|
|
|
|
|
EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher c Owned by the authors, published by EDP Sciences, 2017
|
|
|
|
|
|
|
|
|
|
|
|
arXiv:1701.00087v2 [hep-th] 16 Jan 2017
|
|
|
|
|
|
|
|
|
|
|
|
Anomaly induced transport in non-anomalous currents
|
|
|
|
|
|
Eugenio Meg<65>as1,2,a 1Max-Planck-Institut f<>r Physik (Werner-Heisenberg-Institut), F<>hringer Ring 6, D-80805, Munich, Germany 2Departamento de F<>sica Te<54>rica, Universidad del Pa<50>s Vasco UPV/EHU, Apartado 644, 48080 Bilbao, Spain
|
|
|
|
|
|
Abstract. Quantum anomalies are one of the subtlest properties of relativistic field theories. They give rise to non-dissipative transport coefficients in the hydrodynamic expansion. In particular a magnetic field can induce an anomalous current via the chiral magnetic effect. In this work we explore the possibility that anomalies can induce a chiral magnetic effect in non-anomalous currents as well. This effect is implemented through an explicit breaking of the symmetries.
|
|
|
|
|
|
|
|
|
|
|
|
1 Introduction
|
|
|
|
|
|
|
|
|
|
|
|
The basic ingredients in the hydrodynamic approach are the constitutive relations, which are expressions of the energy-momentum tensor T <20>, and the charge currents J<>, in terms of fluid quantities [1].
|
|
|
|
|
|
These relations are supplemented with the hydrodynamic equations, which correspond to the con-
|
|
|
|
|
|
servation laws of the currents. However, in presence of chiral anomalies the currents are no longer conserved, i.e. <20>T <20> 0 and <20>J<EFBFBD> 0. This leads to very interesting non-dissipative phenomena that already appear at first order in the hydrodynamic expansion: the chiral magnetic effect, responsible
|
|
|
|
|
|
for the generation of an electric current parallel to a magnetic field [2], and the chiral vortical effect, in
|
|
|
|
|
|
which the current is induced by a vortex [3]. The constitutive relation for the charge currents then read
|
|
|
|
|
|
|
|
|
|
|
|
J<EFBFBD> = nu<6E> + BB<42> + V<> + <20> <20> <20> ,
|
|
|
|
|
|
|
|
|
|
|
|
(1)
|
|
|
|
|
|
|
|
|
|
|
|
where n with the
|
|
|
|
|
|
|
|
|
|
|
|
is the charge field strength
|
|
|
|
|
|
|
|
|
|
|
|
density, u<> is the of the gauge field
|
|
|
|
|
|
|
|
|
|
|
|
local fluid defined as
|
|
|
|
|
|
|
|
|
|
|
|
velocity, B<> F<> = D<>A
|
|
|
|
|
|
|
|
|
|
|
|
= -
|
|
|
|
|
|
|
|
|
|
|
|
D21A<EFBFBD><EFBFBD>,aundF<64>is=th<74>emuagDneutic
|
|
|
|
|
|
|
|
|
|
|
|
field, is the
|
|
|
|
|
|
|
|
|
|
|
|
vorticity vector. The transport coefficients responsible for the chiral magnetic and vortical effects, B
|
|
|
|
|
|
|
|
|
|
|
|
and V, have been studied in a wide variety of methods: these include Kubo formulae [4<>6], diagram-
|
|
|
|
|
|
|
|
|
|
|
|
matic methods [7], fluid/gravity correspondence [8<>11], and the partition function formalism [12<31>15].
|
|
|
|
|
|
|
|
|
|
|
|
It is already clear from these studies that the axial anomaly [16] and the mixed gauge-gravitational
|
|
|
|
|
|
|
|
|
|
|
|
anomaly [17] are responsible for the previously mentioned non-dissipative effects.
|
|
|
|
|
|
|
|
|
|
|
|
In this work we are going to focus on the Kubo formalism. The most significant result of anomalies
|
|
|
|
|
|
|
|
|
|
|
|
is that they produce equilibrium currents, and the corresponding conductivities are defined via Kubo
|
|
|
|
|
|
|
|
|
|
|
|
Presented by E. Meg<65>as at the 5th International Conference on New Frontiers in Physics (ICNFP 2016), 6-14 July 2016,
|
|
|
|
|
|
Kolymbari, Crete, Greece. ae-mail: emegias@mppmu.mpg.de
|
|
|
|
|
|
|
|
|
|
|
|
EPJ Web of Conferences
|
|
|
|
|
|
|
|
|
|
|
|
formulae that involve retarded correlators at zero frequency. The Kubo formula for the chiral magnetic conductivity was derived in [16], and it reads
|
|
|
|
|
|
|
|
|
|
|
|
B = lim i pc0 2 pc
|
|
|
|
|
|
|
|
|
|
|
|
abc
|
|
|
|
|
|
a,b
|
|
|
|
|
|
|
|
|
|
|
|
JaJb
|
|
|
|
|
|
|
|
|
|
|
|
|=0 .
