The chiral anomaly, Berry's phase and chiral kinetic theory, from world-lines in quantum field theory

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by  Niklas Mueller

The chiral anomaly, Berry’s phase and chiral kinetic theory, from world-lines in
quantum field theory
Niklas Mueller1,∗ and Raju Venugopalan2,†
1Institut fu¨r Theoretische Physik, Universita¨t Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
2Physics Department, Brookhaven National Laboratory, Bldg. 510A, Upton, NY 11973, USA
(Dated: February 8, 2017)
We outline a novel chiral kinetic theory framework for systematic computations of the Chiral
Magnetic Effect (CME) in ultrarelativistic heavy-ioncollisions. The real part of thefermion deter-
minant in theQCD effective action is expressed as a supersymmetric world-line action of spinning,
colored, Grassmanian point particles in backgroundgauge fields,with equationsof motion that are
covariant generalizations of the Bargmann-Michel-Telegdi and Wong equations. Berry’s phase is
7
obtainedinaconsistentnon-relativisticadiabaticlimit. Thechiralanomaly,incontrast,arisesfrom
1
the phase of the fermion determinant; its topological properties are therefore distinct from those
0
of the Berry phase. We show that the imaginary contribution to the fermion determinant too can
2
be expressed as a point particle world-line path integral and derive the corresponding anomalous

b axial vector current. Our results can be used to derive a covariant relativistic chiral kinetic theory
e includingtheeffectsoftopologicalfluctuationsthathasoverlapwithclassical-statistical simulations
F of theCME at early times and anomalous hydrodynamicsat late times.

7

The possibility that topological sphaleron transitions thekineticdescriptionofchiralcurrentsmustbematched
]
h can be identified in heavy-ion collision experiments has to anomalous hydrodynamics [20, 21].
p
arousedgreatinterest. Besidestheinformationtheypro- There has been a considerable amount of recent work
-
p vide about the non-perturbative real time dynamics of onchiralkinetictheory,andoftherolethereinofthewell
e the QCD vacuum, sphaleron transitions are conjectured known Berry phase [22] and of the chiral anomaly [23–
h
to play a role in electroweakbaryogenesis[1, 2]. A strik- 36]. However, despite significant progress, much work
[
ing manifestation of the role of topology is the Chiral remains to complete a first principles derivation of a rel-

2 Magnetic Effect (CME) where, as a consequence of the ativistically covariant kinetic theory from QCD (or even
v
chiral anomaly, an induced current is generated in the QED). This is especially true for the treatment of topo-
1
3 direction of the external magnetic field [3, 4]. Whether logical fluctuations which the chiral current will experi-
3 suchaneffectisseeninheavy-ioncollisionsisstillunclear enceasittraversesthefireball. Further,asfirstobserved
3 and is a focus of experimentalresearchin the field [5, 6]. by Fujikawa, there is often a conflation of the topology
0 We note that the CME has been observed in condensed ofBerry’sphase with that ofthe chiralanomaly[37–39].
.
1 matter systems [7]. In this letter, we will outline the elements of a first
0
For the CME to be large enough to be observed, it principles world-line derivation of relativistic chiral ki-
7
must be generated at the earliest times in the heavy-ion netic theory. The real part of the fermion determinant
1
: collisionwherethemagneticfieldsareverylargeinitially can be expressed exactly in terms of the supersymmet-
v
before dying off rapidly [8, 9]. Firstprinciples weak cou- ric quantum mechanical path integral for point particle
i
X pling computations of sphaleron transitions [10] in the world-lines[40–47],where the internalspinandcolorde-
r non-equilibrium Glasma matter produced indicate that grees of freedom are expressed in terms of Grassmann
a the sphaleron transition rate is signficantly larger [11] variables[48,49]. WewilldemonstratehowBerry’sphase
than the corresponding equilibrium rate [12]. Because arises in a specific non-relativistic and adiabatic limit of
the occupancies of gluons in the Glasma are large, the corresponding Euler-Lagrange equations of motion
classical-statistical simulations can be employed [13] to for the spinning and colored Grassmanian fields [50–54].
