The Subtleties of the Wigner Function Formulation of the Chiral Magnetic Effect

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by  Yan Wu

The Subtleties of the Wigner Function Formulation of the Chiral Magnetic Effect
Yan Wu,1,2 De-fu Hou,2 and Hai-cang Ren3,2
1 School of Mathematice and Physics, China Geoscience University(Wuhan), Wuhan 430074, China
2 Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE),
Central China Normal University, Wuhan 430079, China
3 Physics department, The Rockefeller University,
1230 York Avenue, New York, New York 10021-6399, USA
(Dated: February 16, 2016)
We assess the applicability of the Wigner function formulation for the chiral Magnetic Effect
and noted some issues regarding theconservation and theconsistency of theelectric current in the
presence of an inhomogeneous and transient axial chemical potential. The problems are rooted in
6
theultraviolet divergence of the underlyingfield theory associated with theaxial anomaly.
1
0
PACSnumbers: 74.20.Fg,03.75.Nt,11.10.Wx,12.38.-t
2

b
e The chiral magnetic effect of electrically charged the ultraviolet divergence embedded in the underlying
F fermions, proposed in [1–4], remains a subject being quantum field theory. While the subtlety does not im-
actively investigated. Because of the axial anomaly, a pactontheexistingresultof[13],wheretheaxialchemi-
4
1 nonzero axial charge density controlled by the chemical cal potential is assumed constant, a number of problems
potential µ in an constant magnetic field B generates showupwhentheaxialchemicalpotentialbecomesinho-
5
] an electric current along the magnetic field, given by mogeneous and/or transient as we shall see. It appears
h
that the Wigner function in its present form is not ap-
p e2
- J= µ B (1) plicable to this more complicated situation. Through-
p 2π2 5 out this letter, we shall stay with the Euclidean metric
he whichisfreefromhigherordercorrectionsbecauseofthe ds2 =dxµdxmuwithallγ matriceshermitian. Inpartic-
[ non-renormalizationtheoremoftheanomaly. Anetaxial ular, x4 =it for a real time t.
charge density can be implemented through topological Following[13], the Wigner function for Dirac fermions
3 charge fluctuations of QCD in a quark-gluon plasma or at the phase space point (x,p) is a 4×4 matrix with its
v
through topological surface modes of certain Weyl met- elements defined by
0
2 als. The experimental indications of CME include the d4y
5 charge separation post off-central heavy ion collisions in Wαβ(x,p)= (2π)4e−ip·y <ψ¯β(x+)U(x+,x−)ψα(x−)>,
6 RHIC[5] and the enhanced electrical conductivity of a Z
0 Weyl metal in a magnetic field[7]. (2)
. where the gauge link
1 The experimental situation is far from ideal in RHIC,
60 wlishioerneisthienhmomagongeetniceofiuesldangdenterraantseidenvtiaanodff-tcheenttrhaelrmcoal-l U(x+,x−)=eieRxx−+dξµAµ(ξ) (3)
v:1 evqesutiilgibartiiuomns,oiffCreMalEizeadr,e isstilllorceaql.uirMedorteo ethneriocrhetiitcsalphine-- <wit..h.>x±d=enxot±esy2aanndenAseµmtbhleegaavuegreagpeo,twenhtiicahl.isTnhoetsynmecbeos-l
i nomenological predictions and solidify its experimental
X sarily in thermal equilibrium. The electric current den-
evidences observed so far. The field theoretic method
sity can be extracted formally from the matrix W(x,p)
r [3, 8–10], holography [11, 12] and kinetic theories [13]
a as
arethe three mainapproachesexploredinthe literature.
