Notes on Relevant, Irrelevant, Marginal and Extremal Double Trace Perturbations

Double trace deformations, that is products of two local operators, define perturbations of conformal field theories that can be studied exactly in the large-N limit. Even when the double trace deformation is irrelevant in the infrared, it is believed to flow to an ultraviolet fixed point. In this note we define the Kallen-Lehmann representation of the two-point function of a local operator O in a theory perturbed by the square of such operator. We use such representation to discover potential pathologies at intermediate points in the flow that may prevent to reach the UV fixed point. We apply the method to an"extremal"deformation that naively would flow to a UV fixed point where the operator O would saturate the unitarity bound. We find that the UV fixed point is not conformal and that the deformed two-point function propagates unphysical modes. We interpret the result as showing that the flow to the UV fixed point does not exist. This resolves a potential puzzle in the holographic interpretation of the deformation.


Introduction
Certain conformal field theories that come in families defined by an integer N simplify in the large-N limit. Notable examples are theories that can be described by weakly coupled gravities via the holographic duality and O(N) vector models. The latter are deformations of N massless free field theories. The deformation is called by abuse of term "double trace." It is defined deformation of the bosonic model is relevant and flows to a nontrivial infrared fixed point [1,2].
The deformation of the fermionic model is irrelevant, so non-renormalizable by power counting.
Nevertheless, a UV fixed point is believed to exist in the large-N limit [3,4,5] and even at finite N 3 . Both models can be solved exactly by using a method that easily generalizes to the case of adjoint theories with holographic semiclassical gravity duals. In the context of holographic duality, the correct treatment of multi-trace perturbations was explained in [7] and further simplified in [8]. The analysis of [8] was further extended and generalized beyond AdS/CFT holography in [9]. We will review the results of [8,9] -for completeness and to fix notations and normalizations-in section 2.
The rediscovery of multi-trace perturbations in the context of AdS/CFT duality makes clear that they can be studied exactly in an appropriately defined large-N limit 4  The possibility of solving the deformed model raises several interesting questions, which we shall try to answer in this paper. One such question arises already in the case of a doubletrace deformation O 2 when 2∆ > d in d spacetime dimensions. In this case the deformation is irrelevant in the IR, that is non renormalizable by power counting. Nevertheless the exact solution of the deformed model at large "N" has a UV fixed point. A difficult question is whether this UV fixed point exists at finite N. A simpler one is whether the fixed point can be connected to the IR by a physical renormalization group (RG) flow. By "physical" we mean the following: the deformed theory has both an IR and a UV fixed point; therefore, it defines a field theory valid at all energies. The theory is thus a UV complete one rather than merely an effective field theory, valid only up to a maximum energy scale. To qualify as a "physical" this theory must be free of tachyons and ghosts. Of course an "unphysical" theory, plagued by ghosts or tachyons, can still have physical UV or IR fixed points. We will see several examples of such behavior in section 3. Such theory may also describe an interesting statistical mechanics system, but not a relativistic, local field theory. We call an RG flow "physical" when it is generated by a deformation that produces a physical relativistic field theory. Ref. [7] shows that the RG flow due to the double trace perturbation O 2 connects a fixed point where the We will recover this pathology in section 3, where we will refine the analysis of refs. [10,11] by giving a complete Källen-Lehmann representation of the two-point correlator O(x)O(0) . Section 3 will also study the case ν = 1, which is perhaps the most intriguing of all double- which therefore would saturate the unitarity bound. In a unitary CFT any operator saturating the unitarity bound must be a free field [12] 5 . This free field should come from a different identification of sources and VEVs in a theory with the same near-boundary behavior of the scalar field and the same bulk action [15]. On the other hand, a standard scalar action does not carry the singleton representation corresponding to a free scalar [16] 6 . So, an obvious question to ask is whether the flow induced by the "extremal" double trace perturbation O 2 , with O of dimension d/2+1, is physical all the way up to the UV and really terminates in a free-field fixed point. We will be able to answer this question (in the negative) at the end of section 3. Final remarks on operator mixing and connections to other papers on double-trace perturbations -in section 4-and an appendix summarizing the AdS/CFT holographic description of multi-trace perturbations conclude the paper.

