Baryon Number-Induced Chern-Simons Couplings of Vector and Axial-Vector
  Mesons in Holographic QCD
EFI-07-08
Baryon Number-Induced Chern-Simons Couplings of Vector and Axial-Vector Mesons
in Holographic QCD
Sophia K. Domokos and Jeffrey A. Harvey
Enrico Fermi Institute and Department of Physics,
University of Chicago, Chicago Illinois 60637, USA
(Dated: April 2007)
We show that holographic models of QCD predict the presence of a Chern-Simons coupling
between vector and axial-vector mesons at finite baryon density. In the AdS/CFT dictionary, the
coefficient of this coupling is proportional to the baryon number density, and is fixed uniquely in the
five-dimensional holographic dual by anomalies in the flavor currents. For the lightest mesons, the
coupling mixes transverse ρ and a1 polarization states. At sufficiently large baryon number densities,
it produces an instability, which causes the ρ and a1 mesons to condense in a state breaking both
rotational and translational invariance.
INTRODUCTION
Models which use the gravity/gauge correspondence to
treat strongly-coupled QCD as a five-dimensional theory
of gravity have progressed dramatically in recent years
[1, 2, 3]. Particularly at high energies, these theories dif-
fer significantly from QCD – yet those models which in-
corporate light quarks [4] and chiral symmetry-breaking
of the form observed in QCD [5] do capture much of the
important low-energy structure of the theory, and give
rise to a spectrum of mesons whose masses, decay con-
stants, and couplings match those of QCD to within 20%.
The gravity/gauge approach includes both top-down
models of QCD arising from D-brane constructions in
string theory [5], and bottom-up phenomenological mod-
els, which attempt to capture the essential dynamics us-
ing a simple choice of five-dimensional metric (AdS5)
and a minimal field content consisting of a scalar X
and gauge fields AaLµ, and A
Rµ [6, 7]. These fields are
holographically dual to the quark bilinear q̄αqβ , and to
the SU(Nf )L × SU(Nf)R flavor currents q̄LγµtaqL and
µtaqR of QCD, respectively.
These holographic models can be used to study QCD
at finite baryon density [8, 9]. In this paper we focus
on a novel effect, in which a Chern-Simons term leads to
mixing between vector and axial-vector mesons. We will
use the model introduced in [6, 7] and for the most part
follow the conventions and notation of [6].
THE MODEL
We work in a slice of AdS5 with metric
ds2 =
−dz2 + dxµdxµ
, 0 < z ≤ zm . (1)
The fifth coordinate, z, is dual to the energy scale of
QCD. We generate confinement by imposing an IR cutoff
zm, and specifying the IR boundary conditions on the
fields. The UV behavior, meanwhile, is governed by z →
In AdS/CFT calculations, boundary contributions to
the action must be treated with care. In the full AdS
space, the only boundary is in the UV (at z = 0). UV-
divergent contributions to the action and to other quanti-
ties are canceled by counterterms. For details see [10, 11].
In the model at hand, the IR boundary at z = zm may
contribute to the action. We follow the approach of [6, 7]
by (1) dropping IR boundary terms, and (2) taking pa-
rameters normally fixed by IR boundary conditions on
the classical solution as input parameters of the model.
We generalize the gauge symmetry to U(Nf )L ×
U(Nf )R and add a Chern-Simons term which gives the
correct holographic description of the QCD flavor anoma-
lies [3]. The Chern-Simons term does not depend on the
metric and on general grounds will be present in any holo-
graphic dual description of QCD. The U(1) axial symme-
try in QCD is anomalous, but in the spirit of the large Nc
approximation we treat it as an exact symmetry of QCD
with massless quarks. Including the anomaly would not
affect our conclusions.
The Lagrangian is thus
d4xdz
|DX |2 + 3|X |2 − 1
(F 2L + F
+SCS .
