Table Of ContentWhat flows in the chiral magnetic effect?
— Simulating the particle production with CP-breaking backgrounds —
Kenji Fukushima
Department of Physics, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Toaddressaquestionofwhetherthechiralmagneticcurrentisastaticpolarizationoragenuine
flow of charged particles, we elucidate the numerical formulation to simulate the net production
of right-handed particles and anomalous currents with CP-breaking background fields which cause
an imbalance between particles and anti-particles. For a concrete demonstration we numerically
imposepulsedelectricandmagneticfieldstoconfirmouranswertothequestionthattheproduced
net particles flow in the dynamical chiral magnetic effect. The rate for the particle production and
the chiral magnetic current generation is quantitatively consistent with the axial anomaly, while
5 they appear with a finite response time. We emphasize the importance to quantify the response
1 time that would suppress observable effects of the anomalous current.
0
2
g I. INTRODUCTION ization [6]. This possibility is nowadays referred to com-
u monly as the local parity violation (LPV). Theoretical
A and experimental challenges are still ongoing about the
In many quantum problems in physics it is highly de-
LPV detection (and no direct evidence has been found
3 manded to establish a numerical framework to simulate
for it). Along these lines the recognition of the inter-
2 full dynamical processes of particle production out of
play with pulsed magnetic field B between two heavy
equilibrium. Unsettled and important problems include
ions was a major breakthrough [7]; B on top of CP-odd
] theleptogenesisandthebaryogenesisfortheexplanation
h backgrounds would induce an electric current j in par-
of the baryon asymmetry of the Universe (BAU). It is
p allel to B, which is summarized in a compact formula
- well-known that Sakharov’s three conditions are needed
p of the chiral magnetic effect (CME) with chiral chemi-
for the BAU; namely, B-breaking process, C and CP vi-
e olation, and out-of-equilibrium. The particle production cal potential µ5 [8]. In a realistic situation of the HIC
h we can anticipate P and CP violation not only from the
[ on top of the electroweak sphaleron [1] accommodates a QCD sphalerons but also from the glasma [9] which is a
4 process with ∆(B +L) (cid:54)= 0, which means that B +L description of the initial condition of the HIC in terms
decays toward chemical-equilibrium [and this is why the
v idea based on the SU(5) grand unified theory [2] does of coherent fields. Then, without introducing µ5 that
0 hashugetheoreticaluncertainty(seeRef.[10]forarecent
notworkoutfortheBAU].Therefore,iftheleptogenesis
4 study with µ ), we can and should directly compute the
9 withsome(B−L)-breakingprocessbeyondtheStandard 5
anomalous currents associated with the particle produc-
1 Model(seeRef.[3]forexample)generatesL(cid:54)=0initially,
tion with CP-odd backgrounds composed from chromo-
0 itwouldamounttoB (cid:54)=0finallyinchemical-equilibrium.
electric and chromo-magnetic fields [11].
1. The particle production associated with the sphaleron
It would be an intriguing attempt to test the CME in
0 transition has a character of topological invariance and
a table-top experiment in a way similar to the quantum
5 the index theorem relates the topological number to the
Halleffect. ThebiggestdifferencebetweentheCMEand
1 change in the particle number. We must recall, how-
: the Hall effect arises from the fact that the carriers of
v ever, that the sphaleron is a special static configuration
electric charge are not the ordinary particles but chiral
i atsaddle-pointandformoregeneralandnon-topological
X particles; that is, a non-zero excess of the right-handed
gauge configurations with C- and CP-odd components
particlenumbertotheleft-handedparticlenumberisre-
r we have to consider microscopic simulations with Weyl
a quired[12]. Itisquitenon-trivialhowtoimplementsuch
fermions. Also, to compute the momentum spectrum
a chiral battery (usually represented by µ (cid:54)= 0) in ma-
of produced particles with arbitrary gauge backgrounds, 5
terials. JustrecentlytheCMEmayhavebeenconfirmed
thereisnolongermathematicaleleganceandsomebrute-
inasystemrathersimilartotheglasma, inwhichchiral-
forcetacticswithnumericalsimulationsisindispensable.
ityimbalanceisimposedbybackgroundfieldsthatbreak
The sphaleron transition rate is proportional to T4 CP symmetry [13].
where T is the temperature [4], and so such topologi- Here, to elucidate the motivation of our present study
cal fluctuations should be abundant also in the strong clearly, let us discuss theoretical subtleties about the in-
interaction when the physical system is heated up to terpretation of the chiral magnetic (and separation) ef-
T > ΛQCD as argued in Ref. [5]. Such high tempera- fect.