computeab initio,inasphaleronbackground,andinthe Incontrast,explicitidentificationofaBerryphaseisnot
presence of external magnetic fields [14, 15], the devel- relevant in ultrarelativistic contexts; the semi-classical
opment of a chiral magnetic, and accompanying “chiral phasespacedescriptionofworld-linetrajectoriesprovides
separation”, wave [16, 17]. all the essential elements in the construction of a covari-
Detailedthermalizationscenarioshoweversuggestthat ant chiral kinetic theory.
thermalization occurs at parametrically later times [18, The fermion determinant has a relative phase that
19]. Because the gluon occupancy is of order unity or is well known to be related to the physics of the chi-
lower in this regime, kinetic theory provides the ap- ral anomaly [55–58]. We will here adapt a trick due to
propriate description of quark-gluon dynamics and the D’Hoker and Gagn´e [59, 60] to express this phase as a
classical-statistical simulations of the CME must there- point particle path integral nearly identical to the one
forebematchedtothisframework. Likewise,atthelater obtained for the real part. The only difference is that
times when the quark-gluon matter is strongly coupled, the gauge fields are multiplied by a parameter that reg-
2
ulates chiral symmetry breaking. We will outline how Σ˜2 admits a manifestly positive definite heat-kernel reg-
the anomaly arises in this context and clearly demon- ularization. Therefore using Schwinger’s proper-time
strate that its origin is distinct from the Berry phase. scheme,therealpartoftheeffectiveactioncanberewrit-
The vector and axial vector current are treated on the ten as
samefootingandareessentialelementsinconstructinga ∞
chiral kinetic theory. Many details of our computations, WR = 1 dT Tr16 e−E2TΣ˜2, (10)
8 T
andseveralnewresults,willbegiveninanaccompanying
Z
0
longer paper [61].
where is the einbein, to be discussed further later.
We begin with the Euclidean action for massless ThisE16-dimensional representation of Σ˜2 is useful be-
fermion fields in the background of a vector (A) and an
causeitis convenientlycastinto apathintegralinterms
auxilliary Abelian axial-vector (B) field [62]
of Grassmanian variables. These variables are eigenval-
ues of coherent states of creation/annihilation operators
S [A,B]= d4xψ¯ i∂/+A/+γ B/ ψ, (1)
F 5 that generate finite dimensional representations of the
Z
(cid:0) (cid:1) internal symmetries of the theory [48, 49]. One obtains
and allow the fermion fields to transform under internal
after some algebra,
(gauge)symmetry. Performingthepathintegraloverthe
∞
fermion fields, one obtains the effective action 1 dT
WR = (T) x ψ λ† λ (λ†λ)
8 T N D D D D J
−W[A,B]=logdet(θ) θ =i∂/+A/+γ5B/ (2) Z0 PZ AZP
T T
The determinant of θ carries a relative phase. One can −Rdτ LL(τ) −RdτLR(τ)
therefore formally split Eq.(2) into real and imaginary e 0 +e 0 . (11)
× 
parts,
 
Some details of this derivation are given in [61]. The
W[A,B]=WR[A,B]+iWI[A,B]. (3)
point particle Lagrangianfor left/right chiralities is
The real part of the effective action can expressed as x˙2 1
(τ)= + ψ ψ˙ +λ†λ˙
1 LL/R 2 2 a a
WR = logdet θ†θ . (4) E i
−2 λ† ix˙ (A B) Eψ ψ F [A B] λ, (12)
µ µ µ ν µν
(cid:0) (cid:1) − ± − 2 ±