The Chern-Simons term is given by
SCS =
[ω5(AL)− ω5(AR)] (3)
where dω5 = TrF
3, Nc is the number of colors, and
AL,R = ÂL,Rt̂ + A
a where ta are the generators
of SU(Nf )L,R normalized so that Tr t
atb = δab/2 and
t̂ = 1/
2Nf is the generator of the U(1) subalgebra
of U(Nf). In what follows, we take Nf = 2 so that
a = 1, 2, 3. We will often work with the vector and axial-
vector fields V = (AL +AR)/2 and A = (AL −AR)/2.
CLASSICAL BACKGROUND
We expand around a nontrivial solution to the classical
equations of motion for the scalar X . Following [6, 7] we
http://arxiv.org/abs/0704.1604v1
find the scalar background
X0(z) =
≡ v(z)
1 (4)
where the coefficient M of the non-normalizable term is
proportional to the quark mass matrix, and Σ is the q̄q
expectation value. We take bothM and Σ to be diagonal:
M ≡ mq1 and Σ ≡ σ1. As shown in [6, 7], we can
fix the five-dimensional coupling g5 by comparison with
the vector current two-point function in QCD at large
Euclidean momentum. This leads to the identification
g25 =
. (5)
The model is thus defined by three parameters: zm, mq
and σ. Note that including the U(1) gauge fields and
Chern-Simons coupling does not mandate the addition
of any new parameters. We use zm = 1/(346 MeV),
mq = 2.3 MeV and σ = (308 MeV)
3, which correspond
to values found through a global fit to seven observables
(Model B) in [6].
A background with a static, constant quark density is
described by a classical solution to the equation of motion
for the time component of the U(1) vector gauge field V̂µ,
which is dual to the quark number current. Solving the
V̂0 equation of motion at zero four-momentum yields
V̂0(z) = A+
Bz2 . (6)
By the general philosophy of AdS/CFT, the coefficient
of the non-normalizable term, A, is proportional to the
coefficient with which the operator dual to V̂0 enters the
gauge theory Lagrangian. Since V̂µ is dual to the quark
number current, A must be proportional to the quark
chemical potential. Meanwhile, the coefficient of the nor-
malizable term, B, is proportional to the expectation
value of the operator dual to V̂0: the quark number den-
sity. We now obtain the normalizations of A and B. The
action evaluated for the background Eq. (6) is given by
a boundary term:
V̂0∂zV̂0|z=0 =
d4x . (7)
At finite temperature and baryon number, the Euclidean
action is equal to the grand canonical potential. Using
Eq. (5), this implies that
nqµq (8)
with nq the quark number density and µq the quark
chemical potential. To fix A we separate U(Nf )L,R into
U(1)L,R and SU(Nf )L,R components and note that the
Chern-Simons term contains the coupling
d4xdzǫMNPQ(ÂL0 TrF
PQ−ÂR0 TrFRMNFRPQ)
where the indices M,N,P,Q run over 1, 2, 3, z and the
trace is over SU(Nf ). Defining the SU(Nf)L,R instanton
numbers by
nL,R =
d3xdzǫMNPQ TrF
PQ (10)
and taking Â
constant, this reduces to the coupling
nL − ÂR0 nR
. (11)
Using the connection between instantons and Skyrmion
configurations of the pion field carrying non-zero baryon
number [12, 13, 14, 15, 16], we can interpret an instanton
with nL = −nR = Nb as a state with baryon number Nb.
Eq. (11) then fixes A = µb/Nc = µq with µq the quark
chemical potential; Eq. (8) fixes B = 24π2nq/Nc.
QUADRATIC ACTION
In vacuum, the spectrum of the theory consists of
towers of scalar, vector, pseudoscalar, and axial-vector
mesons given by mode-expanding the five-dimensional
fields along the holographic (z) direction, and integrat-
ing over z. In this section, we identify the spectrum
of excitations and their dispersion relations at non-zero
baryon density by expanding the action to quadratic or-
der around the background given by Eqns. (4),(6).
We focus on the π mesons and the isospin triplet vector
ρ and axial-vector a1 mesons, ignoring contributions from
heavier mesons, and from the scalar σ which arises from
fluctuations in the magnitude of X . Couplings similar to
those for the ρ− a1 mesons exist for the isoscalar ω and
f1 mesons. For simplicity, we omit these as well.
