JHEP07(2011)023
Published for SISSA by Springer
Received: May 13, 2011
Accepted: June 5, 2011
Published: July 5, 2011
Effective conformal theory and the flat-space
limit of AdS
A. Liam Fitzpatrick,
a
Emanuel Katz,
a
David Poland
b
and David Simmons-Duffin
b
a
Department of Physics, Boston University,
Boston, MA 02215, U.S.A.
b
Department of Physics, Harvard University,
Cambridge, MA 02139, U.S.A.
E-mail:
fitzpatr@physics.bu.edu, amikatz@buphy.bu.edu,
dpoland@physics.harvard.edu, davidsd@physics.harvard.edu
Abstract: We develop the idea of an effective conformal theory describing the low-lying
spectrum of the dilatation operator in a CFT. Such an effective theory is useful when
the spectrum contains a hierarchy in the dimension of operators, and a small parameter
whose role is similar to that of 1/N in a large N gauge theory. These criteria insure that
there is a regime where the dilatation operator is modified perturbatively. Global AdS is
the natural framework for perturbations of the dilatation operator respecting conformal
invariance, much as Minkowski space naturally describes Lorentz invariant perturbations
of the Hamiltonian. Assumin g that the lowest-dimension single-trace operator is a scalar,
O, we consider the anomalous dimensions, γ(n, l), of the double-trace op erators of the form
O(∂
2
)
n
(∂)
l
O. Pu rely from the CFT we find that pertu rbative unitarity places a bound on
these dimensions of |γ(n, l)| < 4. Non-renormalizable AdS interactions lead to violations
of the bound at large values of n. We also consider the case that these interactions are
generated by integrating out a heavy scalar field in AdS. We show that the presence of the
heavy field “unitarizes” the growth in the anomalous dimensions, and leads to a resonance-
like behavior in γ(n, l) when n is close to the dimension of the CFT operator dual to
the heavy field. Finally, we demonstrate that bulk flat-space S-matrix elements can be
extracted from the large n behavior of the anomalous dimensions. This leads to a direct
connection between the spectrum of anomalous dimensions in d-dimensional CFTs and flat-
space S-matrix elements in d + 1 dimensions. We comment on the emergence of flat-space
locality from the CFT perspective.
Keywords: AdS-CFT Correspondence, Field Theories in Higher Dimensions, 1/N Ex-
pansion
ArXiv ePrint: 1007.2412
c
SISSA 2011 doi:
10.1007/JHEP07(2011)023
JHEP07(2011)023
Contents
1 Introduction
1
2 Formalism 5
2.1 Algebra constraints 5
2.2 Unitarity limit 8
2.3 Review of AdS global coordinate wavefunctions 11
2.4 Locality and microcausality in AdS 12
3 Dilatation matrix elements in low-energy ECT 14
3.1 Primary wavefunctions 14
3.2 Normalization of primary two-particle wavefunctions 15
3.3 Example calculation of V
nm
17
3.4 Dimensional analysis with n 18
4 Heavy field exchange 19
4.1 S-channel scalar exchange 20
4.2 Matching between low and h igh energies 22
4.3 T- and U-channels 23
5 Emergence of the flat-space S-matrix from γ(n, l) 24
5.1 Emergence of momentum conservation 25
5.2 Two-particle primaries at large n in AdS
d+1
28
5.3 Examples 30
5.3.1 Example 1: φ
4
30
5.3.2 Example 2: (∇φ)
4
31
5.3.3 Example 3: γ(n, L) at m aximum spin L 31
5.3.4 Example 4: scalar exchange in d = 2 32
6 Conclusion 34
A Check of γ(n, 0) for (∇φ)
4
and (∇
µ
∇
ν
φ)
2
(∇φ)
2
35
1 Introduction
One of the central puzzles of the AdS/CFT correspondence [
1–3] concerns determining
which CFTs have well-behaved AdS descriptions. A well-behaved description is usually
taken to mean an effective theory containing several AdS fields whose interactions allow
a perturbative description over a range of scales. Thus, bulk theories typically contain
fields w hose masses are of order the AdS curvature scale, while their non-renormalizable
– 1 –
JHEP07(2011)023
interactions are suppressed by a scale much larger than the curvature scale. In p articular,
the bulk Planck scale must also be large compared to the AdS curvature scale. Local bulk
scattering of the light fields then satisfies perturbative unitarity until one reaches the scale of
non-renormalizable operators. Though high-energy scattering appears to violate unitarity,
the expectation is that the infinitely many heavy AdS fields will ultimately “unitarize” this
scattering, much as QCD resonances lead to sen sible scattering of pions. The low-energy
bulk description is th erefore valid as long as tree level processes are far from violating the
bounds of perturb ative unitarity.
