On the Chiral Magnetic Effect in Soft-Wall AdS/QCD

0.17 MB·English

by  A. Gorsky

ITEP-TH-07/10
On the Chiral Magnetic Effect in Soft-Wall AdS/QCD
A. Gorsky1, P. N. Kopnin1,2 and A. V. Zayakin1,3
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1 Institute of Theoretical and Experimental Physics,
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B. Cheremushkinskaya ul. 25, 117259 Moscow, Russia
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J 2 Moscow Institute of Physics and Technology,

4 Institutsky per. 9, 141 700 Dolgoprudny, Russia

]
h 3 Fakult¨at fu¨r Physik der Ludwig-Maximillians-Universit¨at Mu¨nchen und
p
Maier-Leibniz-Laboratory, Am Coulombwall 1, 85748 Garching, Germany
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p
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h
[

3
v
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9
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. Abstract
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The essence of the chiral magnetic effect is generation of an electric current along an
1
: external magnetic field. Recently it has been studied by Rebhan et al. within the
v
i Sakai–Sugimotomodel, where itwas shown tobezero. Asanalternative, we calculate
X
the chiral magnetic effect in soft-wall AdS/QCD and find a non-zero result with the
r
a natural boundary conditions. The mechanism of the dynamical neutralization of the
chiral chemical potential via the string production is discussed in the dual two-form
representation.
1 Introduction
Chiral magnetic effect [1, 2, 3] (CME) is best described as a generation of an electric
current by a magnetic field in a topologically nontrivial background. The standard
field-theoretical argumentation is the following. Let us consider QCD with massless
quarks, so that left and right quarks can be dealt with independently, and suppose a
chiralchemical potentialµ ispresent, accountingforacertaintopologicallynontrivial
5
background. The topologically nontrivial field configuration changes chirality, and an
external magnetic field B = (0,0,B) orders spins parallel to itself. Thus arises a
non-zero vector current, which is given by Fukushima, Kharzeev and Warringa [3]
µ B
JV = 5 . (1)
3 2π2 ≡ JFKW
During recent years holography has become one of the main alternative tools for
analyzing non-perturbative QCD. Different conductivities of quark matter, including
chiral magnetic conductivity, have already been analyzed in a variety of holographic
models. Electric conductivity in the D3/D7 model was examined by Karch and
O’Bannon in [4]. Axial, ohmic and Hall conductivity in a magnetic field were calcu-
lated on the basis of the Kubo formula and correlator analysis for the Sakai-Sugimoto
model in [5, 6]. One of the results for electric current in [5] is 1/2 of QCD weak
coupling result (1).
An attempt to describe the chiral magnetic effect for the vector current in Sakai-
Sugimoto model has been made recently in [7]. The result at zero frequency, where
only the Yang–Mills part of the action was used, exactly amounts to the weak cou-
pling QCD effect; non-zero frequencies have also been considered. In [10] a more
sophisticated anomaly subtraction scheme was suggested. It was argued that if one
uses the Bardeen term subtraction, then one gets zero effect for vector current, oth-
erwise one gets 2/3 of the weak-coupling effect. The reason for adding the Bardeen
term to the action was to cure the pathological behavior of the vector anomaly. Note
that a similar effect for the axial current in a theory with the conventional chemical
potential has been discussed in [8, 9].
Experimental status of the problem is discussed in [15]. Presently it is claimed
µ B
that the effect is present, yet the exact proportionality coefficient c in JV = c 5
3 · 2π2
cannotbeinferredfromit. Latticeestimatesarealsocloseto2/3[11]ofweak-coupling
effect. The discussion of the effect in the framework of NJL model can be found in
[12]. Chiral magnetic effect at low temperature was considered in [13]. An analog of
this effect is known in superfluid helium [14].
The present note aims at comparing the calculations of [10] to the chiral mag-
netic effect effect as derived in the framework of soft-wall AdS/QCD. The question
whether the effect is present in a holographic model or not, turns out to be quite
delicate. The paper is organized as follows. In Section 2 we consider the analysis
1
of the gauge sector of the soft-wall model and confirm the result of [10]. In Sec-
tion 3 we discuss the contribution of scalars and pseudoscalars and focus on their
boundary conditions in the 5d equations of motion. Section 4 is devoted to the dual
representation of the chiral chemical potential and the mechanism of its possible dy-
namical neutralization via the Schwinger-type process. The results of this note are
summarized in the Conclusion.