Multi Trace Deformations
As a warm-up example of multi-trace deformations let us consider the interacting O(N) vector model. In three dimensions it was conjectured to be the holographic dual of AdS 4 high spin theories in [18]. Its action is The deformation is relevant in dimension d < 4.
This action can be expressed in terms of the bilinear φ a φ a by introducing an auxiliary field Σ: in which the expectation value Σ = N a=1 N −1 φ a φ a is formally O(N 0 ), i.e. finite in the large-N limit. Integrating out Σ one recovers action (1); integrating out φ a and discarding a Σ-independent constant, one obtains instead a non-local action for Σ The "intensive" action s[Σ] is independent of N. The generator of connected correlators of the In the large-N limit the integration over Σ in eq. (4) reduces to computing a saddle point.
Therefore, the effective action is computed at the stationary point in J and Σ. By computing first the stationary point in J, it is easy to find that effective action of the deformed O(N) model is So, the effect of the deformation is additive in the effective action. Notice that this result follows simply from the fact that at large N the integral in Σ can be evaluated using the saddle point approximation.
As we mentioned before, W [J] = Nw[J] generates connected correlators of the operator N a=1 φ a φ a , which are all O(N). This is another manner of checking that w[J] is independent of N in the large-N limit. Notice that the field O appearing in the free energy is the expectation value of the normalized operator N a=1 N −1 φ a φ a , which differs from the operator sourced by J by the normalization factor N −1 .
We see that the double-trace perturbation is additive in the effective action at leading order in 1/N. This simple result generalizes easily to any multi-trace deformation NU( N a=1 N −1 φ a φ a ), when the function U(x) is independent of N [19], and to any theory admitting a large-N limit [9].

In fact, in all theories with an effective action
Here N * is a (large) number counting the effective degrees of freedom of the theory. In even dimensions, this is proportional to the coefficient one of the the conformal anomalies. For O(N) vector models N * = N while for CFTs with fields in the adjoint representation of a rank-N algebra, such as those that possess holographic duals To prove additivity of the multi-trace perturbation we begin by writing the Feynman integral representation of the free energy in Lorentzian signature, using the functional Fourier transform of the Dirac delta function. We will denote by φ the fundamental fields of the CFT and we will defines the functional Γ U [O] in the perturbed theory in terms of the unperturbed functional So far all manipulations have been formal, but exact in N * . In the large N * limit, any theory in which the free energy is

Two-Point Functions of Double Trace Perturbations and their Källen-Lehmann Representation
Consider an operator O of general conformal dimension ∆ = d/2 + ν. Unitarity requires O is necessarily free [12] (see also [13,14]).
Without any deformation, the connected two-point function of O is The positive pre-factor K is O(1/N * ), when the operator O is normalized as in the previous for non-integer ν.
The cases where ν ∈ Z + need to be considered separately due to the appearance of ln k 2 terms. In particular, for ∆ = d/2 + 1 one finds (see Appendix) with C ′ positive for d > 2 and µ an arbitrary scale that can be changed by adding a contact term proportional to k 2 .
Similarly, for ∆ = d/2, The undeformed effective action is then Now we add a double-trace deformation to the effective action γ[O], where λ is dimensionless. The cases λ > 0 and λ < 0 will be considered separately. This deformation is IR-relevant for ∆ < d/2, marginal for ∆ = d/2 and irrelevant for ∆ > d/2. It is tempting to identify Λ with the cut-off of the theory; the rest of this section will substantiate such identification.
The deformed effective action becomes Thus we obtain the deformed two-point function 3.1 Case 1: 1 > ν > 0 It was pointed out in [7] that for 1 > ν > 0 such a double-trace deformation leads to an RG flow in which the IR and UV fixed points are CFTs in which the operator O has conformal dimension ∆ = ∆ ± = d/2 ± ν respectively. They are the two possible choices of quantization in the AdS/CFT context for a massive scalar in the bulk [15] (see Appendix for a review). This flow can be achieved with λ positive or negative. Recall that, omitting a positive coefficient, which reduces to the original undeformed CFT of ∆ = ∆ + , as expected of an irrelevant deformation.
Upon removing the contact term, this is the two-point function of a CFT with ∆ = ∆ − . Now we come to the heart of our paper. We express the two-point function in Källen-Lehmann form.
Let us introduce now the complex function f (z) = 1 |λ|z ν +1 ; when its branch cut is placed on the negative real axis it is meromorphic in the range (π ≥ arg z > −π).
In this case f (z) has no singularity in the first sheet. This implies that there are no tachyonic or otherwise unphysical one-particle states among the states created by applying O(x) to the vacuum. Using Cauchy's formula, one finds So, the spectral density is positive-finite and the spectrum is free of ghosts and tachyons.
Thus the deformation with λ < 0 may provide a healthy flow between the IR and UV fixed points. Of course there is a (nonperturbative) fly in the ointment here, since λ < 0 means that the potential λO 2 is unbounded from below.