As shown in [59, 60], this can be rewritten as h i
T
−ERdτ p2(τ)
WR = 1logdet(Σ˜2)= 1Trlog(Σ˜2). (5) withthenormalizationN(T)= Dpe 2 0 . Here
−8 −8 ψ = √2 ψ Γ ψ , with a = 1, ,6 are Grassmann
a h | a| i R ···
variables, which are defined over a real vector space,
Here Σ˜2 is a sixteen dimensional matrix given by
where ψ represents a coherent state basis of the Clif-
| i
i ford algebra. Likewise, λ† and λ are independent Grass-
Σ˜2 =(p )2 I + Γ Γ F [ ], (6)
−A 8 2 µ ν µν A manianeigenvaluesrespectivelyofcreationandannihila-
tion Fermion operators that generate finite dimensional
with I the 8-dimensional identity matrix and
8 representation of SU(N ), where N is the number of
c c
0 γ 0 γ 0 iI colors. The factor (λ†λ) = (π)Nc exp[iφ(λ†λ +
Γ = µ , Γ = 5 , Γ = 4 , J T φ
µ (cid:18)γµ 0(cid:19) 5 (cid:18)γ5 0(cid:19) 6 (cid:18)−iI4 0 (cid:19) Nc/2−1)] is required to project intermPediate states in
(7) the path integral on to coherent states with unit occu-
pancy [60]. The labels P and AP denote periodic and
are8 8dimensionalgammamatrices. Wefurtherdefine anti-periodic boundary conditions for the configuration
×
a Γ7 matrix anti-commuting with all other elements of space and Grassmanian variables respectively. Hence-
the algebra, forth, the Abelian reduction of Eq.(11) will be sufficient
for ourpurposes; the extensionofour discussionto color
6 I 0
Γ = i Γ = 4 . (8) degrees of freedom is straightforward[50–52].
7 − A 0 I4 Varying the real part of the effective action with re-
A=1 (cid:18) − (cid:19)
Y
spect to the vector gauge field gives the vector current,
The gauge fields in Eq.(5) that appear explicitly and in
∞
thefield-strengthtensorFµν[A]canbesplitintoleft-right jV(y) = δΓR = i dT x ψjV,cl
chiralstructures with the 2 2 dimensional matrix form h µ i δA (y) −8 T N D D µ
× µ Z Z Z
[63], 0 P AP
T T
= A+B 0 . (9) e−R0 dτLL(τ)+e−R0 dτ LR(τ) , (13)
A 0 A B × !
(cid:18) − (cid:19)
3
with jV,cl(y)= T dτ [ ψ ψ ∂ x˙ ]δ4 x(τ) y . This B =0),
µ 0 E ν µ ν − µ −
satisfies both ∂ jV,cl =0 and ∂ jV =0.
µRµ µh µi (cid:0) (cid:1) m cz m2 i
Theimaginaryrelativephaseintheeffectiveactioncan = R 1+ + ψψ˙ ψ ψ˙
L − 2 m2 2 − 0 0
be written as (cid:18) R(cid:19) (cid:16) (cid:17)
x˙ Aµ(x) iz iz
+ µ ψ0F ψi ψiF ψj, (19)
1 0 θ c − m c 0i − 2m c ij
WI = argdet[Ω], Ω= , (14) R R
−2 θ 0
(cid:18) (cid:19) where m2 =m2+iψµF ψν/c2 is the effective mass for
R µν
where θ is given in Eq.(2) and the matrix Ω is thespinningworld-line. Thespinthree-vectorcanbede-
fined as Si = iǫijkψjψk, where ǫijk is the Levi-Civita
−2
Ω=Γ (p A ) iΓ Γ Γ Γ B . (15) symbol. Likewise, the magnetic field Bi = 1ǫijkFjk and
µ µ µ 7 µ 5 6 µ 2
− − the electric field Ei = F0i. The corresponding equa-
The D’Hoker and Gagn´e [59, 60] trick consists of in- tions ofmotion are the covariantform of the Bargmann-
troducing a parameter that regulates chiral symmetry Michel-Telegdi equations [66] for spinning particles (and
breaking— distinct fromthose employedpreviously[58] likewise, Wong equations for colored particles [67]) [50–
— to write WI as 54]. The last two terms in Eq.(19) are respectively,
S (π E) i
1 ∞ iψ0F ψi = · × ; ψiF ψj =S B. (20)
WI = iE dα dT Tr Mˆe−E2TΣ˜2(α) , (16) − 0i cπ0 −2 ij ·
64
−Z1 Z0 n o Hereπµ pµ Aµ,wherepµisthecanonicalmomentum.