2 The soft-wall model
2.1 Gauge part of the action
Let us consider the gauge field sector of the soft-wall AdS/QCD model [18] taking
into account the Chern-Simons action A F F. We begin our consideration with
∧ ∧
an action of Abelian fields L and R wiRth a coupling g that has the following form:
5
S = S [L]+S [R]+S [L] S [R] (2)
YM YM CS CS
−
1 1
S [A] = e−φF F = dz d4x e−φ√gF FMN (3)
YM −8g2 Z ∧∗ −8g2 Z MN
5 5
k N 1 1
c
S [A] = · A F F A A A F + A A A A A
CS
− 24π2 Z ∧ ∧ − 2 ∧ ∧ ∧ 10 ∧ ∧ ∧ ∧
k N
= · c dz d4x ǫMNPQRA F F . (4)
M NP QR
− 24π2 Z
Here k isan integer that scales the CS term andeffectively the magnetic field. Canon-
ical normalization of the CS term is k = 1, but it will be kept it for the sake of
generality. The metric tensor is the following:
R2 R2
ds2 = g dXMdXN = η dXMdXN = ( dz2 +dx dxµ). (5)
MN MN µ
z2 z2 −
In the A = 0 gauge the YM action acquires the form
z
R e−φ e−φ
S [A] = dz d4x A ((cid:3)ηµν ∂µ∂ν)A +A ∂ ∂ Aµ
YM −4g2 Z (cid:26) z µ − ν µ z(cid:18) z z(cid:19) (cid:27)
5
R e−φ z=∞
+ d4x A ∂ Aµ . (6)
4g2 Z z µ z (cid:12)
5 (cid:12)z=0
(cid:12)
(cid:12)
From the YM part of the action we get
δS [A] R e−φ e−φ
YM = ((cid:3)ηµν ∂µ∂ν)A +∂ ∂ Aµ . (7)
δA −2g2 (cid:26) z − ν z(cid:18) z z (cid:19)(cid:27)
µ 5
2
Varying the volume term of the action one gets
δS [A] k N
CS = · c ǫµνρσ ∂ A F . (8)
z ν ρσ
δA 2π2
µ
R N
c
Taking into account = , one obtains the equations of motion for the
g2 12π2
5
fields L and R
e−φ(z)
∂ ∂ Lµ 24kǫµνρσ ∂ L ∂ L = 0 (9)
z z z ν ρ σ
(cid:18) z (cid:19)−
e−φ(z)
∂ ∂ Rµ +24kǫµνρσ ∂ R ∂ R = 0 (10)
z z z ν ρ σ
(cid:18) z (cid:19)
with the following boundary conditions
L (0) = µ , R (0) = µ , (11)
0 L 0 R
L (0) = j , R (0) = j , (12)
3 L 3 R
1 1
L (0,x ) = x B, R (0,x ) = x B, (13)
1 2 2 1 2 2
−2 −2
L ( ) = R ( ), ∂ L ( ) = ∂ R ( ), (14)
µ µ z µ z µ
∞ ∞ ∞ − ∞
1 1
here µ = (µ +µ ),µ = (µ µ ), and j are the gauge field boundary values,
L R 5 L R L,R
2 2 −
a variation with respect to which gives the currents
δS[L,R] 1 ∂S[L,R]
= = , (15)
L
δL (z = 0) V ∂j J
3 4D L
δS[L,R] 1 ∂S[L,R]
= = . (16)
R
δR (z = 0) V ∂j J
3 4D R
Conditions (14) arise because both left- and right-handed gauge fields are associated
with a single gauge field in the Sakai–Sugimoto model [16, 17]. In that model, regions
of positive and negative values of the holographic coordinate ρ = 1/z correspond
to left-handed D8 and right-handed D¯8 branes respectively. Since the gauge field
is smooth and continuous at ρ = 0, a boundary condition (14) is obtained at z =
1/ρ = . Furthermore, (14) may be understood as a zero boundary condition for
∞
the axial gauge field that is usually imposed at the black hole horizon, which in our
zero-temperature case is located at z = .