|
|
|
|
|
|
|
|
|
|
|
|
(2)
|
|
|
|
|
|
|
|
|
|
|
|
A similar formula involving the energy-momentum tensor was derived in [4] for the chiral vortical conductivity. In this work we use this formalism to study the chiral magnetic effect of anomalous conductivities. We will study the possibility that, under certain circumstances, this effect might be present in non-anomalous currents as well.
|
|
|
|
|
|
|
|
|
|
|
|
2 The chiral magnetic and separation effects
|
|
|
|
|
|
|
|
|
|
|
|
The chiral magnetic (CME) and chiral separation (CSE) effects are examples of anomalous transport.
|
|
|
|
|
|
In the context of heavy ion collisions, a very strong magnetic field produced during a non-central
|
|
|
|
|
|
collision induces a parity-odd charge separation which can be modelled by an axial chemical potential, and as a consequence an electric current parallel to the magnetic field is generated, leading to
|
|
|
|
|
|
the CME [2, 16, 18]. On the other hand, chirally restored quark matter might give rise to an axial
|
|
|
|
|
|
current parallel to a magnetic field, known as CSE [19]. These effects have been predicted as well in condensed matter systems, see e.g. [20, 21].
|
|
|
|
|
|
Let us consider a theory of N chiral fermions transforming under a global symmetry group G generated by matrices (TA) f g. The chemical potential for the fermion f is given by <20> f = A qAf <20>A, while the Cartan generator is HA = qAf f g where qAf are the charges. The general form of the anomalous induced currents by a magnetic field is
|
|
|
|
|
|
|
|
|
|
|
|
Ja = (B)abBb ,
|
|
|
|
|
|
|
|
|
|
|
|
(3)
|
|
|
|
|
|
|
|
|
|
|
|
where Bb is the magnetic field corresponding to symmetry b. The 1-loop computation of the chiral magnetic conductivity by using the Kubo formula of Eq. (2) leads to [6, 16, 17]
|
|
|
|
|
|
|
|
|
|
|
|
(B)ab
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>c dabc 42
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
dabc
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
1 2
|
|
|
|
|
|
|
|
|
|
|
|
[tr(Ta{Tb,
|
|
|
|
|
|
|
|
|
|
|
|
Tc})R
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
tr(Ta{Tb, Tc})L] ,
|
|
|
|
|
|
|
|
|
|
|
|
(4)
|
|
|
|
|
|
|
|
|
|
|
|
where dabc is the group theoretic factor related to the axial anomaly, which typically appears in the
|
|
|
|
|
|
|
|
|
|
|
|
computation of the anomalous triangle diagram corresponding to three non-abelian gauge fields cou-
|
|
|
|
|
|
|
|
|
|
|
|
pled to a chiral fermion. The subscripts R, L stand for the contributions of right-handed and left-
|
|
|
|
|
|
|
|
|
|
|
|
handed fermions. Anomalies are responsible for a non-vanishing value of the divergence of the cur-
|
|
|
|
|
|
|
|
|
|
|
|
rent, that reads in this case [22]
|
|
|
|
|
|
|
|
|
|
|
|
D<EFBFBD> Ja<4A>
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
<20>
|
|
|
|
|
|
|
|
|
|
|
|
dabc 322
|
|
|
|
|
|
|
|
|
|
|
|
F<EFBFBD>b
|
|
|
|
|
|
|
|
|
|
|
|
Fc
|
|
|
|
|
|
|
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
|
|
(5)
|
|
|
|
|
|
|
|
|
|
|
|
Let us particularize Eq. (3) to the symmetry group UV (1) <20> UA(1). Then there are vector and axial
|
|
|
|
|
|
|
|
|
|
|
|
currents induced by the magnetic field of the vector fields, i.e.
|
|
|
|
|
|
|
|
|
|
|
|
JV
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>A 22
|
|
|
|
|
|
|
|
|
|
|
|
BV
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
JA
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>V 22
|
|
|
|
|
|
|
|
|
|
|
|
BV
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(6)
|
|
|
|
|
|
|
|
|
|
|
|
which correspond to the CME and CSE respectively. The question then arises: is it possible to get a chiral magnetic effect for a non-anomalous sym-
|
|
|
|
|
|
metry w? This means to have an induced current in symmetry w, i.e.
|
|
|
|
|
|
|
|
|
|
|
|
Jw 0 with dwab = dawb = dabw = 0 a, b.
|
|
|
|
|
|
|
|
|
|
|
|
(7)
|
|
|
|
|
|
|
|
|
|
|
|
In the rest of the manuscript we will study the possibility that anomalies can induce transport also in non-anomalous currents.
