Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models

We use the conformal bootstrap approach to explore $5D$ CFTs with $O(N)$ global symmetry, which contain $N$ scalars $\phi_i$ transforming as $O(N)$ vector. Specifically, we study multiple four-point correlators of the leading $O(N)$ vector $\phi_i$ and the $O(N)$ singlet $\sigma$. The crossing symmetry of the four-point functions and the unitarity condition provide nontrivial constraints on the scaling dimensions ($\Delta_\phi$, $\Delta_\sigma$) of $\phi_i$ and $\sigma$. With reasonable assumptions on the gaps between scaling dimensions of $\phi_i$ ($\sigma$) and the next $O(N)$ vector (singlet) scalar, we are able to isolate the scaling dimensions $(\Delta_\phi$, $\Delta_\sigma)$ in small islands. In particular, for large $N=500$, the isolated region is highly consistent with the result obtained from large $N$ expansion. We also study the interacting $O(N)$ CFTs for $1\leqslant N\leqslant100$. Isolated regions on $(\Delta_\phi,\Delta_\sigma)$ plane are obtained using conformal bootstrap program with lower order of derivatives $\Lambda$; however, they disappear after increasing $\Lambda$. We think these islands are corresponding to interacting but nonunitary $O(N)$ CFTs. Our results provide a lower bound on the critical value $N_c>100$, below which the interacting $O(N)$ CFTs turn into nonunitary. The critical value is unexpectedly large comparing with previous estimations.

The conformal bootstrap [1-4] provides a non-perturbative approach to solve conformal field theories (CFTs) using general consistency conditions of CFT. It has led to great successes in 2D, such as the seminal work [4] on solving 2D rational CFTs. In recent years the conformal bootstrap has been revived since the breakthrough discovery in [5], which shows that the crossing symmetry and the unitary conditions can provide strong constraints on the operator scaling dimensions without an explicit form of Lagrangian. The crossing symmetry of four-point correlator leads to an infinite set of constraints on the CFT data. These constraints are difficult to be solved analytically, instead, they are truncated to a finite set and reformulated as a convex optimization problem so that they can be solved numerically.
Here the convexity of conformal block functions [6,7] plays a crucial role. Since then the conformal bootstrap has been significantly developed and it becomes a remarkably powerful technique to obtain CFT data, including operator scaling dimensions and operator product expansion (OPE) coefficients in D > 2 dimensions . Review of previous developments on conformal bootstrap is provided in [50].
From conformal bootstrap with single correlator φφφφ , one can obtain bounds on the conformal dimension or OPE coefficient of objective operator. The bounds may exhibit singular behaviors, such as kinks which are believed to be related to unitary CFTs. One can expect to obtain more information on CFTs through bootstrapping mixed correlators like φφφ 2 φ 2 . Conformal bootstrap with mixed operators has been fulfilled in [28,39] for 3D Ising model and critical O(N) vector models and the results are quite impressivethe allowed scaling dimensions are isolated in small islands. The accuracy can be improved further by refining the numerical techniques [37,49]. Studies on the 3D O(N) vector models are strongly motivated by their special importance in physics. For small N 3 they describe second-order phase transitions happened in real physical systems [51]. Besides, its O(N)singlet sector is proposed to be dual to higher spin quantum gravity in AdS 4 with Dirichlet boundary conditions [52]. In the UV side, the 3D O(N) vector model contains N free scalars φ i , i = 1, · · · , N perturbed by quartic coupling (φ i φ i ) 2 . The RG flows to an IR fixed point which is strongly coupled. For the critical O(N) vector models with large N or in D = 4 − ǫ, ǫ ≪ 1 dimensions, one can obtain reliable results using large N expansion or the well-known Wilson-Fisher ǫ expansion. Actually these analytical results have been used as consistency checks of conformal bootstrap in [18,19]. Nevertheless, for the 3D (ǫ = 1) critical O(N) vector models with small N which are more physically attractive, these perturbative methods turn into less effective. In contrast, conformal bootstrap remains useful and has provided the most accurate results up to date [49].  [53,54]. In D = 4 + ǫ such UV fixed point theory is weakly coupled for sufficient small ǫ and it requires a negative quartic coupling coefficient, which may introduce the problem of instability even though the scaling dimensions of the operators are above the unitary bound. A UV-completed formulation of the O(N) model in D > 4 dimensions has been proposed in [55,56] in which the φ i constructs fundamental representation of O(N) and the O(N) singlet σ performs as composite field φ 2 i in the UV side. The theory contains cubic interaction terms which are relevant in space with dimension D < 6. Using the combination of ǫ and large N expansion it has been shown that this theory admits an interacting IR fixed point [55,56], which is unitary for N > N c , while below the critical value N < N c the coupling turns into complex and the IR fixed point theory is nonunitary. At one-loop level the critical value N c is about N c ≈ 1038. For 5D (ǫ = 1) critical O(N) theories, the small ǫ condition for ǫ expansion approach breaks down so the results obtained from ǫ expansion should be treated carefully. Actually the critical value decreases to N c ≈ 64 at three-loop level. In [57] the author has obtained a critical value N c ≈ 400 at four-loop level based on resummation methods. A non-perturbative method is desirable to determine the critical value N c in 5D. The 5D critical O(N) models have been studied using the nonperturbative functional renormalization group equations [58][59][60][61][62], and Refs [58][59][60][61] that no interacting fixed point exists at large N, while the analysis in [62] agrees with the results from the D = 6 − ǫ perturbative approach when ǫ ≪ 1 and predicts the 5D critical value N c = 1. The stability problem of IR fixed point in the cubic model remains according to [62].