∞
3
Denoting β = 12kB one can get the following set of e.o.m.’s
e−φ(z) e−φ(z)
∂ ∂ L = β∂ L , ∂ ∂ L = β∂ L , (17)
z z 0 z 3 z z 3 z 0
(cid:18) z (cid:19) (cid:18) z (cid:19)
e−φ(z) e−φ(z)
∂ ∂ R = β∂ R , ∂ ∂ R = β∂ R , (18)
z z 0 z 3 z z 3 z 0
(cid:18) z (cid:19) − (cid:18) z (cid:19) −
e−φ(z) e−φ(z)
∂ ∂ L = 0, ∂ ∂ R = 0. (19)
z z 1 z z 1
(cid:18) z (cid:19) (cid:18) z (cid:19)
Solution is the following
1 1
L (z) = µ + µ j e−|β|w(z) 1 , L (z) =j µ j e−|β|w(z) 1 ,
0 L 5 5 3 L 5 5
(cid:18) − 2 (cid:19) − −(cid:18) − 2 (cid:19) −
(cid:0) (cid:1) (cid:0) (cid:1)
1 1
R (z) = µ µ + j e−|β|w(z) 1 , R (z) =j µ + j e−|β|w(z) 1 ,
0 R 5 5 3 R 5 5
−(cid:18) 2 (cid:19) − −(cid:18) 2 (cid:19) −
(cid:0) (cid:1) (cid:0) (cid:1)
1 1
R (z,x ) = x B, L (z,x ) = x B, (20)
1 2 2 1 2 2
−2 −2
z e−φ(z)
here j = j +j ,j = j j , and w(z) = du u eφ(u), w′(z) = 1.
L R 5 L R
− z
R0
2.2 On-shell action and symmetry currents
Let us now compute the on-shell action with the gauge fields given by (20) for both
left- and right-handed gauge fields. Its Yang–Mills part is given as
e−λz2 R
S [A] = dz d4x ηABηMNF F
YM −Z z 8g2 AM BN
5
R e−λz2
= dz d4x 2ηαβ∂ A ∂ A +ηαβηµνF F
−8g2 Z z − z α z β αµ βν
5 (cid:8) (cid:9)
R B2 e−λz2
= V dz (21)
−4g2 4 4DZ z
5
The Chern–Simons part of the action is
k N
S [A] = · c dz d4x ǫµνρσ A ∂ A F
CS µ z ν ρσ
6π2 Z
k N
= · c dz d4x (A ∂ A A ∂ A )F . (22)
3π2 Z 0 z 3 − 3 z 0 12
z
Recall that w(z) = du u eφ(u),w(0) = 0,w( ) = . Then the solutions (20) have
∞ ∞
R0
4
the following form
1 1 1 1
L (z) = µ+ j + µ j e−|β|w(z), L (z) = µ + j µ j e−|β|w(z),
0 5 5 5 3 5 5 5
2 (cid:18) − 2 (cid:19) 2 −(cid:18) − 2 (cid:19)
1 1 1 1
R (z) = µ+ j µ + j e−|β|w(z), R (z) = µ + j µ + j e−|β|w(z),
0 5 5 5 3 5 5 5
2 −(cid:18) 2 (cid:19) 2 −(cid:18) 2 (cid:19)
1 1
FL = B, FR = B. (23)
12 2 12 2
Upon substituting (23) into (22) the on-shell CS action becomes
k N
S [L] = · c dz d4x FL (L ∂ L L ∂ L )
CS 3π2 Z 12 0 z 3 − 3 z 0
k N 1 1 1 1
= · cBV µµ +µ2 µj + µ j j2 jj , (24)
6π2 4D(cid:18) 5 5 − 2 5 2 5 − 4 5 − 4 5(cid:19)
and
k N
S [R] = · c dz d4x FR(R ∂ R R ∂ R )
CS 3π2 Z 12 0 z 3 − 3 z 0
k N 1 1 1 1
= · cBV µµ µ2 + µj µ j + j2 jj . (25)
6π2 4D(cid:18) 5 − 5 2 5 − 2 5 4 5 − 4 5(cid:19)
The symmetry currents , are equal to the partial derivatives of the action with
L R
J J
respect to j ,j
L R
1 ∂S 1 ∂j ∂S ∂j ∂S 1 ∂S ∂S
5
= = + = + , (26)
L
J V ∂j V (cid:18)∂j ∂j ∂j ∂j (cid:19) V (cid:18)∂j ∂j (cid:19)
4D L 4D L L 5 4D 5
1 ∂S 1 ∂j ∂S ∂j ∂S 1 ∂S ∂S
5
= = + = , (27)
R
J V ∂j V (cid:18)∂j ∂j ∂j ∂j (cid:19) V (cid:18)∂j − ∂j (cid:19)
4D R 4D R R 5 4D 5
2 ∂S 2 ∂S
= , = . (28)
5
J V ∂j J V ∂j
4D 4D 5
As can be seen from (21), the YM part of the action does not depend on the current
sources. The CS part, on the other hand, equals
k N 1
S = S [L] S [R] = · cBV 2µ2 j2 +µ j µj . (29)
CS CS − CS 6π2 4D(cid:18) 5 − 2 5 5 − 5(cid:19)
From eqs. (28,29) one obtains
k N
c
= · Bµ , (30)
J 3π2 5
k N
c
= · B(µ+j ). (31)
J5 − 3π2 5
5
If one sets k = 1, a standard normalization of the CS action (4) is recovered and
the result (30) is in agreement with [10] without the Bardeen counterterm. The axial
supercurrent introduced in [10] is an equivalent of our j . If it is interpreted as a
5
source for the axial current it has to be set to zero. Minimizing the action with
respect to it is analogous to setting the axial current (31) to zero. It is interesting
that the answer does not depend on j, which probably justifies its absence in [10].