|
|
|
|
|
|
|
|
|
|
|
|
ICNFP 2016
|
|
|
|
|
|
|
|
|
|
|
|
3 Holographic model
|
|
|
|
|
|
|
|
|
|
|
|
The Kubo formula Eq. (2) has been computed in [23<32>25] at strong coupling within a Einstein-Maxwell model in 5 dim. In order to account for the anomalous effects, the model is supplemented with ChernSimons (CS) terms. In this work we will restrict to a pure gauge CS term, which mimics the axial anomaly. The action reads
|
|
|
|
|
|
|
|
|
|
|
|
1 S = 16G
|
|
|
|
|
|
|
|
|
|
|
|
d5 x
|
|
|
|
|
|
|
|
|
|
|
|
-g
|
|
|
|
|
|
|
|
|
|
|
|
R
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
12
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
1 4
|
|
|
|
|
|
|
|
|
|
|
|
FV2
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
1 4
|
|
|
|
|
|
|
|
|
|
|
|
FA2
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
1 4
|
|
|
|
|
|
|
|
|
|
|
|
FW2
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
3
|
|
|
|
|
|
|
|
|
|
|
|
A
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
FV
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
FV
|
|
|
|
|
|
|
|
|
|
|
|
+ S GH ,
|
|
|
|
|
|
|
|
|
|
|
|
(8)
|
|
|
|
|
|
|
|
|
|
|
|
where S GH is the usual Gibbons-Hawking boundary term. VM and AM are vector and axial gauge
|
|
|
|
|
|
|
|
|
|
|
|
fields, respectively, and WM is an extra gauge field associated to the non-anomalous symmetry w. The
|
|
|
|
|
|
|
|
|
|
|
|
anomalous term mixes the VM and AM fields. 1 A computation of the currents with this model leads to
|
|
|
|
|
|
|
|
|
|
|
|
J~V<> J~A<> J~W<>
|
|
|
|
|
|
|
|
|
|
|
|
= = =
|
|
|
|
|
|
|
|
|
|
|
|
S V<> S A<> S W<>
|
|
|
|
|
|
|
|
|
|
|
|
= = =
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
r
|
|
|
|
|
|
|
|
|
|
|
|
16G
|
|
|
|
|
|
|
|
|
|
|
|
FVr<EFBFBD> + 6<>
|
|
|
|
|
|
|
|
|
|
|
|
- lim
|
|
|
|
|
|
r
|
|
|
|
|
|
|
|
|
|
|
|
- 16G
|
|
|
|
|
|
|
|
|
|
|
|
FAr<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
JA<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
r
|
|
|
|
|
|
|
|
|
|
|
|
- 16G
|
|
|
|
|
|
|
|
|
|
|
|
FWr<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
JW<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
|
|
A
|
|
|
|
|
|
|
|
|
|
|
|
FV
|
|
|
|
|
|
|
|
|
|
|
|
= JV<4A> + K<> ,
|
|
|
|
|
|
|
|
|
|
|
|
(9) (10) (11)
|
|
|
|
|
|
|
|
|
|
|
|
J~<7E> stands for the holographic consistent currents, which are not gauge covariant in general. The covariant version of the currents, denoted by J<>, is the usual one appearing in the constitutive relations, and it corresponds to the covariant part, dropping the CS current K<>. An analysis of the divergence of
|
|
|
|
|
|
the covariant currents leads to the "covariant" anomalies:
|
|
|
|
|
|
|
|
|
|
|
|
D<EFBFBD> JV<4A> = -3<>F<EFBFBD>AFV ,
|
|
|
|
|
|
|
|
|
|
|
|
D<EFBFBD> JA<4A>
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
3 2
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
F<EFBFBD>V
|
|
|
|
|
|
|
|
|
|
|
|
FV
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
D<EFBFBD> JW<4A> = 0 .
|
|
|
|
|
|
|
|
|
|
|
|
(12)
|
|
|
|
|
|
|
|
|
|
|
|
From this result, it is and JA<4A>, this is not the one gets the result of
|
|
|
|
|
|
|
|
|
|
|
|
Eccaqles.ea(r6fot)hrfaoJtrW<72>wJ. Vh<56>IinalenadohnoJelA<6C>o,egbxrupatepcahtisvcatchnoeimshepxinuisgttaevtniaoclnueeoofffotahrneJoW<6F>cmo.nadlouucstitvriatniesspworitthefftheicstsmiondJeVl<56>,
|
|
|
|
|
|
|
|
|
|
|
|
3.1 Holographic model with symmetry breaking
|
|
|
|
|
|
|
|
|
|
|
|
In the following we are going to study the possibility that the constitutive relation for JW<4A> receives anomalous contributions. Let us extend the model of Eq. (8) with the contribution (S tot = S + S )
|
|
|
|
|
|
|
|
|
|
|
|
1 S = 16G
|
|
|
|
|
|
|
|
|
|
|
|
d5 x -g - |DM|2 - m22 ,
|
|
|
|
|
|
|
|
|
|
|
|
DM = [M - i(AM - WM)] ,
|
|
|
|
|
|
|
|
|
|
|
|
(13)
|
|
|
|
|
|
|
|
|
|
|
|
where is a scalar field with a tachyonic bulk mass m2 = ( - 4), and 0 4. S produces an explicit breaking of A and W symmetries via the scalar field . From the AdS/CFT dictionary, the
|
|
|
|
|
|
model is the holographic dual of a Conformal Field Theory (CFT) with a deformation
|
|
|
|
|
|
|
|
|
|
|
|
L = LCFT + O ,
|
|
|
|
|
|
|
|
|
|
|
|
(14)
|
|
|
|
|
|
|
|
|
|
|
|
where O is an operator dual of the scalar field with dim O = , and is the source of the operator with dim = 4 - . The near boundary expansion of reads
|
|
|
|
|
|
|
|
|
|
|
|
(r) = 4-r-4 + r- + <20> <20> <20> ,
|
|
|
|
|
|
|
|
|
|
|
|
r ,
|
|
|
|
|
|
|
|
|
|
|
|
1We use the notation for capital indices M {r, t, x, y, z}, and Greek indices <20> {t, x, y, z}.