The kinetic theoryisa powerfultoolto describeasys- d4p
J (x) = ie trW(x,p)γ
temnotinaglobalthermalequilibrium. Atits verycen- µ (2π)4 µ
Z
ter lies the Wigner function that links various hydrody-
namicquantitiesofthesystemtotheGreen’sfunctionsof = ie d4yδ4(y)U(x+,x−)<ψ¯(x+)γµψ(x−)>
the underlying quantum field theory. The Wigner func- Z
= limJ (x,y) (4)
tion was introducedin RHIC physics in [14] andwas ap- y→0 µ
plied recently to the QGP with net axial charge density
with
[13]. Among its successes are the reproduction of the
chiral magnetic current, chiral vortical current and axial Jµ(x,y)=ieU(x+,x−)<ψ¯(x+)γµψ(x−)>, (5)
anomaliesobtainedfromthefieldtheoreticapproachesat
a constant axial chemical potential. The situation with and the UV divergence embedded in the operator prod-
aninhomogeneousandtransientaxialchemicalpotential uct at the same point makes the limit ill-defined. Since
has not been considered yet. the UV divergence corresponds to a length scale much
In this letter, we would like to point out a subtlety shorter than the hydrodynamic scales underlying the ki-
of the Wigner function being used for CME because of netictheory,aglobalthermalequilibriumcanbeassumed
2
for the ensemble average < ... > in (2) to explore its gauge potential A and the axion field A reads
µ 5µ
consequences and the thermal field theory technique be-
comes handy then.
Sab(x−,x+)
Toillustratetheproblem,weconsideramasslessDirac
fieldinanexternalelectromagneticfieldAµ andanaxion = Sab(x−,x+)− d4zSac(x−−z)
field A with the action c Z
5µ X
×γc S (z−x )A (z)
ρ5 cb + 5ρ
S = dt d3xL, (6) + e d4z1 d4z2Sad(x−−z2)γλd5
Z Z Xcd Z Z
×S (z −z )γcS (z −x )A (z )A (z )
dc 2 1 ρ ca 1 + ρ 1 5λ 2
where the Lagrangiandensity is given by
+ e d4z1 d4z2Sac(x−−z2)γρc
L=−ψ¯γ (∂ −ieA −iγ A )ψ (7) Xcd Z Z
µ µ µ 5 5µ ×S (z −z )γd S (z −x )A (z )A (z )(11)
cd 2 1 λ5 da 1 + ρ 2 5λ 1
The axial chemical potential µ in (1) corresponds to
5 with γ1 = γ , γ2 = −γ , γ1 = γ γ and γ2 = −γ γ ,
the temporal component of the axion field, i.e. A = µ µ µ µ µ5 µ 5 µ5 µ 5
5µ whereS (x−y)isthe freeDiracpropagator. Substitut-
(A ,−iµ ). The spatial components are relevant if the ab
5 5 ing (11) and the expansion
axial magnetic effect is related to the topological fluctu-
ation in QCD via A =∂ θ.
5µ µ x+
The most straightforwardmethod to calculate the en- U(x+,x−)=1+ie dξνAν(ξ)+O(A2) (12)
sembleaveragein(2)istheclosedtime pathGreenfunc- Zx−
tion formation which was proposed in [15, 16] and was
systematicallydevelopedin[17]. Thetimeintegralofthe into(5)andmakingappropriateFouriertransformations,
action, (6) consists of two branches, one from −∞ to ∞ we obtain that