≡ −
Totakethenon-relativisticlimit,weexpandtheeffective
with a trace insertion massasm m(1+X)withX = (S (π E)/[cπ0]+
R
≈ − · ×
S B)/(2m2c2). Observing that X (v/c)2, where ~v
Mˆ =Γ7 (2Γ5Γ6[∂µ,Bµ]+[Γµ,Γν] ∂µ,Bν Γ5Γ6)I2, is ·the three-velocity of the spinning w∝orld-line, one can
{ }
(17) expand Eq.(19) to obtain
1 i v
thatis linearinthe axial-vectorfieldanddiagonalinthe = mc2+ mv2+ ψψ˙ ψ ψ˙ A0+ A
NR 0 0
two dimensional field representation space introduced in L − 2 2 − − c ·
Eq.(9). Σ˜2 isidenticaltotheexpressioninEq.(6),with S ( v/c A/(mc2)(cid:16) E) S(cid:17)B v3
(α) + · − × + · +O .
B αB, where α breaks chiral symmetry explicitly for mc m c3
→ (cid:2) (cid:3) !
α= 1.
6 ± (21)
This form of the phase of the fermion determinant is
useful because it has a heat-kernel structure that can be Compactly expressing the non-relativistic action as S =
computed, in a manner identical to the real part, using dt p x˙ + iψ ψ˙ H , the correspondingHamiltonian
· 2 · −
Grassmanian path integrals [48, 49]. The path integral is [68]
R (cid:0) (cid:1)
representation of WI is given in [61]; it is further shown
explicitly there that this representation gives the well- p A 2
H mc2+ − c +A0(x)
known anomaly relation ≡ 2m
(cid:0) (cid:1)
S ( v/c A/(mc2) E) B S
∂ j5(y) ∂ iδWI = 1 ǫµνρσF (y)F (y). − · −2mc × − m· . (22)
µh µ i≡ µδBµ(y) B=0 −16π2 µν ρσ (cid:2) (cid:3)
(cid:12)(cid:12) (18) Expressedinthisform,thenon-relativisticpointparticle
(cid:12) Hamiltonian is familiar [69]; the penultimate term is the
We will now show how Berry’s phase arises from tak- spin-orbit interaction energy from Thomas precession,
ing a non-relativistic and adiabatic limit of the real while the final term is the Larmor interaction energy.
part of the effective action WR. Our starting point In atomic physics their combined effect is of course to
[64] is the world-line Lagrangian in Eq.(12) continued reduce the Larmorenergyby the famous “Thomas1/2”.
to Minkowskian metric (g = diag[ ,+,+,+]). We pro- In the following, we will show how the system de-
ceed by introducing Lagrange mul−tipliers in Eq.(12) to scribed by Eq.(22) contains, in an adiabatic approxima-
i) impose the mass-shell constraint and ii) to project tion, a Berry phase; in this limit, it has the monopole
out unphysical spin degrees of freedom for both mass- form postulated in [23–25, 27]. To recover the expres-
less and massive point particles. After imposing all con- sions in [23, 24] we re-quantize the spin, by promoting
straints, and eliminating thereby unphysical degrees of the spin (phase-space) variables ψ to the Hilbert space
freedom [65], the Lagrangian can be written as S = operators ψ ~σ ψˆ and S ~σ Sˆ ,
0T dτL (setting τ = ct 1−v2/c2, with xµ = (ct,~x), where σ arei t→he Pqau2liim≡atricies and hia→ts in2diica≡te opk-
defining z = √ x˙2 and putting the auxilliary field erators. Further, to describe the finite phase space of
R − p
4
Grasmannianvariablesψ, we define the two dimensional whereA(p) i ψ+(p)∇ ψ+(p) istheBerryconnec-
p
≡− h | | i
Hilbert space for a spin-1/2 particle at every point in tion. The final expression for the path integral is
phase space (p,x) by the eigenstates ψ± = ψ±(p) .