The conformal bootstrap approach has been employed to study 5D critical O(N) models in [25,35,36] following the proposal of the cubic model [55,56]. In [25] the 5D critical O(N) models have been assumed to saturate the minimum of the O(N) current central charge c J for large N and the existence of 5D critical O(N) models are indicated from these minimums obtained from conformal bootstrap. The authors focused on bootstrapping the OPE coefficients rather than the scaling dimensions of conformal primary operators. In 3D conformal bootstrap the interacting O(N) CFTs have been found to lie at the kinks of the bounds for the scaling dimension ∆ σ of the O(N) singlet σ, which appears as lowest dimension operator in the S channel of the correlator φ i φ j φ k φ l . However, in 5D cubic model the lowest dimension O(N) singlet operator σ performs as φ 2 i , ∆ σ = 2∆ φ = 3 at the UV Gaussian fixed point which reduces to ∆ σ = 2 + O(1/N) near the IR fixed point. The IR fixed point is below the upper bound of scaling dimensions ∆ σ so there is no clue on the fixed point theory in the bound of scaling dimensions. This problem has been overcome in [35,36] by imposing a gap on the scaling dimensions of σ and the second lowest O(N) singlet conformal primary scalar. With a reasonable assumption on the gap, the allowed region of the scaling dimensions (∆ φ , ∆ σ ) can be carved out and forms two sharp kinks.
The UV Gaussian fixed point lies at the higher kink while the lower kink agrees with the large N expansion predictions on IR interacting fixed point theories. Furthermore, the kink disappears for small N ≈ 15 which may indicate a small critical value N c [36].
However, one should be careful to consider the kinks in conformal dimension bound or the minimum of central charges as unitary CFTs. From perturbative methods it is known that in D = 6 − ǫ, ǫ ≪ 1 the IR fixed point of cubic O(N) models is endowed with complex critical couplings for N 1000. 1 Nevertheless, in [36] a sharp kink is still generated from conformal bootstrap for D = 5.95, N = 600 which is much lower than the threshold value and should be nonunitary. The reason seems to be that the precision adopted in [36] is not 1 CFTs in fractional dimensions are known to be nonunitary even with real couplings [63,64]. However, the unitarity is violated by operators with high scaling dimensions so they are more difficult to be tested through conformal bootstrap approach. high enough to detect the small violation of unitary. A more powerful bootstrap approach is needed to study the 5D O(N) models, especially on its critical value N c .
In this work, we will study the conformal bootstrap with multiple correlators of conformal primaries φ i and σ: φ i φ j φ k φ l , φ i φ j σσ , σσσσ . Since there are more operators involved in the bootstrap program, it is expected that the results will provide more rigid restrictions on the scaling dimensions of (∆ φ , ∆ σ ). Actually we find that the scaling dimensions (∆ φ , ∆ σ ) obtained from bootstrapping multiple correlators of 5D O(500) model is isolated in a rather small island, which is nicely compatible with the perturbative results. We also study the critical value N c in 5D. In preliminary numerical calculations we find small islands on the allowed scaling dimensions (∆ φ , ∆ σ ) for all N 1. However, these islands disappear after improving the bootstrapping precisions. Taking N ≈ 40 for example, it shows an apparent kink in the bound from bootstrapping single correlator of φ i s [36]. Using multiple correlator conformal bootstrap with small Λ, we obtain an island on (∆ φ , ∆ σ ) plane close to the kink presented in [36], while it vanishes after increasing Λ. We do not find a stable island even for N = 100, therefore our results suggest a rather large critical value N c > 100.