andthe otherfrom∞to −∞anddegreesoffreedomare
therebydoubled. Asaresult,allfieldvariablesacquirean d4q d4q
Ja(x,y) = e2 1 2 ei(q1+q2)·x
additionalindexlabelingthetime branchwheretheyare µ (2π)4 (2π)4
defined. Consequently, a fermion propagatorbecomes Z Z
×Λabc(q ,q )Ab(q )Ac (q ) (13)
µρλ 1 2 ρ 1 5λ 2
S (x,y) S (x,y)
S (x,y)= 11 12 (8) with the kernel
CTP S (x,y) S (x,y)
21 22
(cid:18) (cid:19)
Λabc(q ,q ) = −y δ Kac(q )
µρλ 1 2 ρ ab µλ 2
where each block is a 4 by 4 matrix in Dirac space. We
have −i[Kµ(1ρ)λabc(q1,q2)+Kµ(2ρ)λabc(q1,q2)]
(14)
S (x,y) ≡ <T[ψ (x)ψ¯ (y)]>
11 αβ α β
S (x,y) ≡ −<ψ¯ (y)ψ (x)> where
12 αβ β α
S (x,y) ≡ <ψ (x)ψ¯ (y)>
21 αβ α β d4p
S22(x,y)αβ ≡ <T˜[ψα(x)ψ¯β(y)]> Kµaνb(q) = e−2iq·y (2π)4e−ip·ytrγµaSab(p+q)γµb5Sba(p)
Z
(9)
d4p
Kµ(1ρ)λabc(q1,q2) = e−2i(q1+q2)·y (2π)4e−ip·y
where x = (x,it), T denotes time ordering and T˜ anti- ×trγaS (p+Z q +q )γc S (p+q )γbS (p)
time ordering. It is straightforward to link the LHS of µ ac 1 2 λ5 cb 1 ρ ba
(5) to different components of the CTP propagator,i.e. Kµ(2ρ)λabc(q1,q2) = e−2i(q1+q2)·y (2dπ4p)4e−ip·y
Z
Jµ(x,y) ×trγµaSab(p+q1+q2)γρbSbc(p+q2)γλc5Sca(p)
−ietrS11(x−,x+)γµ ≡Jµ1(x,y) y0 ≥0 (15)
= (10)
(−ietrS22(x−,x+)γµ ≡Jµ2(x,y) y0 <0 and the repeated CTP indices in (14) and (15) are not
to be summed. The momentum representation of vari-
with the trace acting on Dirac indices. The expansion ous components of the free CTP fermion propagator at
of the propagator Sab(x−,x+) to the linear power of the thermal equilibrium suffices to extract the UV behavior
3
and are given explicitly by (q +q ) [K(1)121(q ,q )+K(2)121(q ,q )]
1 2 µ µρλ 1 2 µρλ 1 2
i p/+µγ4 = (e−2i(q1+q2)·y−e−2i(q1−q2)·y)
S (p) = −π
11 p/+i0++µγ4 E × d4p e−ip·ytrγ S (p)γ γ S (p+q )(22)
×[f(E−µ)δ(p¯0−E)+f(E+µ)δ(p¯0+E)] (2π)4 ρ 21 λ 5 12 1
Z
p/+µγ
4
S12(p) = −π {f(E−µ)δ(p¯0−E) and
E
+[f(E+µ)−1]δ(p¯0+E)} (q1+q2)µ[Kµ(1ρ)λ122(q1,q2)+Kµ(2ρ)λ122(q1,q2)]=0 (23)
p/+m+µγ
4
S21(p) = −π [f(E−µ)−1]δ(p¯0−E) fora=1andsimilarequationsfora=2. Wenoticethat
E
only the momentum integrals that diverges linearly in y
+f(E+µ)δ(p¯ +E)}
0 asy →0contributetothelimit(4). Therefore,theterms
−i p/+µγ
S22(p) = p/−i0++µγ4 −π E 4[f(E−µ)δ(p¯0−E) bineSiganbo(pre)dp.rFopurotrhtieornmaolrteo,tthheecdoimstbriibnuattiioonnofufnSc1t2io(pn+s cka1n)
+f(E+µ)δ(p¯ +E)] (16) andS (p+k ) contributes a productofdelta functions,
0 21 2
δ(p¯ +k +E )δ(p¯ +k −E ) which gives rise
wherep=(p,ip ), p¯ =p +µ,E =|p|,p/≡−iγ p and 0 10 p+k1 0 20 p+k2
0 0 0 ν ν toδ(k −k +E +E )whichimposesanupper
f(x) = 1 with β the inverse temperature and µ the 10 20 p+k1 p+k2
eβx+1 limitofthep-integrationforafixedexternalmomentak1
chemical potential associated to the fermion number.
