$\phi^3$ theory with $F_4$ flavor symmetry in $6-2\epsilon$ dimensions: 3-loop renormalization and conformal bootstrap

We consider $\phi^3$ theory in $6-2\epsilon$ with $F_4$ global symmetry. The beta function is calculated up to 3 loops, and a stable unitary IR fixed point is observed. The anomalous dimensions of operators quadratic or cubic in $\phi$ are also computed. We then employ conformal bootstrap technique to study the fixed point predicted from the perturbative approach. For each putative scaling dimension of $\phi$ ($\Delta_{\phi})$, we obtain the corresponding upper bound on the scaling dimension of the second lowest scalar primary in the ${\mathbf 26}$ representation $(\Delta^{\rm 2nd}_{{\mathbf 26}})$ which appears in the OPE of $\phi\times\phi$. In $D=5.95$, we observe a sharp peak on the upper bound curve located at $\Delta_{\phi}$ equal to the value predicted by the 3-loop computation. In $D=5$, we observe a weak kink on the upper bound curve at $(\Delta_{\phi},\Delta^{\rm 2nd}_{{\mathbf 26}})$=$(1.6,4)$.


Introduction
Conformal field theory (CFT) describing interesting infrared (IR) physics usually arises as the fixed point of renormalization group flow. A useful perturbative tool to study such kind of fixed point is the -expansion, which has been applied to explore the IR fixed point of quartic scalar theory in D = 4 − dimensions, including D = 3 Ising model [1] and critical O(N ) vector model (see [2] for a comprehensive review). In 4 < D < 6, the quartic scalar interaction becomes irrelevant and the renormalization group flow can instead be triggered via a cubic scalar interaction. The simplest φ 3 theory in 6−2 has been considered long time ago [3][4][5], with the Lagrangian L = 1 2 (∂φ) 2 + 1 6 gφ 3 . In [5], it was shown that the 1-loop beta function has a non-unitary IR fixed point with imaginary coupling constant g for D < 6.
Continuation of this fixed point to D = 2 describes the Yang-Lee edge singularity [6,7] in the Ising model (this is the (2,5) minimal model [8,9] with negative central charge). Recently there has been a revival of interests to the renormalization of quantum field theory with φ 3 interaction in D = 6 − 2 1 [10][11][12][13][14][15][16], motivated by studying a-theorem in D > 4 or higher spin holography. In particular, the Lagrangian was utilized in [10,12] to investigate the D = 5 critical O(N ) vector model 2 . An interesting phenomenon originally noted in [4] and recently rediscovered in [10] is that there exists a critical value for N , denoted as N crit , above which a stable, unitary fixed point was found in 6 − 2 dimension. One-loop renormalization suggests that N crit ≈ 1038 [4,10]. Later, a 3-loop computation implies a much smaller N crit [12].
As a non-perturbative approach to CFT, conformal bootstrap dates back to the work of Alexander Polyakov [18] and also the work of Sergio Ferrara, Raoul Gatto and Aurelio Grillo [19] in the 1970s . Its later application to two dimensional conformal field theories led to the famous work of Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov [8], which classified D = 2 minimal models. In D > 2, a significant progress was made by [20].
In this work, we explore the possibility of having a CFT in five dimensions with F 4 global symmetry. The exceptional Lie group F 4 known as the compact real form of Lie algebra f 4 , is also the isometry group of the octonionic projective plane OP 2 [43]. It admits a rank-2 and a rank-3 irreducible symmetric invariant tensors denoted by δ ij and d ijk , where the index transforms as the 26 of F 4 . The simplest interacting F 4 theory can be written as a scalar theory with a cubic self-interaction The cubic interaction is relevant in 6−2 dimensions and may drive the theory to a nontrivial IR fixed point. From the 3-loop renormalization of the coupling constant, we indeed observe a stable IR fixed point in D = 6 − 2 . We then employ conformal bootstrap technique to probe such a fixed point in D = 5.95 and D = 5 4 . We observe that in D = 5.95 the boundary of the allowed region in the (∆ φ , ∆ 2nd 26 ) plane exhibits a sharp peak exactly at the value of ∆ φ obtained from the Padé [2,1] resummed 3-loop results. In D = 5, a weak kink is observed near the 3-loop results. The appearance of the kink has to do with fact that when the anomalous dimension of φ i is small, the second lowest scalar primary in 26 is approximately given by d ijk φ j φ k with dimension 2∆ φ . However, when the anomalous dimension φ i is large enough, the theory acquires notable deviation from the free theory.
In the interacting theory, the operator d ijk φ j φ k becomes a conformal descendant of φ i .
The new second lowest scalar primary in 26 should have much higher dimension that 2∆ φ , yielding a sudden change in ∆ 2nd 26 . This paper is organized as follows. In Section 2, we present the renormalization of the theory (1.2) in D = 6 − 2 using -expansion. In particular, we compute the anomalous dimensions of operators such as φ i , φ i φ i and d ijk φ i φ j φ k up to O( 3 ). We also calculate the anomalous dimensions of other φ 2 operators in 26 and 324 representations at 1-loop level.
In Section 3, we derive the set of crossing equations for a CFT with F 4 global symmetry and then apply it to study the fixed points predicted by the loop calculations in D = 5.95 and D = 5, using numerical conformal bootstrap. We discuss future extensions in Section 4.

