$C_T$ for Non-unitary CFTs in Higher Dimensions

The coefficient $C_T$ of the conformal energy-momentum tensor two-point function is determined for the non-unitary scalar CFTs with four- and six-derivative kinetic terms. The results match those expected from large-$N$ calculations for the CFTs arising from the $O(N)$ non-linear sigma and Gross-Neveu models in specific even dimensions. $C_T$ is also calculated for the CFT arising from $(n-1)$-form gauge fields with derivatives in $2n+2$ dimensions. Results for $(n-1)$-form theory extended to general dimensions as a non-gauge-invariant CFT are also obtained; the resulting $C_T$ differs from that for the gauge-invariant theory. The construction of conformal primaries by subtracting descendants of lower-dimension primaries is also discussed. For free theories this also leads to an alternative construction of the energy-momentum tensor, which can be quite involved for higher-derivative theories.


Introduction
It is a truth almost universally acknowledged that there are no non-trivial unitary conformal field theories in more than six dimensions. Indeed for superconformal theories this is a long established result [1], and it is often conjectured that the superconformal (2, 0) theory in six dimensions is the one theory to rule them all and in the light bind them.
However, there is now good evidence that there are interacting conformal theories, which contain an energy-momentum tensor, in more than four dimensions and, indeed, for nonunitary CFTs, for dimensions larger than six. The O(N ) non-linear sigma model has a tractable 1/N expansion without restriction on the spatial dimension d [2], defining a conformal theory with calculable scaling dimensions, at least for sufficiently large N [3]. This suggests a non-trivial five-dimensional CFT which is also accessible by ε-expansion methods starting from 4+ε and 6−ε dimensions. Generalisations to higher dimensions were recently explored in [4,5]. To leading order in 1/N the non-linear sigma model comprises an N -component scalar ϕ i with dimension 1 2 (d − 2) and also a singlet σ with dimension 2, clearly violating the unitarity bound for scalars when d > 6.
Apart from the scaling dimensions for conformal primary operators and the parameters determining the three-point functions and, hence, the operator product expansion, crucial data defining a CFT are given by the correlation functions involving the energy-momentum tensor.
In any CFT C T , the coefficient of the two-point function which is fixed up to an overall constant by conformal invariance, plays a crucial role. The scale of the energy-momentum tensor is determined by Ward identities and with our conventions C T may be regarded as a measure of the number of degrees of freedom. It determines the contribution of the energy-momentum tensor in the conformal partial-wave expansion, and so is readily determined in bootstrap calculations. For the conventional free scalar and fermion theories C T was determined for arbitrary d and for vector gauge theories for d = 4 some time ago in [6], and later C T was calculated for (n − 1)-form gauge in d = 2n dimensions in [7]. For the O(N ) sigma model results for C T to first order in 1/N were obtained by Petkou [8,9] by applying the operator product expansion to the four-point function for φ i , and have recently been rederived by direct calculation and extended to the Gross-Neveu model in [10]. In the non-linear sigma model = (−1) p−1 4 (2p + 1)! (p − 1)! (p + 3)! , (1.4) which is an integer [4]. Thus, C O(N ) T,1 d=4 = 0, since then the theory reduces to N free scalars with σ non-dynamical, whereas C O(N ) T,1 d=6 = 1. The extra 1 was interpreted in [3] as the contribution of the dynamical free scalar σ and for d = 6 − ε the ε-expansion defines a CFT at a fixed point starting from the renormalisable O(N )-invariant Lagrangian with g, λ = O(ε). In higher even dimensions there are corresponding renormalisable Lagrangians with higher-derivative kinetic terms for σ. For d = 8 − ε there is a perturbative fixed point starting from The β functions for the couplings g, λ ′ , λ have recently been calculated by Gracey [5]. In this which we show arises from the higher-derivative σ contribution. A similar narrative emerges for the Gross-Neveu model with N fermion fields ψ i . There is also a self-consistent 1/N expansion as a conformal field theory for any dimension d. To leading order ψ i has dimension 1 2 (d − 1), and there is a singlet scalar field σ with scale dimension 1, which is consequently below the unitary bound for d > 4. The leading 1/N correction to C T has been recently determined in [10], and can be expressed in the form where 1 2 d tr(1) is the contribution to C T for a single fermion with tr(1) the sum over spinorial indices. For general d the result [10] for C GN T,1 is similar to that for the sigma model; for even dimensions it reduces to In this case C GN T,1 d=4 = 1, representing the contribution of the dynamical scalar σ whereas C GN T,1 d=6 = −5. For d = 4 − ε equivalent results can be obtained as a perturbative ε expansion at the RG fixed point starting from the renormalisable Lagrangian 9) with N Dirac fields ψ.
In this paper we calculate the contributions to C T corresponding to higher-derivative scalars, such as that for σ in (1.6), for general d. The energy-momentum tensor is determined from the corresponding local Weyl-invariant actions on curved space quadratic in a scalar field ϕ. The construction of such actions is equivalent to obtaining conformal differential operators starting from powers of the Laplacian. We then determine C T for scalar theories with kinetic terms with 2p derivatives for p = 2, 3, and conjecture a result for general p. In section 3 we consider (n − 1)-form gauge theories with additional derivatives in 2n + 2 dimensions when they also define a CFT and obtain C T in this case. Reflecting the lack of unitarity, C T < 0. We also discuss in section 4 the (n − 1)-form theory, without additional derivatives, extended to define a CFT away from d = 2n dimensions when gauge invariance is lost.
The energy-momentum tensors in the higher-derivative theories are rather non-trivial. In section 5 we discuss an alternative construction which constructs a spin-two conformal primary by successively subtracting the descendants of lower-dimension primary operators. The final expressions thereby obtained are identical with those derived from curved-space Weyl-invariant actions; the subtractions are related to improvement terms which need to be added to the canonical energy-momentum tensor to obtain a tensor which is traceless as well as symmetric.