andk andrendersthe integralUVfinite. Consequently,
2
Now we show two problems coming from the limit in
the only CTP components that contribute to the limit
(4):
y → 0 of (17) correspond to a = b = c = 1 and a = b =
c=2 with the vacuum propagators,i.e. T =µ=0. We
1. The nonconservation of the electric current
obtain that
Taking the divergence of (13), we have
(q +q ) Λ111(q ,q ) = 4(−y q ǫ q −q ·yǫ
∂ d4q d4q 1 2 µ µρλ 1 2 ρ 1µ αµβλ 2β 2 αρβλ
∂x Jµa(x,y) = ie2 (2π)14 (2π)24ei(q1+q2)·x(q1+q2)µ +q1·yǫαρβλq2β)uα(y) (24)
µ
Z Z
×Λabc(q ,q )Ab(q )Ac (q ), (17) and
µρλ 1 2 ρ 1 5λ 2
with (q +q ) Λ222(q ,q ) = 4(−y q ǫ q −q ·yǫ
1 2 µ µρλ 1 2 ρ 1µ αµβλ 2β 2 αρβλ
∗
(q1+q2)µΛaµbρcλ(q1,q2) +q1·yǫαρβλq2β)uα(−y) (25)
= −y δ (q +q ) Kac(q )
ρ ab 1 2 µ µλ 2 where
−i(q1+q2)µ[Kµ(1ρ)λabc(q1,q2)+Kµ(2ρ)λabc(q1,q2)](18) d4p pαe−ip·y yα
u (y)= =− . (26)
UsingthefollowingidentitiesofthefreeCTPpropagator, α (2π)4(p2−i0+)2 8π2y2
Z
S (p+q)/qS (p) = i[S (p)−S (p+q)] Substituting (26) into (24) and (25), we end up with
11 11 11 11
S (p+q)/qS (p) = −iS (p+q)
21 11 21 (q +q ) Λ111(q ,q )=(q +q ) Λ222(q ,q )
1 2 µ µρλ 1 2 1 2 µ µρλ 1 2
S (p+q)/qS (p) = iS (p)
11 12 12
1 y y
λ α
S21(p+q)/qS12(p) = 0. (19) = −2π2 ǫµβλρ+ y2 ǫραµβ q1µq2β. (27)
(cid:18) (cid:19)
and shifting some of the integration momenta, we find
where the Schouten identity
(q +q ) [K(1)111(q ,q )+K(2)111(q ,q )]
1 2 µ µρλ 1 2 µρλ 1 2 y ǫ +y ǫ +y ǫ +y ǫ +y ǫ =0
µ ρλαβ β µρλα α βµρλ λ αβµρ ρ λαβµ
= (e−2i(q1−q2)·y−e−2i(q1+q2)·y) (28)
d4p has been employed. The coordinate representation of
× (2π)4e−ip·ytrγρS11(p)γλγ5S11(p+q1) (27) reads
Z
+ (e2i(q1−q2)·y−e−2i(q1+q2)·y) ∂ J (x,y)
µ
d4p ∂xµ
× e−ip·ytrγ γ S (p)γ S (p+q ),(20)
(2π)4 λ 5 11 ρ 11 2 i yλyα ∂
Z = 8π2 ǫµρβλFµρ(x)F5βλ(x)+2ǫµραβ y2 Fµρ(x)∂x A5λ(x)
(cid:20) β (cid:21)
(q +q ) [K(1)112(q ,q )+K(2)112(q ,q )] (29)
1 2 µ µρλ 1 2 µρλ 1 2
= (e2i(q1−q2)·y−e−2i(q1+q2)·y) where Fµν = ∂∂Axµν − ∂∂Axνµ, F5µν = ∂∂Ax5µν − ∂∂Ax5νµ and the
× d4p e−ip·ytrγ S (p+q )γ γ S (p),(21) CTP indices have been suppressed. Because of the sec-
(2π)4 ρ 12 2 λ 5 21 ond term on RHS, the limit y → 0, does not exist in
Z
4
rigorous sense. If we follow a hand-waving definition of and
the limit by averagingthe direction of y (after continua-
tion to Euclidean space, i.e. y →−iy ), we find [20] K(2)abc(q′,q−q′)−K(1)bac(−q,q−q′)
0 0 µρλ ρµλ
∂ ∂ 3i = e2iq·y−e2iq′·y d4p e−ip·y
∂xµJµ(x)≡ ∂xµJµ(x,0)= 32π2ǫµρβλFµρ(x)F5βλ((x3)0.) ×(cid:16)trγµaSab(p)γρbS(cid:17)bZc(p(−2πq)′4)γλc5Sca(p−q). (37)
It is interesting to note that if the axion potential is a
pure gradient, A = ∂θ , (31) becomes Followingtheargumentsafter(23),thecomponentswith
5µ ∂xµ a=b=c=1 and a=b=c=2 contribute to a nonzero
limit of (35) as y → 0 and the consistency condition is
∂ i y y ∂2θ
J (x,y)= ǫ λ αF (x) (31) thereby violated. We obtain that
∂x µ 4π2 µραβ y2 µρ ∂x ∂x
µ β λ
Λ111(q′,q−q′)−Λ111(−q,q−q′)
µρλ ρµλ
and the limit (30) following the hand-waving definition
vanishes. = Λ2µ2ρ2λ(q′,q−q′)−Λ2ρ2µ2λ(−q,q−q′)