|
|
|
|
|
|
|
|
|
|
|
|
(15)
|
|
|
|
|
|
|
|
|
|
|
|
EPJ Web of Conferences
|
|
|
|
|
|
|
|
|
|
|
|
where 4- is interpreted as the source , and as the condensate O . In the following we will choose = 3, so that the bulk mass is m2 = -3. The explicit breaking of symmetries is realized
|
|
|
|
|
|
via the boundary term limr r <20> (r) = M, where we have identified the mass parameter M with the
|
|
|
|
|
|
source in Eq. (15). Our goal is to study the induced currents
|
|
|
|
|
|
|
|
|
|
|
|
JV = V (M)B , JA = A(M)B , JW = W (M)B ,
|
|
|
|
|
|
|
|
|
|
|
|
(16)
|
|
|
|
|
|
|
|
|
|
|
|
where one expects a dependence of the conductivities in M. A non zero value for W (M) would signal the existence of a non-anomalous current induced by anomalies.
|
|
|
|
|
|
|
|
|
|
|
|
3.2 Background equations of motion
|
|
|
|
|
|
|
|
|
|
|
|
We will work in the probe limit, so that the metric fluctuations are neglected. The equations of motion
|
|
|
|
|
|
|
|
|
|
|
|
of the background (gMN , V M, AM, W M, ) can be solved by considering the AdS Schwarzschild
|
|
|
|
|
|
|
|
|
|
|
|
solution
|
|
|
|
|
|
|
|
|
|
|
|
d s2
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
-r2
|
|
|
|
|
|
|
|
|
|
|
|
f
|
|
|
|
|
|
|
|
|
|
|
|
(r)dt2
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
dr2 r2 f (r)
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
r2
|
|
|
|
|
|
|
|
|
|
|
|
dx2 + dy2 + dz2
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
f (r)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
rh4 r4
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(17)
|
|
|
|
|
|
|
|
|
|
|
|
and the background gauge fields
|
|
|
|
|
|
|
|
|
|
|
|
V = Vt(r)dt , A = At(r)dt , W = Wt(r)dt .
|
|
|
|
|
|
|
|
|
|
|
|
(18)
|
|
|
|
|
|
|
|
|
|
|
|
The chemical potentials are computed as <20>Y Yt(r ) - Yt(rh) with Y = V, A, W. Then, the fields have the following near boundary expansion
|
|
|
|
|
|
|
|
|
|
|
|
lim r <20> (r) = M ,
|
|
|
|
|
|
|
|
|
|
|
|
(19)
|
|
|
|
|
|
|
|
|
|
|
|
r
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
r
|
|
|
|
|
|
|
|
|
|
|
|
Vt(r)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>V
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
Vt (rh )
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
r
|
|
|
|
|
|
|
|
|
|
|
|
At(r)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>A
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
At(rh)
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
r
|
|
|
|
|
|
|
|
|
|
|
|
Wt(r)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>W
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
Wt(rh)
|
|
|
|
|
|
|
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
|
|
(20)
|
|
|
|
|
|
|
|
|
|
|
|
It is convenient to define in the following new fields A1 = A - W and A2 = A + W, so that the covariant derivative writes DM = [M - i(A1)M] . The boundary expansions of A1,2 write as in Eq. (20), with chemical potentials <20>1,2 <20>A <20>W . Then, the equations of motion of the background read
|
|
|
|
|
|
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
Vt
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
3 r
|
|
|
|
|
|
|
|
|
|
|
|
Vt
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
(A2)t
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
3 r
|
|
|
|
|
|
|
|
|
|
|
|
(A2)t
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(21)
|
|
|
|
|
|
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
(A1)t
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
3 r
|
|
|
|
|
|
|
|
|
|
|
|
(A1)t
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
42 r2 f (r) (A1)t
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(22)
|
|
|
|
|
|
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
5 f +
|
|
|
|
|
|
rf
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
(A1)2t r4 f 2
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
m2 r2 f
|
|
|
|
|
|
|
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
|
|
(23)
|
|
|
|
|
|
|
|
|
|
|
|
The solutions of Vt and (A2)t can be obtained analytically, and the result is
|
|
|
|
|
|
|
|
|
|
|
|
Vt(r)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
cV
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>V
|
|
|
|
|
|
|
|
|
|
|
|
rh2 r2
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(A2)t(r)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
c2
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>2
|
|
|
|
|
|
|
|
|
|
|
|
rh2 r2
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(24)
|
|
|
|
|
|
|
|
|
|
|
|
where cV and c2 are integration constants. The solutions of A1 and can be obtained numerically by solving the coupled system of differential equations, Eqs. (22)-(23). From these equations one can easily see that regularity of the solution near the horizon demands (A1)t(rh) = 0.