Higher-derivative Scalar Theories
The actions for higher-derivative free scalars considered here have the form Theories starting from such actions were considered in [11]. A symmetric traceless energymomentum tensor may be obtained by the usual Noether procedure or by extending (2.1) to a general curved space background so as to be invariant under Weyl rescalings of the metric. Assuming diffeomorphism invariance then reducing to flat space ensures that the resulting energy momentum tensor satisfies conformal Ward identities which ensure that it is a conformal primary.
For S 4 the extension to a Weyl invariant form on curved space is equivalent to constructing the Paneitz operator [12] (see also [13] and [14] for the d = 4 version of the Paneitz operator) and for S 6 this involves the generalisation of the d = 6 Branson operator [15] to general d.
A convenient form for the Branson  Varying the metric about flat space gives (an alternative derivation based on a generalised 1 CT,S in (1.3) corresponds to p = 1.
2 See version 3 of [16]. 3 This is then an obstruction to relating Weyl and conformal invariance [18], but it also prevents the existence of a symmetric traceless energy-momentum tensor which is a conformal primary. Related discussions are given in [19] and for d = 2 in [20].
Noether procedure is given in [21]) It is useful to note that The terms in the expressions for T µν ϕ,4 , T µν ϕ,6 involving the second-and higher-order derivative operators D µνσρ , ∂ µ ∂ ν − η µν ∂ 2 and D µνσρ B arise from explicit curvature-dependent terms in the curved-space action and represent improvement terms to be added to the canonical energymomentum tensor. In particular the contribution involving D µνσρ B comes from the reduction of a term in the curved-space result proportional to ∂ µ ϕ∂ ν ϕ B µν , with B µν the Bach tensor, and this gives The results for T µν ϕ,4 and T µν ϕ,6 in (2.2) and (2.4) obey the conservation and trace conditions, which of course vanish on the relevant equations of motion (∂ 2 ) p ϕ = 0.
Correlators and operator products in the free field theories are determined just by .

(2.9)
These are respectively singular when d → 4, 6 but in (2.2), (2.4) the only terms not involving ∂ϕ have overall factors d − 4, d − 6 in each case. From this term, for both T µν ϕ,4 , T µν ϕ,6 , we may verify for the leading term in the operator product The coefficient is determined by Ward identities assuming T µν ϕ,2p is canonically normalised. In free field theories any local operator formed from ϕ with derivatives at the same point can be decomposed in terms of conformal primaries and descendants, or derivatives, of conformal primaries of lower dimension. Since T µν is a conformal primary the result in (1.1) is therefore unchanged for T σρ → T σρ + ∂ τ X σρτ for any local X σρτ expressible as a conformal primary or descendant. This ensures that, dropping also terms which vanish on the equations of motion, We get, with λ as in (2.7), .
There is an obvious generalisation of (2.1) to actions with more derivatives, S 2p formed from (∂ 2 ) r ϕ or ∂ µ (∂ 2 ) r ϕ for p = 2r or p = 2r + 1. On the basis of the results in (2.12) we may guess