ture is indeed realized in the quark-gluon plasma cre-
ated in the relativistic heavy-ion collision (HIC) exper- 1. What carries the electric charge of the current?
iment. Although the strong interaction does not break In the derivation using the Chern-Simons-Maxwell
CP (except for negligibly small strong θ), we can still theory [14] the chiral magnetic current appears
anticipate local violation of P and CP before thermal- in Amp`ere’s law, but it takes a similar form as
2
y
Maxwell’s displacement current. We know that
Maxwell’s displacement current is a source for the y jy
B
magnetic field but no charge carrier flows (though
thePoyntingvectorshowsaflowofelectromagnetic
x
energy) as pointed out in Ref. [15].
2. Whatisthemomentumspectrumofchargedparti- jz
clesthatflowasthechiralmagneticcurrent? Inthe
derivationusingthethermodynamicpotential[8]a jx Bz z
finiteresultremainsafteracancellationbetweenin-
finitely large momentum contributions pz ∼ ±∞, Ez
but it is unlikely in any experiment that particles
withsuchlargemomentaemergefromthevacuum. FIG. 1. Schematic view of currents induced by By in a CP-
Inthekineticderivation[16–18]athermaldistribu- odd domain realized by parallel Ez and Bz. The currents
tion of particles is finally assumed to retrieve the flowinallthex,y,andz directions;jx istheanomalousHall
exact CME coefficient, which provides us with a current and jy is the CME current.
theoretically correct description of the dynamical
current. It is, however, still needed to understand
how such a distribution of particles arises. In gen- kinetic derivation may look like a problem of one-
eralwitharbitrary(non-thermal)initialconditions particle motion, but it should be justified by the
the particle distribution functions are not neces- worldlineformalismonthequantumlevel,andthen
sarily thermal and then the CME current may not the(proper)timemaybegivenameaningdifferent
obey the standard formula, which also motivates from the genuine time.
the numerical simulation.
5. What is the response time for the chiral magnetic
3. Theexpression(cid:104)Ω|j|Ω(cid:105)=(cid:54) 0doesnotalwaysmeana current to get activated? It is unlikely that the
flowing current but it may represent a polarization current suddenly starts flowing as soon as B and
that is a static expectation value of a vector oper- µ are turned on. (This is actually an unavoid-
5
ator j, where |Ω(cid:105) represents a certain pure state of able problem to simulate the chiral magnetic effect
thephysicalsystem. (Wecangeneralizeourdiscus- assuming a certain µ .) The current generation
5
sions to mixed states, but the consideration with a ratehasbeencalculatedonlyinanidealizedsetup,
pure state suffices for our purpose.) In the lattice- andthequantitativeestimateofthisresponsetime
QCDsimulationinEuclidspacetimeasinRef.[19], should be crucial for experimental detection in re-
|Ω(cid:105)istheEuclideanQCDvacuum,andthecurrent alistic and thus disturbed environments.
measuredwith|Ω(cid:105)shouldbeapolarizationbecause
thecurrentisanon-equilibriumandsteadydynam- To answer these question, in this work, we consider
ical phenomenon. Actually the chiral separation a special setup as illustrated in Fig. 1, which is the
effect (cid:104)Ω|j |Ω(cid:105) ∝ µB has a natural interpretation schematic view of the CME setup in the HIC and in the
A
as a spin polarization [20] (see also Ref. [21] for condensedmatterexperiment. TheparallelE andB (in
a recent study on this). To make a possible dif- the z direction in Fig. 1) form P- and CP-odd product
ference explicit we point out an example found in E·B, and the CME current jy is induced in a direction
the computation of the chiral magnetic conductiv- perpendicular to E, which is reminiscent of the anoma-
ityσ (ω,p)thatrepresentsalinearresponseinthe lous Hall current jx. (In this sense one can regard the
χ
presence of spacetime dependent B(ω,p). From CME as a 3D extension of the Hall effect.) In the HIC
Fig. 2 of Ref. [22] it is obvious that the static limit Ez and Bz are Abelian projected chromo-electric and
σ (ω = 0,p → 0) takes a value different from the chromo-magnetic fields [11]. In the sphaleron transition
χ
dynamical limit σ (ω →0,p=0). thesefieldsareprovidedfromtheAbelianprojectedpart
χ
of the sphaleron gauge configuration too.