|
|
|
|
|
|
|
|
|
|
|
|
ICNFP 2016
|
|
|
|
|
|
|
|
|
|
|
|
4 Conductivities in presence of symmetry breaking
|
|
|
|
|
|
|
|
|
|
|
|
The Kubo formulae for the conductivities appearing in Eq. (16) are:
|
|
|
|
|
|
|
|
|
|
|
|
Y
|
|
|
|
|
|
|
|
|
|
|
|
i = lim
|
|
|
|
|
|
pz pz
|
|
|
|
|
|
|
|
|
|
|
|
(JY )x(JV )y
|
|
|
|
|
|
|
|
|
|
|
|
|=0 ,
|
|
|
|
|
|
|
|
|
|
|
|
with Y = V, A, W .
|
|
|
|
|
|
|
|
|
|
|
|
(25)
|
|
|
|
|
|
|
|
|
|
|
|
These involve retarded correlators which can be computed by using the AdS/CFT dictionary, see e.g. [5, 26<32>28]. We will explain in this section the computation in some details.
|
|
|
|
|
|
|
|
|
|
|
|
4.1 Fluctuations
|
|
|
|
|
|
|
|
|
|
|
|
Without loss of generality we consider perturbations of momentum p in the z-direction at zero fre-
|
|
|
|
|
|
|
|
|
|
|
|
quency. To study the effect of anomalies it is enough to consider the shear sector, i.e. transverse
|
|
|
|
|
|
|
|
|
|
|
|
momentum fluctuations,
|
|
|
|
|
|
|
|
|
|
|
|
K<EFBFBD> = K<>(r) + k<>(r)eipzz , <20> = x, y ,
|
|
|
|
|
|
|
|
|
|
|
|
(26)
|
|
|
|
|
|
|
|
|
|
|
|
where K<>(r) refers to the background solutions computed in Sec. 3.2 for any of the fields V, A1,2 (or V, A and W), and k<>(r) are the corresponding fluctuations v, a1,2 (or v, a and w).
|
|
|
|
|
|
In studying the fluctuations it is useful to organize the equations of motion according to their
|
|
|
|
|
|
helicity under the transverse SO(2) rotation, which is a left-over symmetry. The equations for the fluctuations of the gauge fields are then classified into helicity <20>1, and at order O() they read 2
|
|
|
|
|
|
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
v<EFBFBD> +
|
|
|
|
|
|
|
|
|
|
|
|
3 f +
|
|
|
|
|
|
rf
|
|
|
|
|
|
|
|
|
|
|
|
v<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
pz 3r4 f
|
|
|
|
|
|
|
|
|
|
|
|
3pz <20> 4r (A1)t + (A2)t
|
|
|
|
|
|
|
|
|
|
|
|
v<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4 pz 3r3 f
|
|
|
|
|
|
|
|
|
|
|
|
((a1)<29>
|
|
|
|
|
|
|
|
|
|
|
|
+ (a2)<29>) Vt ,
|
|
|
|
|
|
|
|
|
|
|
|
(27)
|
|
|
|
|
|
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
(a1)<29> +
|
|
|
|
|
|
|
|
|
|
|
|
3 f +
|
|
|
|
|
|
rf
|
|
|
|
|
|
|
|
|
|
|
|
(a1)<29>
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
1 r4 f
|
|
|
|
|
|
|
|
|
|
|
|
p2z
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
4r22
|
|
|
|
|
|
|
|
|
|
|
|
(a1)<29>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
8pz 3r3 f
|
|
|
|
|
|
|
|
|
|
|
|
Vtv<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(28)
|
|
|
|
|
|
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
(a2)<29> +
|
|
|
|
|
|
|
|
|
|
|
|
3 f +
|
|
|
|
|
|
rf
|
|
|
|
|
|
|
|
|
|
|
|
(a2)<29>
|
|
|
|
|
|
|
|
|
|
|
|
-
|
|
|
|
|
|
|
|
|
|
|
|
p2z r4 f
|
|
|
|
|
|
|
|
|
|
|
|
(a2)<29>
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
8pz 3r3 f
|
|
|
|
|
|
|
|
|
|
|
|
Vt v<>
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(29)
|
|
|
|
|
|
|
|
|
|
|
|
where the fields are defined as v<> = vx<76>ivy and (a1,2)<29> = (a1,2)x<>i(a1,2)y. In order to obtain the retarded correlation functions we should perform a low momentum expansion of the fluctuation solutions, i.e.
|
|
|
|
|
|
|
|
|
|
|
|
v<EFBFBD> = v(<28>0) + pzv(<28>1) + <20> <20> <20> ,
|
|
|
|
|
|
|
|
|
|
|
|
(a1,2)<29> = (a1,2)(<28>0) + pz(a1,2)(<28>1) + <20> <20> <20> .