Higher-order (n − 1)-Form Gauge Theories
In d = 2n dimensions free conformal theories can be formed from (n − 1)-form gauge fields This may be extended to a general curved background metric γ µν so as to be invariant under Weyl rescalings in the form is the Schouten tensor,R = 1 2(d−1) R a rescaled scalar curvature and W µνλρ the Weyl tensor. The Weyl tensor term is invariant under Weyl rescaling by itself and so, for n > 1, has an arbitrary coefficient. The expression for the action may be written in various forms with the aid of the identity, for d arbitrary, depending on the Bianchi identity for F . 4 Varying the metric in (3.3) determines the corresponding flat space energy momentum tensor for (n − 1)-form gauge fields involving two derivatives and so T µν n,2 is conserved subject to the equation of motion ∂ 2 ∂ µ F µµ 1 ...µ n−1 = 0.
In a Feynman type gauge the action (3.1) reduces to so that in this gauge where is the projector on to rank n antisymmetric tensors. Then, with F defined in (3.1), the two-point function for F , which is gauge independent, is given by with I λν (x) determined by (1.2). The result (3.9) has the expected form for F µ 1 ...µn a conformal primary of dimension n.
For this theory the two-point function of the energy-momentum tensor is determined from for, discarding total derivatives and terms which vanish on the equations of motion, The c W contribution in (3.4) can also be dropped since this term is a conformal primary descendant of a conformal primary and does not contribute to (3.10). Using the two-point function (3.9) the combinatorics for arbitrary n can be handled with the identities and, if n ≥ 1, , p = 0, . . . , n . (3.14) Consistency requires B (n) p−1 . Evaluating (3.10) then gives Note that C gauge T,1,2 = C T,ϕ,4 for d = 4. Large N methods, similar to those for the O(N ) and Gross-Neveu models, have been extended to an Abelian gauge theory coupled to N fermions [23]. For d = 4 + 2p, p = 0, 1, . . . , this becomes equivalent to a renormalisable theory with N fermions and a higher-derivative gauge theory with a Lagrangian − 1 4 F µν (−∂ 2 ) p F µν . Using the large N results for C T and subtracting the free fermion contribution [23], in the notation used above, predicted that for the free gauge theory C gauge T,2,2p = (−1) p 2(p + 2)(2p + 4)!/((p + 1)!(p + 3)!). For p = 0 this is the standard result and the case p = 1 agrees with (3.15) when n = 2.

(n − 1)-Form Theories away from Integer Dimensions
The usual gauge invariant action for (n − 1)-form gauge fields is only conformally invariant in d = 2n dimensions although it may be extended, as in the previous section, in higher even dimensions with additional derivatives. However abandoning gauge invariance the action may be extended to be conformal for general d. The corresponding Weyl-invariant action on a curved-space background was obtained by Erdmenger [24], following from the construction of a conformal second-order differential operator on k-forms obtained by Branson [15], and the corresponding flat-space action for vector fields or one-forms was given in [25].
The curved-space action obtained in [24] may be expressed, with a similar notation to On flat space this is tantamount to a particular choice of a covariant gauge fixing term [25].
The flat space energy-momentum tensor is then This satisfies and so the energy-momentum tensor is conserved and traceless on the equations of motion.
Of course for d = 2n, T µν n,0 reduces to the usual gauge invariant form. The two-point function for the (n−1)-form field determined by the action (4.1) on flat space was calculated in [24] by inverting the Fourier transform of the kinetic differential operator and then returning to x-space giving which has the form required by conformal invariance for A µ 1 ...µ n−1 a conformal primary. From and This has the conformally invariant form in terms of the inversion tensor only when d = 2n and is identical in this case to the expression obtained for gauge choices for A µ 1 ...µ n−1 other than that implicit in (4.4) [21].
As in other cases, evaluating the energy-momentum tensor two-point function can be simplified to T µν n,0 (x) T σρ n,0 (0) = T µν n,0 (x) Y σρ n (0) , This then determines dimensions, whose energy-momentum tensor is obtained just from the F F terms in (4.3), is C gauge T,n,0 = 2 n 2 (2n − 2)!/(n − 1)! 2 , [7,21]. This is not equal to C T,n in (4.8) when d = 2n, although (4.2) apparently reduces to the required form for this d. 5 The difference arises since 5 If the energy-momentum tensor (4.2) is restricted to the gauge-invariant F F terms and we use (4.6) then the resulting two-point function is not of the required conformal form (1.1). If CT is identified through the coefficient of the xxxx terms for n = 2 then CT,2 = 1 2 d 2 (d − 2) as obtained in [23]. This prescription in general gives CT,n = 1 2 d 2 (d − 2) · · · (d − n)/(n − 1)!.
the AA two-point function in (4.4) is also singular when d = 2n. The representation of the conformal group generated from a conformal primary A µ 1 ...µ n−1 is reducible when d = 2n and an irreducible representation for the associated gauge theory is obtained by quotienting by the invariant subspace corresponding to gauge transformations. Since C T is related to the number of degrees of freedom it is expected to differ between the gauge theory and that corresponding to the (n − 1)-form A µ 1 ...µ n−1 .