Defining n = p (sinθcosφ,sinθsin|φ,coisθ),|one hais T(p ,p ,+)= x p exp i dt x˙ p H˜ ,
|p| ≡ f i D D · −
two choices Z (cid:16) Z h i(cid:17)(28)
N 1 cosθ
|ψ+(1)(p)i= 2 (1+n·σ)(cid:18)0(cid:19)=(cid:18)eiφsin22θ(cid:19) (23) witEhq.H(˜28=)miscc2l+ose(lpy−2Arme/lca)t2e+d Ato0(ax)si−mpi˙la·rAfo(prm).ulation in
N 0 e−iφcosθ
ψ(2)(p) = (1+n σ) = 2 , (24) [23–25, 27].
| + i 2 · 1 sinθ
(cid:18) (cid:19) (cid:18) 2 (cid:19) We note a few crucial points in our derivation and in-
terpretation for systems where the chemical potential is
for the “spin up” + basis vectors (where N is a normal-
much smaller than the rest mass. Firstly, as we showed,
ization factor) and similarly for the “spin down” basis
the structure of Berry’s phase is restricted to the non-
vectors (see also [70]). The two choices of basis vectors
relativistic adiabatic limit where the Larmor interaction
are not defined globally for all p with Eq.(23) (Eq.(24))
energy is much smaller than the rest energy. It is ill-
ill defined for the south (north) pole for θ = π (0). One
defined in the massless case albeit the spin basis vectors
set can however be used for the northern hemisphere
in Eq.(23) look similar to Weyl spinors. An exception
and the other for the southern one, and are related as
holds for massless systems with a large chemical poten-
|ψ+(1)(p)i=eiφ|ψ+(2)(p)i [70]. tial; the latter in that case takes over the role of the
These basis states allow us to derive a path integral mass [61]. Our derivationfurther makes it clearthat the
formulationintheadiabaticlimitofthetheorydefinedby topologicalstructureofBerry’sphase[71]isdistinctfrom
Eq.(22). The transition amplitude for the Hamiltonian that of the chiral anomaly. The former arises from real
operator corresponding to Eq.(22) from an initial state part of the QED/QCD effective action while the latter
ψ+(pi) at time ti to the state with momentum pf at can be traced to its relative phase [72].
| i
finite time tf is Since the CME dynamics in heavy-ioncollisionsis rel-
ativistic,andfarfromadiabatic,kinetictheoryconstruc-
T(p ,p ,+) p ,ψ+(p )e−iHˆ(tf−ti) p ,ψ+(p ) . tions that explicitly incorporate the Berry phase along
f i f f i i
≡h | | i
(25) thelinesofEq.(28)areinsufficientforthiscase[73]. Our
world-line expressions in Eq.(13) for the vector current,
The construction of the path integral for this amplitude and for the axial vector current j5(y) iδWI
requiresinsertionsofcompletesetsofintermediatestates h µ i ≡ hδBµ(y)i B=0
satisfying I = d3xk |xkihxk| = d3pk |pkihpk|, as iwnoErlqd.-(li1n8e)kairneetkiecythinegorreyd.iWenthsilienBaerrreya’ls-tpimhaesemiasnnyo-bt(cid:12)(cid:12)(cid:12)ordely-
well as one for the two dimensional spin-Hilbert space:
I = ψ+ ψ+ +Rψ− ψ− . TheadiabRaticapproximation evant, the effects of the axial anomaly are transparently
c2orre|sponihds to| B|·S ih 0.|Therefore in this limit we can introduced through jµ5(y).