This paper is organized as follows. In section 2 we briefly review the cubic model of According to above 1/N expansion, the conformal dimension of φ i is above the unitary bound (∆ φ > 3/2 for scalar fields) given N > 35. The critical value N c = 35 can be significantly modified by by higher order corrections. Actually the 5D 1/N expansions converge much slower than those in 3D [55].
Alternatively, the 5D quartic theory can also be studied using ǫ expansion [76]. Conformal dimensions of φ i and φ 2 (σ) have been calculated up to five-loop [77]: where where Taking ǫ = −1 the results can be interpolated to 5D. For large N the higher order coefficients c i s are of order 1/N. In this case the ǫ expansion performs worse asymptotically in 5D comparing with the large N expansion. While for small N it is not clear at this stage which approach can provide better estimation. These perturbative results will be useful to estimate the conformal dimension gap which can be applied in the conformal bootstrap to improve the numerical efficiency.
Both the large N expansion and the ǫ expansion contain negative terms at higher loop level. For small Ns these negative contributions may play dominating roles in the perturbative expansion and result in negative anomalous dimension. Specifically the fiveloop result (5) shows the conformal dimension ∆ φ < 3/2 for N 14 [55]. In [36] the conformal bootstrap with single correlator has been applied to generate bound on ∆ σ .
As usual, above perturbative result is not sufficient to make a solid estimation on 5D (ǫ = 1) N c due to its asymptotic performance. It is tempting to evaluate the critical value N c using non-perturbative method. Besides the above interacting IR fixed point, the cubic model also admits extra fixed points with different critical value N ′ c ; however, they are not corresponding to the classical interacting quartic fixed point and will not be studied in this work.

III. CONFORMAL BOOTSTRAP WITH MULTIPLE CORRELATORS
Conformal bootstrap with multiple correlators has been developed in [28,39] [39]. More details on this program are provided in [37]. 2 The renormalization group approach suggests the cubic model admits an extra RG relevant direction with positive critical exponent at the IR fixed point [62]. In this sense the universality class of the quartic O(N ) model is a subset of that of cubic O(N ) model.

A. Bootstrap Equations from Crossing Symmetry
Conformal partial wave function is the crucial ingredient for conformal bootstrap. In even dimensions D = 2, 4, 6, the conformal partial wave functions have been solved analytically [6,7]. In odd dimensions there is no analytical expression for conformal partial wave function; however, it can be calculated recursively with arbitrary precision [18,28,78]. 3 The general four-point function of scalar operators can be expanded in terms of conformal partial waves where σ i s are scalar operators with conformal dimension ∆ i (∆ ij = ∆ i − ∆ j ) and O is the conformal primary operator appears in the OPE expansion of σ 1 σ 2 ∼ λ 12O O (and also and v = The four-point function can be evaluated equivalently in different channels, as suggested by crossing symmetry, and it leads to the following equations in which To study the 5D O(N) vector model, we apply the crossing relations for correlators   [39]. These equations can be written in a compact form [39] which are presented below for later reference Explicit forms of the 7-component vectors V S , V T , V A , V V are provided in the Appendix.

B. Bounds from Crossing Relations
The equations from crossing symmetry (12) provide nontrivial constraints on the CFT data. The numerical approach to study these equations was first proposed in [5] and the following developments show this method is extremely powerful. The logic of numerical conformal bootstrap is firstly to make assumptions on the CFT spectra. If the assumptions are physical they are required to satisfy the crossing relations (12) and the unitary condition.
Numerical conformal bootstrap provides a systematical way to check the consistency between the assumptions and general constraints on CFTs. Bounds on the CFT data, including conformal dimensions of primary operators and OPE coefficients can be obtained by falsifying possible assumptions on the CFT spectra.
Specifically for any hypothetical spectra (∆ φ , ∆ σ ) above the unitary bounds, they should be consistent with the crossing relations (12). However, if there are linear functionals α = (α 1 , α 2 , · · · , α 7 ) satisfying then the crossing relations (12) can never be satisfied and initial assumption on the spectra (∆ φ , ∆ σ ) have to be abandoned as unphysical. In the bootstrap conditions (13) (13), it is the summation of contributions in S (from σ) and V channels (from φ) that is required to be positive-semidefinite due to the equality of OPE coefficients The bootstrap conditions in (13) are not the only way to break the crossing relations (12). In particular, to bootstrap certain OPE coefficient of operator (∆ 0 , ℓ 0 ) in channel X, one may set α · V X,∆ 0 ,ℓ 0 = 1 instead of choosing the unit operator as in (13). The bootstrap conditions are further refined in [49]. The lower bounds ∆ * S,0 and ∆ * V,0 introduced in (13) are necessary to isolate the conformal dimensions (∆ φ , ∆ σ ) in a small island. A higher but remaining physical lower bound can improve the numerical efficiency to carve out the allowed parameter space. For sufficient large N, these lower bounds can be justified from perturbative expansions. The O(N) singlet scalar next to φ 2 is φ 4 in the quartic model, and its conformal dimension can be evaluated through the large N expansion (4). In the cubic theory (1) this is given by a mixing of σ 2 and φ 2 . One of the linear combination of σ 2 and φ 2 is actually the descendent of σ, while another orthogonal mixing constructs a primary O(N) singlet that shares the same conformal dimension as obtained from quartic theory [55,56].