4. Howcantheparticlemotionbealignedtothemag- Then, we can address the questions in a very concrete
netic field? The conventional explanation (i.e., the way and we shall present short answers in order:
correlation between the helicity and the spin align-
mentunderastrongB)isbasedonastaticpicture 1. The produced particles from the vacuum fluctua-
corresponding to the polarization phenomenon. If tions form a current. The right-handed particles
there are free right-handed fermions, say, in the andanti-particleshaveasymmetricmomentumdis-
direction perpendicular to B, they just move on tributions due to E · B (cid:54)= 0, where the particle
a circle with the Larmor radius classically. The number is conserved with the opposite excess from
spin receives a torque from the Berry’s curvature theleft-handedsector(andthecurrentisdoubled).
term, but the spin cannot be aligned to B with- Therefore, the chiral magnetic effect can be real-
out dissipation. Also we make a comment that the ized in a non-equilibrium environment and is most
3
relevant to the early-time dynamics of the HIC in- states where the interaction is turned off. In general a
volving the particle production. constant background A may be coupled even without
µ
interaction. In the present work we set a gauge that
2. Produced particles are accelerated by E and the
makes A =0 for a technical reason. We can then write
momentum should have a peak around an impulse 0
positive-energyparticlesolutionsasφ =u (p;A)e−ip·x
given by the product of the Lorentz force times R R
with
theduration,whichistheordinarysituationinthe
Schwinger mechanism with pure E. In the same (cid:18) (cid:112)|p |+pz (cid:19)
wBay(cid:54)=w0ecaasnwceolml apsutEet(cid:54)=he0ditshtraitbuistioloncafulinzcetdioinnwmitoh- uR(p;A)= eiθ(pA)(cid:112)A|pA|−A pzA (2)
mentum space and so regions with infinitely large
in a certain gauge [or a convention for the overall U(1)
momenta are completely irrelevant.
phase]. Here we defined p ≡ p∓eA and the phase
±A
3. The particle production process is the real-time factor is:
phenomenon out of equilibrium. In this sense, ac-
px +ipy
ccolerdpirnogdutoctioounr,u(cid:104)Ωnd|jers|tΩa(cid:105)n∝dinµgBbacasendboeninttheerppraertetid- eiθ(pA) = (cid:112)(pxA)2+(Apy)2 . (3)
A A A
as a dynamical flow of the chirality, which results
simply from a different combination of the right- We can identify the anti-particle state from φR¯ =
handed and the left-handed sectors of the present −iσ2φ∗R, which leads to a relation: uR¯(p;A) =
calculation. This fact supports the realization of uR(−p;A). In the same way we can find the
non-zero quadrupole moment of the electric charge negative-energy particle and anti-particle solutions
suggested from the chiral magnetic wave. We note with φR/R¯ = vR/R¯(p;A)e+ip·x, which results in
thatif(cid:104)|Ω|jA|Ω(cid:105)wereaspinpolarizationoramag- vR(p;A) = −e−iθ(p−A)uR(p;−A) and vR¯(p;A) =
netization,therewouldbenomovementofchirality −e−iθ(pA)uR(−p;−A). We note that uR(p;A) and
in space, which is not the case in our fully dynam- vR¯(p;A) have an energy ±|pA|, while other two,
ical setup. uR¯(p;A) and vR(p;A), have an energy ±|p−A|.
For the problem of particle and anti-particle produc-
4. The particle motion has a peculiar angle distribu-
tion we evaluate an amplitude for the transition from a
tionorasymmetricmomentumdistribution(which
negative-energy state (with momentum p and vector po-
willbedemonstratedinFig.2)reflectingtheconfig-
tential A) to a positive-energy state (with momentum q
urationsofEandB. Thus,unliketheconventional
andvectorpotentialA(cid:48)),whichwecanexplicitlyexpress
explanation, there is no need to bend the motion
as
of classical particles nor the direction of the spin
polarization with B only.