|
|
|
|
|
|
|
|
|
|
|
|
(30)
|
|
|
|
|
|
|
|
|
|
|
|
From Eqs. (27)-(29) one gets the corresponding equations at order O(p0z ) and O(pz), which can be solved by imposing the appropriate boundary conditions for the fluctuation fields. In general we should consider regularity of the fields up to the horizon, and the sourceless condition, which means
|
|
|
|
|
|
that the fluctuations cannot modify the background fields at the boundary. Then
|
|
|
|
|
|
|
|
|
|
|
|
k<EFBFBD>(rh) = finite , k<>(r ) = 0 ,
|
|
|
|
|
|
|
|
|
|
|
|
(31)
|
|
|
|
|
|
|
|
|
|
|
|
for any of the fluctuation fields k<> = v<> or (a1,2)<29>. The only fluctuation that can be computed analytically is (a2)<29>, and the solution at order O(pz) reads
|
|
|
|
|
|
|
|
|
|
|
|
(a2)(<28>1)(r)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
c 4 3
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>V rh2
|
|
|
|
|
|
|
|
|
|
|
|
log 1
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
rh2 r2
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(32)
|
|
|
|
|
|
|
|
|
|
|
|
2The equation for the scalar field is only affected by these perturbations at order O(2).
|
|
|
|
|
|
|
|
|
|
|
|
EPJ Web of Conferences
|
|
|
|
|
|
|
|
|
|
|
|
where c is an integration constant. In a near boundary expansion, the fields behave as
|
|
|
|
|
|
|
|
|
|
|
|
v(<28>1)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
C0,v
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
C2,v r2
|
|
|
|
|
|
|
|
|
|
|
|
log r
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
C2,v r2
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD><EFBFBD><EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(33)
|
|
|
|
|
|
|
|
|
|
|
|
(a1)(<28>1)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
C0,a1
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
C2,a1 r2
|
|
|
|
|
|
|
|
|
|
|
|
log r +
|
|
|
|
|
|
|
|
|
|
|
|
C2,a1 r2
|
|
|
|
|
|
|
|
|
|
|
|
+<2B><><EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(34)
|
|
|
|
|
|
|
|
|
|
|
|
(a2)(<28>1)
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
C0,a2
|
|
|
|
|
|
|
|
|
|
|
|
+
|
|
|
|
|
|
|
|
|
|
|
|
C2,a2 r2
|
|
|
|
|
|
|
|
|
|
|
|
log r +
|
|
|
|
|
|
|
|
|
|
|
|
C2,a2 r2
|
|
|
|
|
|
|
|
|
|
|
|
+<2B><><EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
|
|
(35)
|
|
|
|
|
|
|
|
|
|
|
|
The sourceless condition for the fluctuations imply C0,v = C0,a1 = C0,a2 = 0. The other coefficients C2, C2 can be obtained from a numerical solution of the equations of motion of the fluctuations.
|
|
|
|
|
|
|
|
|
|
|
|
4.2 Conductivities
|
|
|
|
|
|
|
|
|
|
|
|
The correlators of Eq. (25) are contained in the coefficients C2,v, C2,a1 and C2,a2 . In particular, we have the following contributions
|
|
|
|
|
|
|
|
|
|
|
|
i
|
|
|
|
|
|
|
|
|
|
|
|
i
|
|
|
|
|
|
|
|
|
|
|
|
i
|
|
|
|
|
|
|
|
|
|
|
|
C2,v = lim
|
|
|
|
|
|
pz 0
|
|
|
|
|
|
|
|
|
|
|
|
pz
|
|
|
|
|
|
|
|
|
|
|
|
Y
|
|
|
|
|
|
|
|
|
|
|
|
JV
|
|
|
|
|
|
=V,1,2
|
|
|
|
|
|
|
|
|
|
|
|
JY
|
|
|
|
|
|
|
|
|
|
|
|
|=0 ,
|
|
|
|
|
|
|
|
|
|
|
|
C2,a1
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
pz 0
|
|
|
|
|
|
|
|
|
|
|
|
pz
|
|
|
|
|
|
|
|
|
|
|
|
J1
|
|
|
|
|
|
Y=V,1,2
|
|
|
|
|
|
|
|
|
|
|
|
JY
|
|
|
|
|
|
|
|
|
|
|
|
|=0 ,
|
|
|
|
|
|
|
|
|
|
|
|
C2,a2
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
pz 0
|
|
|
|
|
|
|
|
|
|
|
|
pz
|
|
|
|
|
|
|
|
|
|
|
|
J2
|
|
|
|
|
|
Y=V,1,2
|
|
|
|
|
|
|
|
|
|
|
|
JY
|
|
|
|
|
|
|
|
|
|
|
|
|=0 .