Pions arise as Nambu-Goldstone modes associated
with the breaking of U(Nf )L × U(Nf )R to U(Nf )V .
We write X(x, z) = X0(z) exp(i2π
ata) and expand
to quadratic order in πa. The four-dimensional pion
field is obtained by writing πa(x, z) = πa(x)ψπ(z).
Similarly, the ρa and a1 mesons appear by writing
V aµ (x, z) = g5ρ
µ(x)ψρ(z), A
µ(x, z) = g5a
µ(x)ψa(z). The
wave functions ψπ(z), ψρ(z), and ψa(z) are solutions of
the quadratic equations of motion for fields with four-
momentum q2 = m2 and with boundary conditions
ψ(0) = ∂zψ(zm) = 0. For details see [6, 7].
Making the above substitutions and expanding to
quadratic order yields the four-dimensional action
a∂µπa − 1
aπa − 1
(ρaµν)
(aaµν)
+µǫijk (ρai ∂ja
k + a
i ∂jρ
, (12)
with ρµν , aµν the field strengths for ρµ, aµ. The Chern-
Simons term with coefficient µ mixes the ρ and a1
mesons. It arises from reduction of a term of the form
dV̂ TrAdV in the expansion of Eq. (3).
As usual, to obtain Eq. (12) one must remove the mix-
ing between aaµ and ∂µπ
a by performing the transforma-
tion aaµ → aaµ+ξ∂µπa and then rescaling the pion field to
obtain a canonical kinetic energy term [17]. This leads to
a pion contribution to the Chern-Simons term. A total
spatial derivative, it does not contribute to the equations
of motion and may be dropped.
Since the ρ has JPC = 1−− and the a1 has J
1++, the Chern-Simons coupling is even under P and odd
under C. This is indeed consistent with a background
having non-zero baryon number, which preserves P and
violates C: the coupling is rotationally invariant, but not
Lorentz invariant due to the preferred rest frame of the
baryons.
We can deduce the existence of the Chern-Simons cou-
pling in four-dimensional terms as follows. The reduction
of the five-dimensional Chern-Simons term [5, 22] gives
rise to the usual gauged WZW action [18, 19, 20], as well
as a set of couplings which arise from inexact bulk terms.
These include a ρ − a1 − ω coupling which, in the pres-
ence of a coherent ω field in nuclear matter, gives rise to
a coupling of the form given in Eq. (12). The ρ− a1 − ω
coupling has been considered previously in a general dis-
cussion of chiral effective Lagrangians [25], and is implicit
in the formulae of [23]. Related terms appear in [21, 22].
In AdS/QCD, different forms of the gauged WZW action
can be obtained by the addition of UV counterterms [24],
but these will not cancel the Chern-Simons coupling and
lead to explicit breaking of chiral symmetry beyond that
given by the quark mass term in Eq. (4).
The mass of the ρ meson is given by mρ = 2.405/zm,
while ma1 must be determined from a numerical solution
of the equation of motion. Model B of [6] finds mρ =
832 MeV, ma1 = 1200 MeV which should be compared
to the experimental values mρ = 775.8 ± 0.5 MeV and
ma1 = 1230 ± 40 MeV [26]. The parameter µ in the
Chern-Simons coupling is given by
µ = 18π2nbz
mI (13)
where I is the dimensionless overlap integral
dzzψρ(z)ψa1(z) . (14)
Numerical evaluation of the integral gives I = 0.54.
A typical baryon density in nuclear matter, n0b ≃
0.16/(fermi)3 , gives
µ ≃ 1.05 GeV
. (15)
PHENOMENOLOGICAL APPLICATIONS
We now outline two potentially observable conse-
quences of the Chern-Simons coupling between the ρ and
a1. Details will appear elsewhere.