β =(cid:90) d3xu†R(qA(cid:48))ei|qA(cid:48)|x0+iq·x g (x0,x),
5. One might think that the asymmetric momentum q,p (cid:112)2|q | −p
A(cid:48)
distributionoccursfromthebeginningoftheparti- (4)
cleproductionandsothereisnodelayforthetopo- β¯ =(cid:90) d3xu†R¯(q−A(cid:48))ei|q−A(cid:48)|x0+iq·x g¯ (x0,x)
logical currents to flow. The plane wave, however, q,p (cid:112)2|q | −p
−A(cid:48)
shouldbedeformedintotheLandauwave-function
after a finite B is switched on. The response time for the particles and the anti-particles, respectively. We
isgovernedbythedevelopmentintheamplitudeof here introduced new functions, g (x) and g¯ (x), as so-
p p
theLandauwave-function. Inparticulartheonsets lutions of the particle and the anti-particle equations of
of the particle production and the current genera- motionsatisfyingthefollowingnegativeenergyboundary
tion may look different due to different rates. conditions:
g (x) −→ vR(p−A)ei|p−A|x0−ip·x ,
II. THEORY OF THE PARTICLE AND p (cid:112)
2|p |
CURRENT PRODUCTION −A (5)
g¯ (x) −→ vR¯(pA)ei|pA|x0−ip·x
In this work we focus on the right-handed sector only p (cid:112)2|p |
A
and the current (net particle) density should be doubled
(canceled) once the left-handed sector is taken into ac- for x0 around the initial time in the past. It should be
count. Theright-handedWeylfermionshouldsatisfythe noted that the equation of motion for the anti-particle
following equation of motion: is not Eq. (1) but e is replaced with −e and σµ with
(cid:0)iσµ∂ −eσµA (cid:1)φ =0 (1) σ¯µ. Finally we can express the net particle number (i.e.
µ µ R
the particle number minus the anti-particle number in
with σµ = (1,σ). We can readily construct a complete the right-handed sector) as well as the spatial currents
set of plane-wave solutions of Eq. (1) for the asymptotic in the following manner (after dropping the zero-point
4
oscillation term from J0): where(cid:15)¯(t)=(cid:82)t (cid:15)(t(cid:48))dt(cid:48). WenotethatAz(t=−∞)=0
−∞
and Az(t = ∞) = −E T (cid:54)= 0 and this is why we should
J0 =e(cid:90) d3q (cid:2)f(q)−f¯(q)(cid:3), keep pA in all the ex0pressions for produced particles.
(2π)3 The above vector potentials actually lead to the mag-
(6)
(cid:90) d3q (cid:20) q q (cid:21) netic fields as almost desired:
J =e A(cid:48) f(q)− −A(cid:48) f¯(q) ,
(2π)3 |q | |q |
A(cid:48) −A(cid:48) Bx(t)=0, By(t)=B (cid:15)(t), Bz(t)=B (cid:15)(t), (11)
⊥ (cid:107)
wherethedistributionfunctionsf(q)andf¯(q)aboveare
defined with the amplitudes integrated over all incoming whereas the electric fields have some extra components:
momenta, i.e.,
Ex(t)=−B (cid:15)(cid:48)(t)z+B (cid:15)(cid:48)(t)y , Ey(t)=0,
⊥ (cid:107)
(cid:90) d3p (cid:90) d3p (12)
f(q)≡ (2π)3 |βq,p|2, f¯(q)≡ (2π)3 |β¯q,p|2 . (7) Ez(t)=E0(cid:15)(t).
We chose the gauge (10) so that we can have Ey(t) = 0
These functions are naturally given an interpretation as
because we are interested in jy. This also indicates that
the distribution functions of particles and anti-particles.
the current jx has not only the anomalous component
We note that J0 above is a formula equivalent to that
but a finite contribution induced by Ex (cid:54)=0.
as utilized in Ref. [23]. We also make a comment on the
We note that in our present calculation we neglect the
relation to the chiral kinetic theory where the current is
defined with x˙ and the equation of motion solves b˙x in back-reaction and so this jx would not affect jy. Be-
cause we start the simulation with the null initial condi-
terms of q and an additional contribution from Berry’s
tion with zero particle, this approximation for the back-
curvature. The appearance of the canonical momenta
q (shifted by A(cid:48)) in Eq. (6) corresponds to using b˙x reaction is expected to hold well especially for our setup
A(cid:48) with pulsed fields. In principle our formulation can be
on the level of the equation of motion. Still, it is highly
upgraded to the classical statistical simulation contain-
non-trivialhowtoseetheanalyticalrelationbetweentwo
ingback-reaction[27]ortheDiracequationscanbecom-
approaches, which will be an interesting future problem.
bined with a numerical solution of Maxwell’s equations.
The classical evolution suffices for our present purpose
to figure out the net particle production and associated
III. SIMULATION SETUP
CME current generation.