Conformal Primary Operators
The energy-momentum tensor is a conformal primary operator. The detailed expressions in (2.2) and (2.4) are necessary to ensure this and we show here how they can be recovered by requiring T ϕ,n,µν to be a conformal primary, and that this determines the parameter λ in accord with (2.7), although this term is conserved and traceless by itself.
For a local tensor operator X α 1 ...αn formed from multinomials in ϕ and derivatives at x = 0 we define where ∆ X is determined just by counting the number of derivatives and fields ϕ in X α 1 ...αn .
For any conformal primary X A , A = {α 1 . . . α n }, then K b X A = 0 (this is of course the usual condition K µ X A (0) = 0). Otherwise X A is not a conformal primary and generates a reducible representation of the conformal group. Acting with K b removes derivatives so that for some finite N , K b N +1 X A = 0 and hence we can . Extending {f A,rI (b)} to include all possible rotationally covariant forms, with f A,rI = 0 for some r if necessary, M is a square matrix and we may solve for D A,rI (b) unless {O I } have conformal primary descendants with N derivatives so that det M = 0. For generic ∆ X this does not arise. Iterating this construction then gives the conformal primary X A − Y Y A which is the lowest weight state for an irreducible representation. If det M = 0, for particular ∆ X , the representation space obtained from X A is reducible but not decomposable.
The result (5.1) can be extended successively to multiple derivatives. For a scalar conformal primary ϕ with scale dimension δ, so that D ϕ = δ ϕ, and an arbitrary vector a, we have from which, by acting with ∂ a · ∂ a , n r (−1) r (δ) r (δ) n−r (a · ∂) r ϕ (a · ∂) n−r ϕ (5.5) satisfies K b Φ n (a) = O(a 2 ) and so taking a 2 = 0, which projects out the traces, this demonstrates that Φ n (a) defines a symmetric traceless conformal primary with ∆ Φn = 2δ + n and twist 2δ [26]. For higher twist results are more complicated. In the following we will work out O(a 2 ) terms in a few examples and obtain some results for higher twist. Note that (5.5) of course gives Φ n (a) = 0 for n odd.
In the remainder of this section we apply the procedure outlined above for constructing conformal primaries. Initially we construct a conformal primary starting from ∂ µ ϕ ∂ ν ϕ. Using and also Hence, a symmetric tensor conformal primary with dimension ∆ O 2 = 2δ + 2 and twist 2δ is given by With more derivatives the construction becomes more lengthy as the descendants of more conformal primaries have to be subtracted. To construct conformal primary symmetric tensors with four derivatives we start from This can be cancelled by terms involving four derivatives acting on ϕ 2 so that where It is straightforward to check that K b O 4,µνρσ = 0, as guaranteed by the fact that there are no primaries with three derivatives and two ϕ's. By acting with ∂ a · ∂ a we obtain a spin-two primary with twist 2δ + 2, as well as a scalar primary, For six derivatives (5.11) is extended to K b 6 (a · ∂) 6 ϕ ϕ = 90δ(δ + 1)(δ + 2) 8(δ + 3)(δ + 4)(δ + 5) (a · b) 6 − 60(δ + 3)(δ + 4) (a · b) 4 a 2 b 2 Then, with (5.21) Further, we have
A four-index as well as a two-index and a scalar conformal primary can be obtained from O 6 by acting with ∂ a · ∂ a . To avoid even more lengthy expressions we only list here the scalar primary, Thus, we see that the requirement that T ϕ,6,µν be a conformal primary determines λ, independently and consistently with the result (2.7) obtained from the curved-space action contribution of the Bach tensor.