λ > 0
Consider next the case where λ > 0: which has a pole of positive residue at k 2 /Λ 2 = λ −1/ν on the positive real axis, signaling a (nonghost) tachyon mode. The Källen-Lehmann representation shows that the continuum part of the spectral density is positive definite, so the tachyon is the only unphysical feature of the deformed theory: Two limits are worth mentioning. The first is Λ → ∞. In this case the tachyon moves to infinite mass and the perturbation disappears (since it becomes irrelevant at all energy scales).
The second limit is less trivial. It is the UV limit in which Λ → 0. One interesting question is whether the UV limit may exist as a CFT even if the deformation leading to it is unphysical.
The answer is yes, because in that limit In other words, the tachyonic mode decouples.

Case 2: 0 > ν > −1
One natural question to ask is: Do we get a flow similar to the one above by adding a doubletrace deformation for the operator associated to the alternative quantization d/2 > ∆ > d/2−1 (0 > ν > −1)?
In this case is IR-relevant (and UV-irrelevant).

λ > 0
For λ > 0, In Källen-Lehmann form, Hence, the spectral density is positive-finite and the spectrum is free of ghosts and tachyons, thus providing a flow between the IR and UV fixed points. One can check that the IR fixed point of this flow is a CFT with an operator with ∆ = d/2 + |ν|, while at the UV fixed point ∆ = d/2 − |ν|. This is expected because the double-trace deformation is UV-irrelevant.

λ < 0
Now for λ < 0, The two-point function has a pole of positive residue at k 2 /Λ 2 = |λ| −1/ν on the positive real axis, signaling a (non-ghost) tachyon mode, while the smooth part of the spectral density is positive definite.

Case 3: ν > 1
The case ν > 1 ∈ Z + has been studied in [10] and [11]. Those papers consider a UV completion of the double-trace perturbation obtained by coupling a massive scalar to O. In our analysis we do not introduce any such scalar or any other ad hoc UV completion. We use instead a Källen-Lehmann representation, which can be obtained from those used in the previous subsection by replacing (k 2 /Λ 2 ) ν with ±(k 2 /Λ 2 ) ν , depending on the value of ν.
For λ > 0, the simple poles appear at complex values of k 2 /Λ 2 in conjugate pairs. The corresponding residues also form complex conjugate pairs, signaling tachyonic ghost modes.
For λ < 0, there is exactly one simple pole at real positive k 2 /Λ 2 , with negative residue, i.e. tachyonic ghost.
These results are in agreement with the results of [11] for 2 > ν > 1.
For λ > 0, there is exactly one simple pole at real positive k 2 /Λ 2 , with positive residue, i.e.
For λ < 0, the simple poles are again at complex values of k 2 /Λ 2 and appear in conjugate pairs with, conjugate residues. At least one pair has a negative real part, signaling a tachyonic ghost modes.
Therefore, the deformed theory is not physical, as expected because at the putative UV

Introducing a (marginal) double-trace deformation U[O]
The renormalization scale µ can be removed, as in [7,9], by making the coupling constant λ run with µ. A convenient renormalization condition on λ is to require that it diverges at some fixed scale Λ. This defines an RG flow of λ and so By performing the wave function renormalization O = ZO R , Z = ln(Λ 2 /µ 2 ), we obtain a µ-independent renormalized two-point function There is a simple pole at k 2 = Λ 2 with positive residue, i.e. a tachyon mode. This is expected because this theory has a Landau pole for λ > 0 at Λ under our renormalization condition. For µ > Λ, the coupling constant λ(µ) is negative and asymptotically free, while λ is positive and the theory is IR free for µ < Λ. All of this is of course in agreement with well known results for the λφ 4 theory in four dimensions.
Recall that The two-point function of the alternative quantization ∆ = d/2−1 obtained by a naive Legendre Notice that the value of µ in equation (36) So, eq. (37) is not the two point function of a ∆ = d/2−1 conformal field. In fact it is altogether unphysical, because it decays faster than 1/k 2 at large k 2 . The origin of this unphysical feature can be seen by representing the two-point function in Källen-Lehmann form, because such representation makes it manifest that there exists a simple pole at k 2 = µ 2 with negative residue: In other words, the spectrum contains a tachyonic ghost mode. Notice that µ is a physical scale, not an auxiliary one that can be removed by local counterterms. In fact µ is physical even at the UV fixed point, as pointed out earlier.
Now add the double-trace deformation The branch-cut in the complex k 2 /Λ 2 plane is logarithmic. The spectral density is positivedefinite, but now there are two simple poles.
For λµ 2 /Λ 2 > e, the poles are at real and positive k 2 . The pole at larger k 2 has a negative residue and that at the smaller k 2 a positive residue, with the latter pole approaching 0 as For λµ 2 /Λ 2 < e, the poles are complex and are conjugates of each other. The residues have a positive real part and complex conjugate imaginary parts.
A contour integration gives in which the spectral density in the second term is positive. The pole is at real and positive k 2 with negative residue, signaling the propagation of a ghost tachyon mode.
Therefore, one concludes that the flow to the theory with ∆ = d/2−1 in the UV is unphysical for all values of λ. Moreover, the very fact that the two-point function in eq. (37) is non-unitary shows that the UV limit Λ → 0 is meaningless in this case.
The singleton point is reached the limit λ → 0, µ exp(−1/λ 2 ) = constant. This limit does decouple all ghost and physical states and leads to a two-point function ∝ 1/k 2 , but it cannot be achieved as an RG trajectory. A different singular limit leading to the singleton is described in [20].