′
= 4(y ǫ +y ǫ )u (y)(q−q )
α µβλρ λ ραµβ α β
2. An inconsistency
1 y y
λ α ′
The electric current, being a functional derivative of = −2π2 ǫµβλρ+ y2 ǫραµβ (q−q )β, (38)
the quantum effective action, should satisfy the consis- (cid:18) (cid:19)
tency condition:
which implies that
δJµ(x) δJν(x′) δJ (x,y) δJ (x′,y) i y y ∂A
= (32) µ ν λ α 5λ
δAν(x′) δAµ(x) δA (x′) − δA (x) = 4π2 ǫµρβλ+ǫµραβ y2 ∂x
ν µ (cid:18) (cid:19) β
(39)
which is generalized to
in coordinate space. Like the case with the current di-
vergence(31), the limit y →0 does notexists rigorously.
δJµa(x) = δJνb(x′) (33) For the limit defined in (30), we find that
δAb(x′) δAa(x)
ν µ
δJ (x) δJ (x′) 3i
µ ν
− = ǫ F (x). (40)
in CTP formulation because of the doubling of degrees δA (x′) δA (x) 16π2 µρβλ 5βλ
ν µ
of freedom. The consistency condition dictates the sym-
metry property of the current-current correlator as well Similartothecurrentdivergence,thefirstterminsidethe
as the relationship between the retarded and advanced parenthesesonRHS of(39) does notcontribute if A is
5µ
Green’s function of linear response. To the linear order a pure gradient and the RHS of (40) vanishes then.
in the external gauge potential and axion potential, the This inconsistencyisrelatedto the nonconservationof
consistency condition requires the electric current. As the Wigner function is explicitly
gauge invariant. The current extracted from it would be
lim Λabc(q′,q−q′)−Λbac(−q,q−q′) =0. (34) conserved if the current were a functional derivative.
y→0 µρλ µρλ
We have also considered a static case with A =
(cid:2) (cid:3) 5µ
(0,0,0,−iµ ) with the axial chemical potential spatially
This is, however, not the case because of the UV diver- 5
inhomogeneousandwiththe limit (4)takenalongaspa-
gence. It follows from (14) that
tial direction only. The Matsubara formulation can be
Λabc(q′,q−q′)−Λbac(−q,q−q′) employeddirectlyandthereisnocomplicationswiththe
µρλ µρλ
CTP indices. Starting from the definition
= −δ [y Kac(q−q′)−y Kbc]
ab ρ µλ µ ρλ
= −i[Kµ(1ρ)λabc(q′,q−q′)−Kρ(2µ)λbac(−q,q−q′) Jia(x,y) = ie2 (d23πq)14 (d23πq)24ei(q1+q2)·x
+K(2)abc(q′,q−q′)−K(1)bac(−q,q−q′)] (35) Z Z
µρλ ρµλ ×Λij(q1,q2)Aj(q1)µ5(q2), (41)
UponshiftingtheintegrationmomentumofK(2),wefind The analysis proceeds parallely. In particular, we find
that that
K(1)abc(q′,q−q′)−K(2)bac(−q,q−q′) (q +q ) Λ (q ,q )=−4(q ×q ) y·v (42)
µρλ ρµλ 1 2 i ij 1 2 1 2 j
= e−2iq·y −e−2iq′·y d4p e−ip·y and
(2π)4
(cid:16)×trγaS (p+q)γc(cid:17)ZS (p+q′)γbS (p) (36) Λ (q′,q−q′)−Λ (−q,q−q′)=−ǫ (q−q′) y·v (43)
µ ac λ5 cb ρ ba ij ji ijk k
5
with by the Pauli-Villars scheme, there should be additional
terms on RHS of (2) with ψ replaced by its correspond-
d3p e−ip·yp y
v =iT k =− k ing Pauli-Villars regulators, which may play the role of
k (2π)3[p2+π2T2(2n+1)2]2 8π2y2 the Bardeen term introduced in [10]. The conservation
n Z
X
(44) andconsistency of the electric currentwouldbe restored
as y→0. Consequently, we find then.