Conclusion
In this paper we have computed C T for various free field theories outside the usual range. Although such theories involving higher derivatives in general correspond to non-unitary quantum field theories, they appear to be relevant in understanding some CFTs for large N numbers of component fields where the 1/N expansion remains valid for arbitrary dimension d. Of course there are additional parameters, such as those associated with the energy-momentum tensor three-point function, one of which is related to C T by Ward identities. For the usual free CFTs these were also calculated in [6]. The theories discussed here might also be extended to determine the energy-momentum tensor three-point function, but the complexity of the expressions for T µν makes this a rather formidable task, as are the corresponding large N calculations.
In general in even dimensions C T in a CFT is related to a particular term quadratic in the Weyl tensor in the energy-momentum tensor trace on a curved space background. In four and six dimensions, where the trace anomaly coefficients are c and c 3 , the relations are [16] with a normalisation chosen so that for conventional free field theories c, c 3 are given by 5! c = n S + 3 n W + 12 n A , 7! c 3 = 2 n S + 40 n W + 180 n B with n S scalars, n W Weyl fermions and n A , n B the number of vector, 2-form gauge fields in four, six dimensions. There is of course complete agreement between the results for C T and the curved space results based on using the heat kernel for second-order conformal differential operators. The results obtained here allow some contributions to the heat kernel for higher-order operators on curved backgrounds to be obtained. These have been discussed for Ricci flat backgrounds in [27].
Heat kernel techniques allow a perturbation expansion for arbitrary curved backgrounds so as to determine the leading corrections to c, c 3 . In six dimensions for a cubic interaction 1 6 λ ijk φ i φ j φ k then results in [16,28] give to lowest order For the theory defined by (1.5), where φ i → (σ, ϕ i ),λ ijkλijk = 3Nĝ 2 +λ 2 and to lowest order 3 , so that at the fixed point, to leading order in ε, 1/N ,ĝ 2 = 6ε/N,λ 2 = 6 3 ε/N and hence in (1.3), (1.5) gives for the CFT at d = 6 − ε for large N agreeing with the perturbative flat space calculation in [10] and also the expansion of the large N result. The corresponding results for four-dimensional renormalisable theories were obtained some time ago. For n A gauge fields, with a simple gauge group and coupling g, Dirac fermions and Yukawa, scalar interactionsψY i ψ φ i , 1 24 λ ijkl φ i φ j φ k φ l the results obtained in [29] by expanding about flat space and, using heat kernel methods for a curved background, [30,31] give 6 For scalar and fermion theories large N methods allow non-trivial CFTs to be formally defined for general dimensions d which interpolate between physical theories for d an integer.
The situation is less clear for gauge theories since maintaining conformal invariance and gauge invariance is more difficult, as was demonstrated in section 4. For a gauge-invariant quantum field theory the energy-momentum tensor in general contains contributions arising from the gauge fixing and ghost terms in the action. However these are BRS exact and do not contribute to correlation functions for gauge-invariant operators so that the calculations of C T in [6,7] did not take account of them. In section 4 the gauge fixing terms made a contribution to the energy-momentum tensor (4.2) whose effect did not disappear in correlation functions when they notionally decoupled. It is an open question whether the ghost contributions to T µν 6 In terms of some previous literature c = −16π 2 βa. In [30] in the final result a misprint in (4.16) is corrected by taking 1 9 → 2 9 .
could be extended to general d while maintaining conformal invariance. Their contributions to C T should account for the difference between C T in (4.8) and the corresponding gauge theory result when d = 2n.

Note Added
For higher-derivative scalar theories calculations of C T have also been carried out by Guerrieri et al. [32], who further considered the theory with eight derivatives. Prompted by their discussion there is a quick derivation of C T,ϕ,2p , agreeing with (2.13) for any p, based on known results for conformal partial-wave expansions of four-point functions as a sum over conformal primaries. Using Mellin transform methods Fitzpatrick and Kaplan [33] obtained a conformal partial-wave expansion for the four-point function of generalised free fields in any dimension d. Their results give the expansion c τ,ℓ G 2δ+2τ +ℓ,ℓ (u, v) , (6.5) where u, v are the standard conformal invariants and G ∆,ℓ (u, v) are the conformal partial waves for a conformal primary operator with scaling dimension ∆ and spin ℓ and its descendants.