Letusnowgointosomemoretechnicalparts. Itiscon-
We here consider pulsed fields approximated by step
venient to keep the reflection symmetry of axes in order
functions for a duration T. We choose the origin of the
toavoidnon-vanishingcurrentsduetolatticeartifact. A
time so that we start solving Eq. (1) numerically from
perturbationtheoryintermsofAi wouldgiveatermlike
t=0. Introducing a temporal profile defined by
∝z,y that should be vanishing after the spatial integra-
(cid:40) tions. Thiscanbenon-perturbativelyrealizedonthelat-
1 (−T/2 ≤ t−t ≤ T/2)
(cid:15)(t)≡ 0 (8) ticebytheperiodicityofthelinkvariables. Alternatively,
0 (|t−t |>T/2),
0 ifwetakethespatialsitesni(thatgivesxi =niawiththe
lattice spacing a) from −N to +N for i=x,y,z, these
we can explicitly specify the electric and magnetic fields i i
terms ∝ z,y trivially disappear. The spatial volume is
that we want to consider here as
thus (2N +1)·(2N +1)·(2N +1)·a3 in the present
x y z
Ez(t)∼Bz(t)∼By(t)∝(cid:15)(t). (9) simulation. The corresponding momenta are discretized
as pia = 2πki/(2N +1). We also note that we impose
i
Thischoiceofthetemporalprofileismotivatedbasedon the anti-periodic boundary condition to avoid the singu-
theglasmadynamicsinwhichboththeexternalmagnetic larity at |p | = 0 in Eq. (2) or in Eq. (3). This means
A
field and the chromo-fields decay within the same time thatki shouldtakehalfintegralvaluesfrom−N +1/2to
i
scale of ∼ 0.1 fm/c [24, 25]. Theoretically speaking, to +N +1/2. Weemphasizeherethat,ifthespatialvolume
i
define the produced particle number uniquely, we should is large enough, it does not matter whether the bound-
setuptheasymptoticstateswhereinteractionisswitched ary condition is periodic or anti-periodic. This is simply
off. We also mention another possibility to prescribe the aprescriptionnottohitthesingularityat|p |=0. One
A
particlenumberinatransientstate(mostlyrelyingonan mightthinkthatonecouldinsertasmallregulatorinthe
adiabatic approximation) as studied in Ref. [26], though denominator like the i(cid:15) prescription. Indeed, one could
we do not adopt it. adopt the periodic boundary condition and could take
Itis,however,technicallynon-trivialhowtorealizethe anaverageofcontributionsfromsixneighbors, px =±(cid:15),
A
external fields as Eq. (9) strictly. Here, we will make a py = ±(cid:15), pz = ±(cid:15) around |p | = 0. As seen in mo-
A A A
simpler choice of the gauge potentials: mentum space, however, such an averaging procedure is
effectivelyidenticaltoimposingtheanti-periodicbound-
Ax =B (cid:15)(t)z−B (cid:15)(t)y , Ay =0,
⊥ (cid:107) ary condition. In any case we should carefully treat the
(10)
Az =−E (cid:15)¯(t), singularity at |p | = 0 because this infrared singularity
0 A
5
Particle 2nd-orderRunge-Kuttamethod(withwhichwecarefully
Anti-particle
checkedthattheaccuracyisgoodenoughforourpresent
1.0
simulation).
nn
utioutio 0.8 WedidnotquantizethefieldstrengthunlikeRef.[29].
DistribDistrib 0.6 Aqucatuntailzlyedwfieeldcosm(ipnartehde ruensuitltsofw2iπth/NuxnNqyuafnotrizBedz eatncd)
cle cle 0.4 and found that this would make only negligible differ-
artiarti 0.2 ence. Becauseourcalculationisclassicalforgaugefields,
PP
non-quantization of the field strength is harmless for our
−π/2 numericalresults,andmoreover,technicallyspeaking,we
−π/4 π/2
qz 0 π/4 −π/4 0 qy π/4 dfioeldnsotuhnalevsestoweimspolovseetahbeoeuqnudaatriyoncoonfdmitiootniofnorctohnetagianuingge
π/2 −π/2
the derivatives of the gauge fields.