The candidate of next O(N) vector scalar is φ 2 φ i (or σφ i in the cubic theory). However, as argued in [39] for the 3D theories, in D = 6 − ǫ, ǫ ≪ 1 dimension the quartic theory generates the following equation of motion for φ i : which suggests that the operator φ 2 φ i is a descendent of φ i rather than a conformal primary scalar. One can get the same conclusion in cubic theory (1) with replacement φ 2 → σ.

C. Numerical Implementation of Conformal Bootstrap
Equations from crossing symmetry (12) provide an infinite set of constraints (13) on the CFT data. For the numerical implementation the constraints need to be truncated to a large but finite set. In (13) the constraints are parameterized by (∆, ℓ). The spins ℓ construct an infinite tower of spectra while in conformal bootstrap only these spectra with small ℓ will be considered. Contributions from operators with large spin are exponentially suppressed.
The linear functionals α can be expanded as where (z,z) are defined in terms of (u, v) through: u = zz, v = (1 − z)(1 −z). Moreover, for the linear functional α i , the number of derivatives is also truncated up to Λ. Taking higher order of derivatives in (15), we have more chances to find the linear function satisfying (13).
As a result, the conformal bootstrap program can exclude larger regions in parameter space.
In practice the parameter Λ is restricted by computation power. The setups of parameter Λ and spins used in this work are as follows The problem to find the linear functions α under truncated constraints can be solved with SDPB program [37].

A. Bootstrapping 5D O(500) Vector Model
The 5D O(500) vector model has been studied in [25,35,36]  These predictions will be compared with the results obtained from conformal bootstrap.
In Figure 1 we present the bounds on (∆ φ , ∆ σ ) obtained through bootstrapping the single correlator φ i φ j φ k φ l (light blue region) and the multiple correlators (dark blue island). To bootstrap the single correlator we have assumed that the next O(N) singlet scalar has dimension above the gap ∆ * S,0 = 3.965, which can be justified from the large N expansion result (4): ∆ σ 2 ≈ 3.972. This gap is also employed in [36]. The upper part of light blue region is similar to the bound provided in [36]. Besides, there is an extra kink in the lower region and the whole region actually forms a sharp tip like presented in [35], although a  as discussed before.
Remarkably, the allowed region of (∆ φ , ∆ σ ) obtained from the multiple correlator bootstrap is enclosed in a small island, which is colored in dark blue in Figure  However, it should be careful to make statement based on results from conformal bootstrap with lower order of derivatives. Actually in preliminary study we have obtained isolated islands even for N = 1 with Λ ∼ 15; however, they disappear after increasing Λ.
As to the model with N = 500, we have checked the performance of the island with larger Λ. The results are provided in Figure 2. The allowed regions shrink notably from Λ = 21 to Λ = 25. Interestingly, the fixed point predicted by large N expansion remains located in the center of the small island even though the allowed region has contracted significantly. We have searched the allowed regions on (∆ φ , ∆ σ ) plane for N 100. The isolated islands can be obtained for small Ns with mild assumptions on the dimension gaps (∆ * S,0 , ∆ * V,0 ). However, these islands disappear after increasing the number of derivatives Λ. For N ∼ O(10) or smaller, the perturbative approaches cannot provide an approximate estimation on the conformal dimension ∆ σ 2 . One may argue that the islands disappear due to the reason of the unphysical dimension gaps (∆ * S,0 , ∆ * V,0 ) used in the bootstrap program instead of the nonunitarity of the CFTs. While for larger Ns the perturbative predictions are expected to provide rough estimations on the fixed point. This can be seen from the fact   Figure 3 is rather close to the kink from single correlator bootstrap presented in [36]. In [36]  and the correspond spins provided in (16). We apply a dimension gap ∆ * S,0 = 3.6 in the S-channel. Besides, an extra dimension gap ∆ * V,0 = 5 has been used in the V-channel for bootstrapping multiple correlators. The black dot and cross relate to the predictions from ǫ expansion and large N expansion, respectively. subtle to be discovered in numerical conformal bootstrap. In the 5D O(N) single correlator conformal bootstrap [36], sharp kinks are also generated in the fractional dimension D = 5.95 with N = 600, notably lower than the critical value N c ≈ 1000. We have studied this model through bootstrapping multiple correlators. There remains isolated allowed region even at Λ = 21, though it is quite small. The uncertainty on ∆ σ shown in the island is about 2 × 10 −3 , while as shown in [36], the magnitude of imaginary part in ∆ σ is of the same order ∼ 1.5 × 10 −3 so it is expected that current conformal bootstrap program cannot capture the tiny unitarity violation unless the numerical accuracy can be improved significantly.