2.3 The divergence of the vector current
In this section the general formula for the left and right symmetry currents will
L,R
J
be derived.
We are going to use a different approach to calculating currents, yet the results will
be identical to the previous section. The current definition is the following (for the
current 3 it is the same as in (28)):
JL,R ≡ JL,R
δS δS
µ(x) = , µ(x) = . (32)
JL δL (z = 0,x) JR δR (z = 0,x)
µ µ
The variation of the action δS = δS + δS can be split into two parts – one
YM CS
proportional to the equations of motion and one reducible to a surface term
R e−φ(z)
δS [A] = δSvol [A] d4x ∂ AµδA , (33)
YM YM − 2g2 Z z z µ(cid:12)
5 (cid:12)z=0
k N (cid:12)
δS [A] = δSvol[A]+ · c d4x ǫµνρσA F δA(cid:12) . (34)
CS CS 6π2 Z ν ρσ µ(cid:12)
(cid:12)z=0
(cid:12)
(cid:12)
The 5D parts are equal to zero on-shell, so that one gets
R e−φ(z) k N
µ(x) = ∂ Lµ + · c ǫµνρσ L FL, (35)
JL −2g2 z z 6π2 ν ρσ
5
R e−φ(z) k N
µ(x) = ∂ Rµ · c ǫµνρσ R FR. (36)
JR −2g2 z z − 6π2 ν ρσ
5
R N
c
Recallingthat = andV = L +R thefollowingexpressionforthedivergence
g2 12π2 µ µ µ
5
of the vector current is obtained
N e−φ(z) k N
∂ µ = ∂ ( µ + µ) = c ∂ ∂ Vµ + · c ǫµνρσ (∂ L FL ∂ R FR)
µJ µ JL JR −24π2 z z µ 6π2 µ ν ρσ − µ ν ρσ
N e−φ(z) k N
= c ∂ ∂ Vµ + · c ǫµνρσ ∂ V ∂ A . (37)
z µ µ ν ρ σ
−24π2 z 3π2
6
To express the divergence of the vector field V another equation of motion generated
µ
δS
by will be needed
δA
z
δS [A] R e−φ(z) N e−φ(z)
YM = ∂ ∂ Aµ = c ∂ ∂ Aµ,
δA 2g2 z z µ 24π2 z z µ
z 5
δS [A] kN δ
CS = c d4x dz ǫµνρσ(A F F 4A F F ) =
z µν ρσ µ zν ρσ
δA −24π2δA Z −
z z
kN
= c ǫµνρσ ∂ A ∂ A . (38)
µ ν ρ σ
−2π2
The corresponding e.o.m.’s assume the form:
e−φ(z) e−φ(z)
∂ ∂ Lµ = 12kǫµνρσ ∂ L ∂ L , ∂ ∂ Rµ = 12kǫµνρσ ∂ R ∂ R ,
z µ µ ν ρ σ z µ µ ν ρ σ
z z
e−φ(z)
∂ ∂ Vµ = 12kǫµνρσ ∂ V ∂ A . (39)
z µ µ ν ρ σ
z
Thus the divergence in (37) equals:
kN
∂ µ = c ǫµνρσ ∂ V ∂ A . (40)
µ µ ν ρ σ
J −6π2
2.4 The Bardeen counterterm
The Bardeen counterterm has a dimensionless prefactor c that is determined by the
following condition
∂ µ = ∂ µ +∂ µ = 0, (41)
µJsubtracted µJ µJBardeen
where the counterterm has the form
S = c d4xǫµνρσL R (FL +FR). (42)
Bardeen µ ν ρσ ρσ
Z
It may be interpreted as a surface counterterm in the spirit of the holographic renor-
malization. In our case
S = 2c d4x(L R L R )∂ V = 2cBV (L R L R ) =
Bardeen − Z 0 3 − 3 0 2 1(cid:12) 4D 0 3 − 3 0 |z=0
(cid:12)z=0
= 2cBV (µ j µ j ) = 2cBV(cid:12) (µ j µj ). (43)
4D L R R L (cid:12)4D 5 5
− −
hence
= 4cBµ , = 4cBµ. (44)
Bardeen 5 Bardeen 5
J J −
7
The general expression for the currents is the following:
δS = d4x( µ δL + µ δR ,)
Bardeen Z JBardeen L µ JBardeen R µ
µ = 2cǫµνρσ(R ∂ R +2R ∂ L +L ∂ R ),
JBardeen L ν ρ σ ν ρ σ σ ρ ν
µ = 2cǫµνρσ(L ∂ L +2L ∂ R +R ∂ L ),
JBardeen R − ν ρ σ ν ρ σ σ ρ ν