Even if B = B = 0, a finite E would induce the
FIG. 2. Distribution of the produced particles and anti- (cid:107) ⊥ 0
particlesasfunctionsofqyandqz(andintegratedwithrespect pair production of particles and anti-particles under the
to qx; see the text). The lattice size is N = N = N = 12 Schwinger mechanism (see Ref. [30] for a comprehensive
x y z
and the magnetic fields are B =B =E /2. review). Because we deal with Weyl fermion, the pair
(cid:107) ⊥ 0
productionisalwayspossibleforanyE (cid:54)=0,whichleads
0
to a finite distribution of particles with −E T (cid:46) pz (cid:46) 0
0
is responsible for the axial anomaly, which is manifested and px = py = 0 (and with 0 (cid:46) pz (cid:46) E0T for anti-
in the representation with Berry’s curvature [16–18]. particles). If B(cid:107) or B⊥ is finite, this distribution is
Ifweintegrateoverallmomenta,wecannotavoidpick- smearedinmomentumspaceasclearlyobservedinFig.2
ing up all contributions from the doublers and the net in which we present (cid:82)(dqx/2π)f(q) and (cid:82)(dqx/2π)f¯(q)
particle production should be then absolutely zero be- as functions of qy and qz in the case with B(cid:107) = B⊥ =
cause the axial anomaly is exactly canceled according E0/2. In addition to smearing, it is evident in Fig. 2
to the Nielsen-Ninomiya theorem [28]. It is thus cru- that the particle is more enhanced over the anti-particle
cial to get rid of the doubler contributions properly. To becauseoftheCP-oddbackground,whichsignalsforthe
this end we limit our momentum integration range to net productionofright-handedparticles. (Forthevector
the half Brillouin zone only, i.e., ki from −N /2+1/2 gauge theory the total particle number is conserved as a
i
to +N /2 − 1/2 (which is used in Ref. [23] too). We result of summing right-handed and left-handed contri-
i
note that the axial anomaly is correctly captured from butions up.)
the singular contributions in the wave-function around Toclosethissectionweshallmakesomeremarksabout
|p | = 0, though it usually appears near the ultraviolet the lattice convention. Taking the sum over dimension-
A
cutoff in a diagrammatic approach. (We have numeri- less momenta ki corresponds to the phase-space integral
cally checked that our results are robust against small V (cid:82) d3p/(2π)3. In our definition βq,p has a mass dimen-
shiftsatthemomentumedges, whichisalsounderstand- sion of V and so f(p) has a mass dimension of V too.
able from Fig. 2; produced particles are centered with Then, J0 and J become dimensionless quantities. In
small momenta.) One should not be confused with the thesefinalexpressions,however,theprescriptiontoavoid
situation in Euclidean, i.e., static lattice simulation in the doubler contributions makes the phase-space volume
which the whole Dirac determinant should be evaluated smaller by a factor N3/(2N +1)3, which should be cor-
toidentifythestatisticalweightforgaugeconfigurations. rected in the end. In this paper we would not show the
For the present purpose, in contrast, we need the infor- lattice version of expressions, but for example, qi in
±A
mation on each momentum mode, and so the computa- Eq. (6) should be understood as sin[(qi∓Ai)a].
tion is much more time consuming than evaluating the
determinant, but it is easier to get rid of the doublers on
the mode-by-mode basis. IV. NUMERICAL RESULTS
Whenwechangethelatticesizeandthelatticespacing
a, weshouldchangeadimensionlessE0a2 accordinglyto Inwhatfollowswenormalizej0 =J0/V andj =J/V
fix E in the physical unit. In this work we choose a
0 by the following quantity:
reference lattice size as N =N =N =8 (i.e., 173 lat-
x y z
ticevolume)andscalephysicalquantitiessuchasE0a2 ∝ E2
172/(2N+1)2 withvaryingNx =Ny =Nz =N. Tosim- n0 = 4π03 ·T , (13)
plify the notation let us absorb e in the field definition
and omit a hereafter. Then, we take E0(N = 8) = 0.1 which is the pair production rate E2/4π3 in the
0
and change B(cid:107) and B⊥ from 0 to E0. We set the dura- Schwinger mechanism multiplied by the duration T,
(cid:112)
tion as T = 10/E (that is, T = 10 for N = 8). Our which gives us a natural order-of-magnitude estimate.
0
choice of t is t =0.6T and we continue solving Eq. (1) Then, j0/n should remain to be a reasonable number,
0 0 0
up to t = 1.2T with time spacing ∆t = 0.02T using the which is confirmed in Fig. 3.