From the AdS effective theory perspective, it appears therefore that what is essential
for the simplicity of description is simply the existence of a small sector of the theory that is
lighter than the Planck scale and most other states.
1
Since the AdS/CFT dictionary relates
dimensions of operators to masses of fields in the bulk, a natural conjecture, proposed
by [
4], is that any CFT with a few low dimension operators separated by a hierarchy from
the dimension of other operators will have a well-behaved dual. However, as any C FT
contains an energy-momentum tensor (dual to the graviton in AdS), there must also be an
additional condition to suppress gravitational interactions in the bulk. In most known cases
this condition follows from the existence of a large number of degrees of freedom in the
CFT (typically, one takes the large N limit of an SU(N) gauge theory). The large N limit
suppresses the connected pieces of higher-point correlation fun ctions as compared to two-
point functions. 1/N thus behaves as a natural expansion parameter for bulk interactions,
and allows one to distinguish between operators du al to single-particle bulk states, and
those dual to multiple-particle bulk states. The idea suggested by [
4] is that having a
hierarchy in dimensions and a parameter such as N in a CFT is sufficient to construct
a sensible AdS effective theory. This theory describes well the correlation functions of
low-dimension operators.
A natural question to ask is then what is the CFT interpretation of the bulk effective
field theory. In particular, there must be an effective conformal theory (ECT) description
which includes only low-dimension CFT operators as states. This ECT must be ab le to
distinguish between renormalizable and n on -r en ormalizable bulk interactions. It must also
obey a condition equivalent to bulk perturbative unitarity w hich sets the range of its valid-
ity. Finally, following standard effective field theory mythology, it would be satisfying, if in
the case that the non-renormalizable terms come from “integrating out” a high-dimension
operator with renormalizable interactions, that perturbative unitarity is restored on the
CFT side. We will see that the ECT indeed has these features once we determine the
appropriate CFT condition for perturbative unitarity.
For simplicity, followin g [
4], we will consider a scenario where the lowest-dimension
operator is a scalar operator, O(x), with dimension ∆. We will refer to O(x) as a “single-
trace operator” in analogy to large N gauge theories with adjoint representations, but it is
not necessary for the operator to have this origin. Other single-trace operators are taken
to have much larger dimensions. We assume that there is a parameter such as N so that
1
For instance, supersymmetry does not appear t o have a d irect role in ensuring that the bulk effective
theory is well behaved, although it might be important for selecting which low-energy bulk descriptions
have actual UV completions.
– 2 –
JHEP07(2011)023
at zero-th order in 1/N the primary operators appearing in the O × O operator product
expansion (OPE) are the “double-trace operators”, which have the schematic form
O
n,l
(x) ≡ O(
↔
∂
ν
↔
∂
ν
)
n
↔
∂
µ
1
. . .
↔
∂
µ
l
O(x) − traces. (1.1)
Here,
↔
∂ =
←
∂ −
→
∂, where the arrows indicate which of the two operators the derivative acts
upon. At zero-th order in 1/N the dimension of this operator is given by 2∆ + 2n + l. We
will be interested in computing the correction to this dimension, γ(n, l), arising from bulk
interactions. For previous work on computing the anomalous dimensions of double-trace
operators in the context of AdS/CFT, see e.g. [
4–15].
In order to develop an ECT, we need to specify a notion of energy in the CFT that
will map nicely to energies in the bulk theory. As the ECT is s upposed to describe low-
dimension operators, a natural notion of energy is the dimension itself. The Hamiltonian
for which we are developing the ECT is the dilatation operator, and the ECT is then
intended to capture its low-lying s pectrum. In that sense, for fixed spin, one can think of
energy, E, as E ∼ 2n. It will be important to keep in mind that this notion of energy
corresponds to the dimensions of CFT operators and is distinct from Poincar´e energy. From
the CFT perspective, the task is to start from a d ilatation operator, D
(0)
, whose spectrum
contains a hierarchy, and perturb it by adding a small correction, V , suppressed by N.
The new dilatation operator, D = D
(0)
+ V , is taken to act on the low-dimension sector of
D
(0)
. In our simplified scenario, this includes multi-trace operators containing only O and
derivatives. Calculating γ(n, l) thus amounts to diagonalizing D
(0)
+ V in perturbation
theory. Purely from the CFT, we will show that perturbative un itarity places a bound on
the anomalous dimensions of |γ(n, l)| < 4. We will then turn to calculating the anomalous
dimensions for particular choices of V , corresponding to local bulk interactions in AdS.