µ = 2cǫµνρσ(R ∂ R L ∂ L 3L ∂ R +3R ∂ L ). (45)
JBardeen ν ρ σ − ν ρ σ − ν ρ σ ν ρ σ
The divergence of the Bardeen current equals:
∂ µ = 2cǫµνρσ∂ V ∂ A (46)
µJBardeen − µ ν ρ σ
Based on (40, 41, 46) one gets
kN
c
c = . (47)
−12π2
As a result, the subtracted current turns out to be zero (30, 44, 47):
kN kN kN
c c c
= + = Bµ +4cBµ = Bµ +4 Bµ = 0.
Jsubtracted J JBardeen 3π2 5 5 3π2 5 (cid:18)−12π2(cid:19) 5
(48)
3 Scalars and Pseudoscalars
3.1 Kinetic term and potential
Let us consider now the scalar–pseudoscalar sector, which was first omitted from our
considerations. The bilinear part is
1 3
S = d4xdz e−φ R3 (DµX)†DµX + X 2 (49)
X
Z (cid:20)z3 z5| | (cid:21)
where D = ∂ X iL X +iXR ; field X is related to pion field via
µ µ µ µ
−
πata 1
X = exp i v(z). (50)
(cid:18) f (cid:19)2
π
What is crucial for our case is that there is a scalar interaction with gauge fields. It
can be thought of in two different ways. If one works in A = 0 gauge (both L = 0
z z
and R = 0), than pion is identified with phase of field X as in (50), and interaction
z
term is
N ∂ π
S = c tr d4xdz ∂ V ∂ V α ǫµνλρα. (51)
XAA µ ν λ ρ
24π2 Z f
π
8
If, however, one does not impose this gauge, then the holonomy of the axial field
A dz is itself the pion field, and the term (51) arises directly form Chern–Simons.
z
NR ote there is no double-counting here: when dealing with Chern-Simons solely (as
was the case in the Sakai-Sugimoto model of [10]), A can always be set to zero.
z
This is impossible without inducing a phase of X in the true AdS/QCD model by
Karch–Katz–Son–Stephanov [18] which we work in.
Taking action (51) and differentiating it over F one gets the following contri-
z3
bution to current
N 1 ∂ π(x)
c 0
= B . (52)
XAA
J 2π23 f
π
The special care concerns the boundary conditions. We would like to argue that
the linear in time “rotating” boundary conditions are appropriate. Let us remind the
PCAC relation connecting the axial current and the pion field
¯
Ψγ γ Ψ f ∂ π (53)
ν 5 π ν
⇔
which implies that we add the following term in the pion lagrangian
µ f ∂ π. (54)
5 π 0
This term changes the pion canonical momentum and condition of the vanishing of
the canonical momentum yields the rotating boundary condition
P = ∂ π +µ f = 0 (55)
0 5 π
Collecting all the terms we get
= + + = 1
Jfull, subtracted J JBardeen JXAA 3JFKW (56)
= + =
full, nonsubtracted XAA FKW
J J J J
3.2 Chern-Simons action with scalars
The result of the previous section can be justified from a somewhat different point of
view. Let us once more consider the Chern–Simons action
kN
c
S = − L dL dL R dR dR . (57)
CS
24π2 (cid:18)Z ∧ ∧ −Z ∧ ∧ (cid:19)
Its gauge transformation (L L+dα , R R+dα ) is proportional to a surface
L R
→ →
term
kN
c
S S + − dα dL dL dα dR dR . (58)
CS CS L R
→ 24π2 (cid:18)Z ∧ ∧ −Z ∧ ∧ (cid:19)
9