6
173
16 213
253
12 1.0
jj00 // nn00 8 onon 0.8
utiuti
4 stribstrib 0.6
DiDi
1 0.8 0.6 0.4 0.4 0.6 0.8 1 Particle Particle 00..24
B|| / E0 0.2 0 0 0.2 B⊥ / E0
−π/2
FasIGa.f3u.ncNtieotnpoafrtBic(cid:107)leadnednsBit⊥y.j0Tnhoerlmatatliiczeedsibzeyins0Ninx E=qN. (y13=) q−zπ/4 0 π/4 π/2 −π/2 −π/4 0 qy π/4 π/2
N =8,10,12 from the bottom (solid) to the top (dotted).
z
FIG. 5. Distribution of the produced particles as functions
of qy and qz (and integrated with respect to qx) for different
lattice sizes 173, 213, and 253. The magnetic fields are B =
(cid:107)
5 B⊥ =E0/2 and the duration is 0.5T.
4
3
jjyy // nn means E ·B = 0, and so, no net particle production is
00
2 allowed (even though the pair production is possible as
1 longasE (cid:54)=0). Thecurrentisvanishingthenduetothe
lack of electric carrier.
0 1
1 0.8
0.8
0.6
0.6
B|| / E0 0.4 0.2 0 0 0.2 0.4 B⊥ / E0 V. LATTICE-SIZE DEPENDENCE AND THE
RESPONSE TIME
FIG. 4. Net current density jy normalized by n which is to
0
be interpreted as the CME current. The lattice-size dependence that we have seen is the
ultraviolet extrapolation, for which the box size is fixed
and the lattice spacing is decreased. Therefore, E in
0
Figure 3 clearly evidences for a non-zero value of the the unit of the lattice spacing was rescaled to fix the ob-
net particle production for E · B = E B (cid:54)= 0. We servables in the physical unit. In the present case the
0 (cid:107)
can make it sure that the net particle production is van- numerical results should be stable for the ultraviolet ex-
ishing when B = 0. This should be trivially so but a trapolation. We can understand this from Fig. 5. As
(cid:107)
non-symmetric treatment of coordinates would pick up mentioned before, the distribution function is well local-
unphysical artifact, which is not the case in the present ized in momentum space, and so the contributions from
simulation. In Fig. 3 the lattice size is N =N =N = the high-momentum region simply attach a tail in the
x y z
8,10,12(i.e.,173,213,253 latticevolumes)fromthebot- particle distribution as shown in Fig. 5.
tom(solid)tothetop(dotted). WecanconcludethatN In our definition f(q) does not go to zero for |q|→∞
i
dependenceisrathermild,whileitisstillthere. Infactj0 when E (cid:54)= 0 and B (cid:54)= 0. If the subtraction of the zero-
shows moderate dependence on B , which results from pointoscillationenergyisnormalizedatq|→∞wecould
⊥
the lattice artifact. In the steady system with constant removeanirrelevantshiftinthedistributionfunction. In
electromagneticfieldstheanomalousparticleproduction principle, however, such a constant shift does not affect
rate has only weak dependence on B and it slightly in- j0 andj: theparticleandtheanti-particlecontributions
⊥
creases with increasing B . We can see a tendency that cancel for J0 and there is no contribution to the angle
⊥
j0 at smaller (larger) B is pushed down (up) as N in- integration of (cid:82) d3p for j. From this consideration we
⊥ i
creases. canexpectbetterultravioletconvergenceforj becausea
Now we are going to discuss j/n obtained in our nu- constant shift vanishes for each of the particle the anti-
0
merical calculation. Here let us focus only on the chiral particle parts, while j0 needs a delicate cancellation. We
magnetic current jy and postpone discussions about the will see that this expectation is indeed the case in our
anomalousHallcurrentjxandOhm’scurrentjz toasep- numerical simulation.
arate publication [31]. It is interesting that our results Itwouldbeofpragmaticimportancetoconsiderj0and
in Fig. 4 surely represents a non-zero CME current as jy asafunctionofthepulseduration,thatis,thelifetime
(cid:112)
theoretically expected. We see that jy = 0 when either of the external fields. So far, we fixed it as T = 10/E
0
B or B is vanishing. Because jy is the CME current, and let us vary it now. Theoretically speaking, for a
(cid:107) ⊥
it is naturally vanishing if B =0. Furthermore, B =0 sufficiently large duration, we should expect linear de-
⊥ (cid:107)
7
scaling behavior. Since we need to go far into techni-
1.5
cal details to address this problem, we will leave it for
1.2 j0 / n another publication that will be focused on the lattice
0
formulation and the systematic checks of the lattice-size
dependence.