2m ≈ The real-time formulation [74] of a pseudo-classical
neglect the second term ψ− ψ− , thereby constraining
| ih | kinetic theory from a world-line action was worked
the dynamical spin degrees of freedom.
out for spinless colored particles in [44]; the non-
The transition matrix element can thus be written as
Abelian Boltzmann-Langevin B¨odeker kinetic theory of
N−1 N hotQCD[75,76]includingbothnoiseandcollisionterms
T(p ,p ,+)= d3p d3x is recovered. In a follow-up work, the formalism devel-
f i k l
Z k=1 ! l=1 ! opedherewillbeworkedoutalongsimilarlinestoderive
Y Y
N the analogous “anomalous” B¨odeker theory [77–79]. As
1
e−ixj·(pj−pj−1)−iHj∆ ψ+(p )ψ+(p ) , alluded to previously, these can then be matched to re-
× (2π)3 h j | j−1 i
j=1 sults from classical-statistical simulations at early times
Y
(26) and to anomalous hydrodynamics at late times.
where ∆ (t t )/N and H is Eq.(22) evalu-
f i j
≡ − ACKNOWLEDGMENTS
ated at (x ,p ). Taylor expanding ψ+(p ) =
j j j−1
1+[p p ] ∇ ψ+(p ) + , it|is straighitfor-
j j−1 p j
{ − · }| i ···
ward to show in the continuum limit that one obtains RV thanks the Institut fu¨r Theoretische Physik, Hei-
Berry’s phase, delberg for their kind hospitality and the Excellence Ini-
tiativeofHeidelbergUniversityforaGuestProfessorship
N duringtheperiodwhenthisworkwasinitiated;wethank
ψ+(pj)ψ+(pj−1) exp i dtp˙ A(p) . (27) Juergen Berges, Jan Pawlowski, and Michael Schmidt
h | i→ ·
jY=1 (cid:16) Z (cid:17) for encouraging this effort. We thank Cristina Manuel
5
and Naoki Yamamoto for valuable comments. We thank [12] G. D. Moore and M. Tassler, JHEP 1102, 105
FiorenzoBastianellifor manyhelpful discussionsandfor (2011) doi:10.1007/JHEP02(2011)105 [arXiv:1011.1167
sharing his deep insights into the world-line formulation [hep-ph]].
[13] N. Tanji, N. Mueller and J. Berges, Phys. Rev. D 93,
ofquantumfieldtheory. RVwouldalsolike tothank the
no. 7, 074507 (2016) doi:10.1103/PhysRevD.93.074507
attendees of a seminar on this work at Stony Brook for
[arXiv:1603.03331 [hep-ph]].
their helpful comments; in particular, he would like to
[14] N. Mu¨ller, S. Schlichting and S. Sharma,
thank Dima Kharzeev, Ho-Ung Yee, Yi Yin and Ismail Phys. Rev. Lett. 117, no. 14, 142301 (2016)
Zahed. doi:10.1103/PhysRevLett.117.142301 [arXiv:1606.00342
[hep-ph]].
NM acknowledges support by the Studienstiftung des
[15] M. Mace, N. Mueller, S. Schlichting and S. Sharma,
Deutschen Volkes and by the DFG Collaborative Re- arXiv:1612.02477 [hep-lat].
search Centre SFB 1225 (ISOQUANT). This material [16] D. T. Son and A. R. Zhitnitsky, Phys. Rev. D
is partially based upon work supported by the U.S. De- 70, 074018 (2004) doi:10.1103/PhysRevD.70.074018
partment of Energy, Office of Science, Office of Nuclear [hep-ph/0405216].
[17] D. E. Kharzeev and H. U. Yee, Phys. Rev. D
Physics,under contractNo. DE-SC0012704,andwithin
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