions
Superconducting condensate (charged ρ condensate):
Superfluid condensate (neutral ρ condensate)
Condensates of ρ mesons, solutions
Superconducting condensate
(charged rho mesons)
Superfluid condensate
B = 1.01 Bc
(neutral rho mesons)
New objects, topological vortices, made of the rho-condensates
The phases of the rho-meson
fields wind around vortex
centers, at which the
condensates vanish.
similar results in holographic
approaches by M. Ammon,
Y. Bu, J. Erdmenger, P. Kerner,
J. Shock, M. Strydom (2012)
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
V.Braguta et al, 2013
Instead of a regular lattice structure we see an irregular vortex
pattern (liquid/glass?) The vortices move as we move the slice.
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
More or less understood:
1) Explicit breaking of flavor (u ≠ d)
2) Dimensional reduction (3+1)D → (1+1)D
3) Enhancement of chiral condensate (“magnetic catalysis”) at T = 0
4) 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?
Bad news: No solid/full picture so far …
Good news: No solid/full picture so far … a very active (surging) area of research!
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:
P-parity
Right handed
Left handed
The chirality (left/right) is a conserved number.
The CME in heavy-ion collisions (II)
Visual picture:
Red: momentum
Blue: spin
Electric charges:
u-quark: q=+2e/3
d-quark: q= - e/3
P-invariant
plasma
of quarks
(and gluons)
Sphaleron
(instanton)
=
topological
charge
Role of topology:
uL → uR
dL → dR
P-odd
plasma
of quarks
(and gluons)
The CME in heavy-ion collisions (II)
Visual picture:
Red: momentum
Blue: spin
Electric charges:
u-quark: q=+2e/3
d-quark: q= - e/3
P-invariant
plasma
of quarks
(and gluons)
Sphaleron
(instanton)
=
topological
charge
Role of topology:
uL → uR
dL → dR
P-odd
plasma
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)
Typical density of the electric current
eB = 0
eB = (780 MeV)
eB = (500 MeV)
2
eB = (1.1 GeV)
2
2
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:
[all formulae are written
for one flavor of fermions]
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.
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, chirallyimbalanced 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