|
|
|
|
|
|
|
|
|
|
|
|
(36)
|
|
|
|
|
|
|
|
|
|
|
|
Using the relation between the conductivities in the (A, W) and (A1, A2) basis
|
|
|
|
|
|
|
|
|
|
|
|
1,2
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
pz 0
|
|
|
|
|
|
|
|
|
|
|
|
i pz
|
|
|
|
|
|
|
|
|
|
|
|
J1,2 JV
|
|
|
|
|
|
|
|
|
|
|
|
= A W ,
|
|
|
|
|
|
|
|
|
|
|
|
(37)
|
|
|
|
|
|
|
|
|
|
|
|
one can express them as
|
|
|
|
|
|
|
|
|
|
|
|
V
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
pz 0
|
|
|
|
|
|
|
|
|
|
|
|
i pz
|
|
|
|
|
|
|
|
|
|
|
|
JV JV
|
|
|
|
|
|
|
|
|
|
|
|
|=0 ,
|
|
|
|
|
|
|
|
|
|
|
|
A,W
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
lim
|
|
|
|
|
|
pz 0
|
|
|
|
|
|
|
|
|
|
|
|
i 2 pz
|
|
|
|
|
|
|
|
|
|
|
|
(
|
|
|
|
|
|
|
|
|
|
|
|
J2 JV
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
J1 JV ) |=0 ,
|
|
|
|
|
|
|
|
|
|
|
|
(38)
|
|
|
|
|
|
|
|
|
|
|
|
where the sign <20> corresponds to the case A and W respectively. We show in Fig. 1 the result for the chiral conductivities of Eq. (38) after solving numerically the equations of motion of the fluctuations. We find a non zero value for W (M) in presence of explicit symmetry breaking, i.e. M 0. In particular, we observe the following properties:
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD> W (0) = 0 .
|
|
|
|
|
|
|
|
|
|
|
|
<EFBFBD>
|
|
|
|
|
|
|
|
|
|
|
|
W ()
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
A()
|
|
|
|
|
|
|
|
|
|
|
|
=
|
|
|
|
|
|
|
|
|
|
|
|
1 2
|
|
|
|
|
|
|
|
|
|
|
|
A
|
|
|
|
|
|
|
|
|
|
|
|
(0)
|
|
|
|
|
|
|
|
|
|
|
|
.
|
|
|
|
|
|
|
|
|
|
|
|
The first property is just the expected behavior of the conductivity of the non-anomalous symmetry W
|
|
|
|
|
|
|
|
|
|
|
|
in absence of symmetry breaking. The second one can be understood intuitively in the following way: In the basis (A1, A2) the CS term in the action is
|
|
|
|
|
|
|
|
|
|
|
|
1 S CS = 16G
|
|
|
|
|
|
|
|
|
|
|
|
d5x -g
|
|
|
|
|
|
|
|
|
|
|
|
6
|
|
|
|
|
|
|
|
|
|
|
|
(A1
|
|
|
|
|
|
|
|
|
|
|
|
FV
|
|
|
|
|
|
|
|
|
|
|
|
FV
|
|
|
|
|
|
|
|
|
|
|
|
+ A2
|
|
|
|
|
|
|
|
|
|
|
|
FV
|
|
|
|
|
|
|
|
|
|
|
|
FV)
|
|
|
|
|
|
|
|
|
|
|
|
,
|
|
|
|
|
|
|
|
|
|
|
|
(39)
|
|
|
|
|
|
|
|
|
|
|
|
so that both symmetries A1 and A2 have a CS interaction. The scalar field however breaks only A1. At M = 0 there will be CSE for both A1 and A2, but for large M, A1 will be badly broken and the CSE in the J1 current goes to zero. 3 Since 1 vanishes at M , we find from Eq. (37) the second property.
|
|
|
|
|
|
3See [29] for a similar effect.
|
|
|
|
|
|
|
|
|
|
|
|
ICNFP 2016
|
|
|
|
|
|
|
|
|
|
|
|
V , A , W V , A , W
|
|
|
|
|
|
|
|
|
|
|
|
0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000
|
|
|
|
|
|
0
|
|
|
|
|
|
|
|
|
|
|
|
V
|
|
|
|
|
|
|
|
|
|
|
|
A W
|
|
|
|
|
|
|
|
|
|
|
|
5
|
|
|
|
|
|
|
|
|
|
|
|
10
|
|
|
|
|
|
|
|
|
|
|
|
15
|
|
|
|
|
|
|
|
|
|
|
|
20
|
|
|
|
|
|
|
|
|
|
|
|
25
|
|
|
|
|
|
|
|
|
|
|
|
MT
|
|
|
|
|
|
|
|
|
|
|
|
0.035
|
|
|
|
|
|
0.030
|
|
|
|
|
|
V
|
|
|
|
|
|
0.025
|
|
|
|
|
|
0.020 A
|
|
|
|
|
|
0.015
|
|
|
|
|
|
0.010
|
|
|
|
|
|
0.005 W
|
|
|
|
|
|
0.000 0 2 4 6 8 10 12 14
|
|
|
|
|
|
M V
|
|
|
|
|
|
|
|
|
|
|
|
Figure 1. Vector V , axial A and induced W conductivities as a function of M/T (left) and M/<2F>V (right). Left panel: we have considered <20>V = 0.4, <20>A = 0.5, <20>W = 0.6 and T = 1. Right panel: We have considered the same values of the chemical potentials, for temperatures T = 1 (solid), T = 0.5 (dashed) and T = 0.3 (dot dashed).