To summarize, the numerical conformal bootstrap provides a powerful approach to falsify assumptions on unitary CFTs. However, it is premature to validate the unitary CFTs using conformal bootstrap due to these "pseudo" unitary solutions. As to the 5D O(500) model, although our results have provided strong evidence, they are still not sufficient to make a strict conclusion on its unitarity. On the other hand, it is surprising that the 5D O(N) vector model is nonunitary even for N = 100. As a result, the critical value N c > 100, which is considerably larger than the value estimated before.

V. CONCLUSIONS
In this work, we have studied the interacting 5D CFTs with global O(N) symmetry using the conformal bootstrap with multiple correlators. The multiple correlator conformal bootstrap has been developed in [28,39] [55,56]. Evidence of such fixed point has already been shown in the single correlator conformal bootstrap studied in [25,35,36]. The island obtained in this work is rather close to the kink in the bound of conformal dimension ∆ σ obtained from bootstrapping correlator of four φ i s [36]. We have studied the performance of the island under higher order of derivatives Λ. The island shrinks notably from Λ = 19 to Λ = 25, while the large N expansion predictions remain staying in the center of the allowed region. Such coincidence is surprising in considering of that only crossing symmetry and unitary conditions are employed to generate the allowed region. Besides we only input the O(N) global symmetry for this model while even did not use its Lagrangian at all.
We are particularly interested in the critical value N c of 5D O(N) vector model below which the interacting fixed point theory loses unitarity. The problem on the critical value N c can also be seen from the perturbative expansions of conformal dimension ∆ φ , that below the critical value the scalar φ i acquires conformal dimension smaller than the unitary bound and breaks the unitary condition. However, in 5D the perturbative expansions converges much slower comparing with these of 3D. In [55,56] the critical value N c has been evaluated based on large N expansion in D = 6 − ǫ spacetime. The critical value N c ≃ 1038 at oneloop level; however, it oscillates drastically order by order. Conformal bootstrap provides a nonperturbative approach to study CFTs, and it has been applied to estimate N c in [36].
The authors found that the pronounced kink in the bound of ∆ σ disappears near N ∼ 15, which may suggest N c ∼ 15 in view of the observation that the singular behaviors, like kink in the dimension bound usually relate to unitary CFTs. In 3D such observation has helped to numerically solve the Ising model [23] and O(N) vector model [18]. However, the unitarity condition becomes subtle for 5D CFTs and the unitarity violation may be too small to be detected by the bootstrap program with low order of derivatives. Therefore a kink does not necessarily guarantee unitarity, instead, it may relate to an interacting but nonunitary CFTs.
We have searched the allowed regions using multiple correlator conformal bootstrap for 1 N 100. The isolated regions on the (∆ φ , ∆ σ ) plan can be obtained from conformal bootstrap program with lower order of derivatives. Moreover, the islands actually locate in the position close to the predictions from perturbative approaches given the Ns are not too small. However, the islands disappear after increasing the number of derivatives in bootstrap program. We believe these islands relate to interacting while nonunitary CFTs and the violation of unitarity can not be observed unless the program is equipped with sufficient high precision. In particular, our results suggest the critical value N c 100, much larger than the value estimated before. The bounds of N c is expected to be improved further using conformal bootstrap. However, for larger N the unitarity violation in O(N) fixed point theory gets smaller and more difficult to be detected. It requires higher accuracy in the bootstrap program to determine the critical value N c and we leave this problem for future work. On the other hand, for a sufficient large N c , the large N expansion approach is validated. The critical value N c can be effectively studied based on this perturbative approach as well.
Due to the asymptotic behavior of perturbative expansions in 5D, probably one needs to calculate several orders higher than in [55][56][57] to get a sufficient good estimation.