Mixing of transverse ρ and a1 states
We consider plane-wave solutions to the equations of
motion resulting from Eq. (12), dropping the pion fields
and focusing on the ρ and a1 dispersion relation and po-
larization vectors. Without loss of generality, we consider
propagation along x3:
ρµ(x) = ǫ
µ(q)e
−iq·x, aµ(x) = ǫ
µ(q)e
−iq·x (16)
with q = (q0, 0, 0, q3). For convenience, we suppress the
SU(2) indices in the following. The components ρ0, ρ3,
a0, and a3 have standard dispersion relations, unaffected
by the Chern-Simons coupling. The transverse compo-
nents ρ1, ρ2, a1, and a2 mix through a derivative cou-
pling. The equations of motion yield the dispersion rela-
tion for the transverse polarizations
(m2ρ +m
(m2a1 −m2ρ)2 + 16µ2q
The lower sign in Eq. (17) gives a state which is pure ρ
as q3 → 0. At non-zero q3, it is a mixture of transverse
ρ and a1 states with orthogonal polarization vectors:
iM2(q3)
= − iM
2(q3)
where we have defined ∆2 = m2a1 − m
ρ and M2(q3) =
∆4 + 16µ2q2
− ∆2)/2. The upper sign in Eq. (17)
gives a pure a1 state for q3 = 0, while for non-zero q3,
= − iM
2(q3)
iM2(q3)
. (19)
For µ greater than some momentum-dependent critical
value, the dispersion relation Eq. (17) leads to tachyonic
modes (modes having dq0/dq3 > 1). For very large mo-
menta, this critical value becomes
µcrit =
(m2ρ +m
)/2 ≃ 1.09 GeV . (20)
For a range of µ below µcrit the dispersion relation with
the lower sign in Eq. (17) exhibits interesting anomalous
behavior, the analysis of which is beyond the scope of
this letter.
It would be interesting to explore signatures of these
mixed polarization states in the quark-gluon plasma and
in nuclear matter.
Vector Meson Condensation
To identify the tachyonic instability which occurs for
µ > µcrit we start with the energy density corresponding
to Eq. (12) for the diagonal component of the ρ and a
fields, aa = aδa3, ρa = ρδa3. Completing the square and
dropping the terms involving the electric components of
the field strengths, which play no role in the instability,
we find
H = 1
(m2a − µ2)~a · ~a+
(m2ρ − µ2)~ρ · ~ρ
( ~Ba − µ~ρ)2 +
( ~Bρ − µ~a)2 (21)
where ~Bρ = ~∇× ~ρ , ~Ba = ~∇× ~a.
Applying the ansatz
~a = v cos(µx3)x̂2, ~ρ = v sin(µx3)x̂1 , (22)
the last two terms in Eq. (21) vanish, while the average
of the first two terms over x3 is negative for µ
2 > µ2
leading to an instability to v 6= 0. Understanding the sta-
bilization of the configuration Eq. (22) requires general-
izing H to include higher order terms. Note that Eq. (22)
breaks both rotational and translational symmetry, ex-
hibiting a structure similar to the smectic phase of liquid
crystals which includes an interesting set of topological
defects.
The critical value Eq. (20) is remarkably close to the
estimate Eq. (15) for µ at ordinary nuclear densities. If
this model is accurate then there should be a conden-
sate of vector and axial-vector mesons in nuclear matter
with baryon densities at or slightly above n0b . In ordi-
nary nuclei, there are finite size effects as well as other
corrections to the ρ and a1 interactions which will have
to be included to determine whether this condensate oc-
curs. Neutron stars are more likely to produce such a
condensate, as they are thought to contain matter at a
density somewhat greater than n0b . The interplay be-
tween this condensate and other conjectured effects in
nuclear matter, such as pion condensation and color su-
perconductivity, deserves further study.
ACKNOWLEDGEMENTS
We thank O. Lunin and J. Rosner for helpful conversa-
tions. JH thanks the Galileo Galilei Institute in Arcetri,
Florence for hospitality while this work was being com-
pleted. The work of SD and JH was supported in part by
NSF Grant No. PHY-00506630 and NSF Grant 0529954.
Any opinions, findings, and conclusions or recommenda-
tions expressed in this material are those of the authors
and do not necessarily reflect the views of the National
Science Foundation.
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