Summary
After reviewing the method that allows to find the two-point function of certain primary op- We also found that the Källen-Lehmann representation automatically contains extra massive scalars, signaled by simple scalar poles in the two-point function. We can say that the Källen-Lehmann representation "integrates in" massive scalar. One result of our study is that, when the perturbation O 2 is relevant, the extra massive scalar found using the Källen-Lehmann representation is physical when λ > 0 and tachyonic when λ < 0. The example of the bosonic O(N) vector model in three dimensions shows that this is the expected behavior, because the potential (λ/2N)( N a=1 φ a φ a ) 2 is stable only for λ > 0. On the other hand, the massive scalar used in [10,11] to define a UV completion of double-trace deformations has the opposite behavior: it is tachyonic for λ > 0 and physical for λ < 0. One possible reason for the disagreement is that the UV completion used in [10,11] can be pathological in the IR. This is manifest in the case of the O(N) vector model, where the scalar potential is unbounded below for either sign of λ.
An especially interesting case is ν = 1, because the conformal dimensions allowed by the alternative quantization of ref. [15]  Finally we should remark that our analysis agrees with ref. [21], which studies double-trace deformations involving two different operators. Here we will restrict our analysis to the most interesting case that one of the two operators, O 1 , saturates the unitarity bound (∆ 1 = d/2 −1) while the other, O 2 , has dimension ∆ 2 > ∆ 1 , ∆ 2 < d/2. Ref. [21] studies a relevant flow from the UV, where O 1 saturates the unitarity bound, to the IR. It is thus quite different from the situation considered in this paper, which considers an irrelevant flow to a putative UV fixed point. Nevertheless, the flow can be studied easily using the methods described in this paper.
The deformation studied in [21] is One can check that when g 1 g 2 > f 2 , g 1 > 0, g 2 > 0 the flow is physical 7 ; on the other hand, whenever g 1 = 0 one is simply giving a mass to a free scalar, so the flow is rather trivial: a massive scalar decouples in the IR. So, let us consider the case g 1 = 0. 7 We thank M. Bertolini for pointing this out to us.
At large N the deformation changes the two point functions Õ i (k)O j (0) , i, j = 1, 2 as In the extreme infrared, The matrix in (47) has rank one and is independent of g 2 , meaning that All this is in perfect agreement with ref. [21].
We conclude by observing that, while some of the UV-complete theories with g 1 > 0, g 2 > 0 are physical, those with g 1 = 0 are plagued by unphysical states, since the two point function in (46) has, among other unpleasantnesses, a tachyonic pole for any value of g 2 . The RG flow generated by the deformation (45) with g 1 = 0 is therefore unphysical, according to our general definition.
Evaluating the on-shell action and discarding possible contact terms, one finds for ∆ = In the case ∆ = d/2, This asymptotic behavior implies that [22] S The AdS/CFT correspondence then reads We have multiplied the exponent by a factor of N * ≫ 1 such that φ 0 and O are both O(1). The two-point function in momentum-space representation in the case ∆ = d/2 + 1 contains gamma functions in the prefactor that appear divergent, see eq. (12). However, one notes that For d/2 + 1 ≥ ∆ > d/2, the alternative quantization in which ∆ = ∆ − is also allowed by unitarity. To obtain a correspondence to another CFT where the operator O has conformal dimension ∆ − , one needs to exchange the roles of φ 0 and A. Since φ 0 and A are conjugate variables, the exchange is done by a Legendre transformation [15].
Define the effective "intensive" action the effective action of the (undeformed) CFT is