e2
∇·J= ∇µ ·B (45)
2π2 5 Acknowledgments
and
We are indebted to Hui Liu for her early participation
δJi(x) − δJj(x′) = e2 δ3(x−x′)ǫ ∂µ5 (46) of this project. We thanks K. Fukushima, L. McLer-
δA (x′) δA (x) 8π2 ijk∂x ranand Qun Wang for helpful discussions. The interest-
j i k
ing comments of K. Landsteiner, M. Chernodub and O.
Eq.(45)appearsobviousif(1)couldbegeneralizedtothe Ruchayskiy are warmly acknowledged. This research is
case with aninhomogeneous µ . In fact, one cannotfind partly supported by the Ministry of Science and Tech-
5
any local terms in ∇µ and B that can be added to (1) nology of China (MSTC) under the ”973” Project No.
5
to render the currentdivergencefree. It was pointedout 2015CB856904(4). Y. Wu is supported by the Funda-
in [19] that a spatially varying µ potential is consistent mental Research Funds for the Central China Normal
5
with the notion of local thermal quilibrium only in the Universities, and the QLPL under grantNo. 201507. D-
situation where ∇µ ⊥B. f. Hou and H-c. Ren are partly supported by NSFC
5
In summary, we noted some problems of the Wigner under Grant Nos. 11375070,11135011and 11221504.
function formalism because of the UV divergence of the
underlying quantum field theory. We find that in the
presence of an electromagnetic gauge potential and a
non-constant axial vector potential, the electric current
extractedfromtheWignerfunctiondefinedby(2)isnei-
ther a functional derivative of some effective action with
respect to the gauge potential nor conserved. The issue
is closely related to the axial anomaly. Therefore, the
present form of the Wigner function formulation needs
to be revisedto be applicable to a generalchiralplasma.
Thesubtletydiscoveredinthisworkdisappearsforacon-
stant axial vector potential, leaving the result in [13] in-
tact and the definition appears appropriate.
Technically, the process of the limit y → 0 applied to
the axial current extracted from the Wigner function
d4p
J = i trW(x,p)γ γ
5µ(x) (2π)4 µ 5
Z
= ilimU(x+,x−)<ψ¯(x+)γµγ5ψ(x−)>(47)
y→0
is equivalent to the point-splitting regularizationscheme
of the axial vector vertex of the triangle diagram in
textbooks and this explains why the the standard ax-
ial anomaly is generated from the Wigner function (2)
in [13]. However, the same limiting procedure for the
electric current (4) amounts to a point-splitting regular-
ization of one of the vector vertex of the same triangle
diagram and thereby violates the Bose symmetry. For
a non-chiral theory or a chiral theory with a constant
axialvectorpotential,this isnotaproblemandthe elec-
tric current remains conserved and consistent as can be
shown explicitly. In more general situations, an unam-
biguous limit (4) may be realized with a more cautious
definition of the Wigner function that is consistent with
the regularization scheme of the underlying field theory.
For example, if the underlying field theory is regularized
6
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