For such calculations we find it most natural to work in global AdS, since the energy
conjugate to global time is associated with th e dilatation operator. Indeed, we will show
that local bulk interactions in global AdS automatically lead to a V which is consistent
with conformal symmetry. We will then demonstrate that using old-fashioned perturbation
theory in global AdS gives a very efficient method of computing the anomalous d imen sions
γ(n, l). This is because these anomalous dimensions are just the correction to the energy
in global coordinates
2
of two-particle AdS states due to bulk interactions. Previously,
obtaining γ(n, l), involved extracting the anomalous dimensions from four-point correlation
functions using sophisticated techniques limited to even CFT dimensions. Our method is
simpler and applies for any dimension.
As expected from AdS, the above unitarity bound will be violated by terms in V
coming from non-renorm alizable bulk interactions. Ind eed, as would follow from the above
identification of n with energy, we find that a local bulk term suppressed by Λ
p
, will lead
to a growth in γ(n, l) ∼ n
p
.
3
Thus, the value of n at w hich the bound is violated sets a
natural boundary for the validity of the ECT. The existence of a useful ECT description
2
The Hamiltonian of AdS in global coordinates is more useful for our purposes than the Hamiltonian in
the Poincar´e patch. This is because translations in global AdS time correspond to dilatations in the CFT,
whereas time in the Poincar´e patch corresponds to Poincar´e time in t he CFT.
3
This growth was found earlier by [
4] using other methods.
– 3 –
JHEP07(2011)023
is then the statement that pertur bative unitarity is not violated over a wide range of n’s.
This is related to locality of interactions which include only the field dual to operator O in
the bulk theory.
To make connection with the conjecture of [4], and to verify standard effective the-
ory lore, we also consider the generation of non-renormalizable bulk interactions via the
exchange of a heavy scalar, dual to a CFT operator O
Heavy
(where ∆
Heavy
≫ ∆). At
n ≪ ∆
Heavy
we reproduce the exact contributions to γ(n, l) one would expect from th e
leading non-renormalizable interactions generated by integrating out the heavy state, sup-
pressed by the appropriate powers of ∆
Heavy
. This resu lt is suggestive that a hierarchy in
the dimension of operators leads to a large range for the ECT. This example also shows
explicitly how putting a large-dimension operator back into the ECT “unitarizes” γ(n, l).
In fact, just as one would expect from effective field theory, we will see that the growth in
γ(n, l) turns into a resonance at n ∼ ∆
Heavy
/2, before decreasing at large n.
At energies much larger than the inverse AdS radiu s it is expected th at one can make
contact with flat-space scattering amplitudes. An important goal that has been pursued
using a variety of methods [
16–24] is to understand how these amplitudes arise from CFT
data. Here we will show that it is in fact possible to extract the flat-space S-matrix elements
of the bulk theory from the large n behavior of γ(n, l). Stated simply, we w ill argue that
at leading order for bulk φ-particle scattering,
M(s, t, u)
d+1
flat space
∼
E
n
(E
2
n
− 4∆
2
)
d−2
2
X
l
[γ(n, l)]
n≫l
r
l
P
(d)
l
(cos θ) , (1.2)
where r
l
P
(d)
l
(cos θ) are the appropriate polynomials in d-dimensions, the total flat-space
energy, E
n
, is given in units of the AdS radius by E
n
= 2∆ + 2n, and [γ(n, l)]
n≫l
indicates
that one needs to take the large n limit of γ(n, l), keeping l fixed. In other words, the
γ(n, l)’s f orm the partial wave exp ansion of the higher dimensional flat-space S-matrix.
4
By “flat-space S-matrix”, one means simply the scattering amplitudes one ob tains f rom
the Lagrangian of the bulk th eory, but applied in Minkowski space. It is interesting that
there seems to be such a direct connection between CFT quantities an d flat-space matrix
elements. Note that this connection is only possible if the ECT including O and O
Heavy
obeys perturbative unitarity for n sufficiently large. Therefore, a hierarchy in dimensions
and a parameter such as N are essential for flat space to emerge.
This paper is organized as follows. In section 2 we will introduce the general formalism
concerning perturbations of the dilatation operator and discuss the constraints arising from
perturbative unitarity. We will then review th e construction of scalar wavefunctions in
global AdS, and discuss why local bulk interactions lead to a sensible perturbation of the
dilatation operator. In section 3 we will derive the general form of the wavefunctions
corresponding to primary operators in the CFT, and use this to calculate the anomalous
dimensions of primary double-trace operators arising from various bulk quartic interactions.
In section 4, we will consider integrating out a h eavy scalar field in AdS, and we will compare
4
This sharpens the relation between M and γ found previously for local bulk operators and neglecting
mass terms [
4].
– 4 –