0.9
0.6 jy / n
0 VI. SUMMARY AND OUTLOOK
0.3
We formulated the production of particles and anti-
0 particles and checked its validity by looking for a right-
handed particle excess on CP-odd backgrounds. We lim-
-0.3 ited our main concern to the CME current at present,
0 0.2T 0.4T 0.6T 0.8T T but this type of the calculation should be applicable for
Duration a wider range of physical problems. For example, the
(pair) particle creation at the horizon of the acoustic
FIG.6. Profileofj0andjy asafunctionofthepulseduration blackhole(seeRef.[32]forapedagogicalarticle)attracts
normalized by n0. The lattice size is N = 8 (solid line), 10 attention, and it would be feasible to combine it with
(dashed line), 12 (dash-dotted line). The magnetic fields are
CP-odd (or T irreversible) background effects, for which
chosen as B = B = E /2. The dotted line represents the
(cid:107) ⊥ 0 we can apply our method. Of course, as mentioned in
analytically estimated rates for the particle production and
the beginning, possible applications should cover prob-
the current generation (with a common offset −0.25).
lems of sphaleron-like transitional processes in the elec-
troweak and the strong interactions on top of arbitrary
gauge fields including non-topological ones.
pendence assuming constant rates of particle production
Here we emphasize the impact of the agreement be-
and current generation. In contrast, for a small dura-
tween our numerical results and the analytical expres-
tion, it is very difficult to predict what should happen a
sions. Itisverynon-trivialthatadirectevaluationofthe
priori, and this is why we must perform the numerical
currentaccordingtoEq.(6)couldbeconsistentwiththe
simulations.
quantum anomaly. Also we emphasize the importance
Our numerical results in Fig. 6 (for N = 8,10,12 cor- ofreal-timecharacterofthechiralmagneticcurrentgen-
responding to 173,213,253) where B(cid:107) = B⊥ = E0/2 is eration. The physical interpretation and the theoretical
fixed clearly confirm linear dependence for large dura- estimateof µ arequiteproblematic andsometimesmis-
5
tion (apart from small oscillation which is characteris- leadingintheliterature. Oursuccessfulsimulationisthe
tic to magnetic phenomena). Although j0 shows some first step toward a realistic simulation in experimental
lattice-size dependence (due to the constant shift in the setups without µ in the system.
5
ultraviolet region of the produced particle distribution), In this kind of approach to solve the equation of mo-
two results for jy with different Ni’s are nearly overlaid tion, the anomaly arises from the infrared singularity
to each other. This clearly indicates that, because the around |p | = 0. The major part of the discretization
constant shift drops off in (cid:82) d3q, jy has no contamina- error mighAt be attributed to underestimating the singu-
tion from the ultraviolet fluctuations. lar contributions with coarse mesh. This implies that
We find for small duration a region of delay in which there may be a hybrid way to extract the singular terms
j0 andjy remainalmostvanishing. Thepresenceofsuch analytically and to calculate non-singular parts numeri-
delay is intuitively understandable: it should take some cally [31]. This would reduce the lattice-size dependence
time for the wave-function to transform from the free and we can make theoretical estimates with improved
plane wave to the Landau-quantized one after a sudden reliability. Though we have confirmed that our CME
switch-on of the magnetic field. We would emphasize current is close to the one in the continuum limit, it
that what seems non-trivial in Fig. 6 is that the delay wouldbeworthdevelopingsuchahybridalgorithm. An-
for jy is more than three times larger than that for j0. other direction for the improvement is to include back-
In our interpretation this difference in the onsets comes reactionfromgaugefluctuations. Oncetheback-reaction
from the difference in the rates. In Fig. 6 we showed the is taken into consideration, it would capture the effect of
analytically estimated rates (by the dotted line) using the chiral plasma instability [33]. These are works under
the formula in Ref. [11] with a common offset −0.25. progress [31].
The onset for jy is more delayed than that for j0 since
its rate ∂ jy is smaller.
t
Finallywebrieflycommentonthesensitivitywhenwe ACKNOWLEDGMENTS
change the box size, i.e., the infrared extrapolation. We
kept the lattice spacing (and so E a2) and changed the ThisworkwaspartiallysupportedbyJSPSKAKENHI
0
lattice size to find that there is a subtle feature in the Grant Number 15H03652 and 15K13479.
8
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