|
|
|
|
|
|
|
|
|
|
|
|
5 Discussion and outlook
|
|
|
|
|
|
In this work we have studied the role played by the axial anomaly in the hydrodynamics of relativistic fluids in presence of external magnetic fields. The anomalous conductivities have been computed by using Kubo formulae. The most interesting result is the characterization of a novel phenomenon related to the possibility that, when symmetries are explicitly broken, anomalies can induce transport effects not only in anomalous currents, but also in non-anomalous ones, i.e. those with a vanishing divergence. We have studied this phenomenon at strong coupling in a holographic Einstein-Maxwell model in 5 dim supplemented with a pure gauge Chern-Simons term. The symmetry breaking is introduced through a scalar field which is dual to an operator O with dim O = 3.
|
|
|
|
|
|
We plan to extend this study to other anomalous coefficients, like the chiral vortical conductivity. In addition, it would be interesting to check whether the mixed gauge-gravitational anomaly could induce any effect as well in non-anomalous currents. These and other issues will be addressed in detail in a forthcoming publication [30].
|
|
|
|
|
|
Acknowledgements
|
|
|
|
|
|
I would like to thank S.D. Chowdhury, J.R. David, K. Jensen, and especially K. Landsteiner for valuable discussions. This work has been supported by Plan Nacional de Altas Energ<72>as Spanish MINECO grant FPA201564041-C2-1-P, and by the Spanish Consolider Ingenio 2010 Programme CPAN (CSD2007-00042). I thank the Instituto de F<>sica Te<54>rica UAM/CSIC, Madrid, Spain, for their hospitality during the completion of the final stages of this work. The research of E.M. is supported by the European Union under a Marie Curie Intra-European fellowship (FP7-PEOPLE-2013-IEF) with project number PIEF-GA-2013-623006, and by the Universidad del Pa<50>s Vasco UPV/EHU, Bilbao, Spain, as a Visiting Professor.
|
|
|
|
|
|
References
|
|
|
|
|
|
[1] P. Kovtun, J. Phys. A45, 473001 (2012) [2] K. Fukushima, D.E. Kharzeev, H.J. Warringa, Phys. Rev. D78, 074033 (2008) [3] D.T. Son, P. Surowka, Phys. Rev. Lett. 103, 191601 (2009) [4] I. Amado, K. Landsteiner, F. Pena-Benitez, JHEP 1105, 081 (2011)
|
|
|
|
|
|
|
|
|
|
|
|
EPJ Web of Conferences
|
|
|
|
|
|
[5] K. Landsteiner, E. Megias, F. Pena-Benitez, Lect. Notes Phys. 871, 433 (2013) [6] S.D. Chowdhury, J.R. David, JHEP 11, 048 (2015) [7] J.L. Manes, M. Valle, JHEP 01, 008 (2013) [8] S. Bhattacharyya, V.E. Hubeny, S. Minwalla, M. Rangamani, JHEP 0802, 045 (2008) [9] J. Erdmenger, M. Haack, M. Kaminski, A. Yarom, JHEP 01, 055 (2009) [10] N. Banerjee et al., JHEP 01, 094 (2011) [11] E. Megias, F. Pena-Benitez, JHEP 05, 115 (2013) [12] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla, T. Sharma, JHEP 09, 046
|
|
|
|
|
|
(2012) [13] K. Jensen, Phys. Rev. D85, 125017 (2012) [14] K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz, A. Yarom, Phys. Rev. Lett. 109, 101601
|
|
|
|
|
|
(2012) [15] E. Megias, M. Valle, JHEP 11, 005 (2014) [16] D.E. Kharzeev, H.J. Warringa, Phys. Rev. D80, 034028 (2009) [17] K. Landsteiner, E. Megias, F. Pena-Benitez, Phys. Rev. Lett. 107, 021601 (2011) [18] B. Keren-Zur, Y. Oz, JHEP 06, 006 (2010) [19] G.M. Newman, D.T. Son, Phys. Rev. D73, 045006 (2006) [20] G. Basar, D.E. Kharzeev, H.U. Yee, Phys. Rev. B89, 035142 (2014) [21] K. Landsteiner, Phys. Rev. B89, 075124 (2014) [22] T. Kumura, Prog. Theor. Phys. 42, 1191 (1969) [23] K. Landsteiner, E. Megias, L. Melgar, F. Pena-Benitez, JHEP 09, 121 (2011) [24] K. Landsteiner, E. Megias, L. Melgar, F. Pena-Benitez, J. Phys. Conf. Ser. 343, 012073 (2012) [25] K. Landsteiner, E. Megias, F. Pena-Benitez, Phys. Rev. D90, 065026 (2014) [26] D.T. Son, A.O. Starinets, JHEP 09, 042 (2002) [27] C.P. Herzog, D.T. Son, JHEP 03, 046 (2003) [28] M. Kaminski, K. Landsteiner, J. Mas, J.P. Shock, J. Tarrio, JHEP 02, 021 (2010) [29] A. Jimenez-Alba, K. Landsteiner, Y. Liu, Y.W. Sun, JHEP 07, 117 (2015) [30] E. Megias, work in progress (2016)
|
|
|
|
|
|
|
|
|
|
|
|
|