3-Loop Renormalization of generic φ 3 theory in 6 − 2 Dimensions
The 3-loop renormalization of generic φ 3 theory in D = 6 − 2 was studied long time ago by [45,46] using the modified minimal subtraction (MS) scheme. Recently [15] has extended the results to 4 loops. Here we only utilize the 3-loop results. The 3-loop beta function is given by 5 The 3-loop anomalous dimension of φ takes the form (1.4) and the anomalous dimension of operator O ∼ φ i φ i (for simplicity, from now on we will denote the F 4 singlet φ 2 operator by φ 2 ∈ 1 , where 1 means the singlet representation of (1.5) 5 Throughout this paper, we mainly follow the convention used in [15], except that the sign of g 2 has been reversed. Compared with [10,12], there is a factor of 2 difference in the definition of β(g), and the numerical factor Area(S5)/(2π) 6 = (4π) 3 has been included in g 2 .
In the above expressions, the constants {T 2 , T 3 , T 5 , T 71 , T 72 } are defined as [15] Using γ φ and γ φ 2 ∈1 , the critical exponents η and ν can be computed via The identities in (1.6) can be represented by the Birdtrack [47] diagrams as shown in Figure   1. Some formulas used here can be found in [47] (see Chapter 16). The 26 representation of F 4 group has the following properties: • There exists a symmetric invariant rank-2 tensor δ ij and a totally symmetric invariant rank-3 tensor d ijk carrying indices in this representation; • Higher rank invariant tensors carrying index in this representation are decomposable in terms of products of δ ij and d ijk using tree diagrams; • The symmetric invariant rank-3 tensor d ijk satisfies [48]: where the indices {i, j, k, . . . } range from 1 to n. This relation holds for all the represenations belonging to the F 4 family (See Table 3). In particular, for F 4 , n = 26.
The normalization of d ijk is represented by From now on we will set α = 1. Notice Contracting (2.1) with from the left and applying (2.2) and (2.3), we get Contracting (2.1) with from the right, one gets = n − 2 4(n + 2) where the empty box in the Birdtrack diagram means symmetrization of the two external indices. On the other hand, the antisymmetrization is given by In the first step, we have used (2.1) to replace the top d ijm d mkl pair, and in the second step, we used the fact that the diagram vanishes because it is symmetric with respect to the two open indices on the left. Combining (2.5) and (2.6), we obtain = n − 2 4(n + 2) + + 10 − n 4(n + 2) Notice this equation reduces the number of vertices by two, thus one can apply such an equality to calculate all the T -constants. The results are given by Substituting the values of T -constants to (1.4), we obtain the beta function up to three loops which implies a unitary fixed point resides at (2.10) At the fixed point g = g * , the anomalous dimension of φ i is given by (2.11) The scaling dimension of O ∼ φ i φ i is given as Taking n = 26, we have Finally, we can use Padé approximation to resum these results  [2,1] and Padé [1,2] . We will use choose Padé [2,1] to estimate (2.15) in the following section, because the other choice gives rise to a negative ∆ φ 2 ∈1 at D = 5. We will also provide 1-loop result when necessary.
2.2 1-Loop Renormalization of φ i × φ j operators in 26 and 324 represen- In this section, we shall compute the 1-loop anomalous dimensions of φ 2 operators transforming nontrivially under F 4 . Minimal subtraction scheme is adopted in the calculation.
We need the following projectors of F 4 (n = 26) [47] ijkl = 8 n + 10 ijkl = n + 2 n + 10  Using Feynman rules, these two diagrams are transferred into The index I is not summed in the equations above. Integrals I 1 and I 2 have been evaluated in Appendix A of [10]. Notice that g 2 = (4π) 3 g 2 , with the numeric factor Area(S 5 )/(2π) 6 absorbed in g 2 . In our case, the 1/ pole is canceled by counterterms with the coefficients from which After some calculations, one can check the following relations hold where the index I is not summed, and , A 324 = 2 n + 2 , The scaling dimensions of various operators quadratic in φ can then be computed from where γ φ (g * ) is given in (2.11) and Notice that as a consequence of the equation of motion ijkl φ k φ l becomes a conformal descendant operator of φ i . From (2.11), (2.25) and also Table 1, one can see explicitly that ∆ φ 2 ∈26 = ∆ φ + 2 at 1-loop, which should also hold at higher loop level.

Conformal Bootstrap
In conformal field theories, the structures of two and three point functions are completely fixed by conformal symmetry. A four point function in CFT with F 4 global symmetry can be decomposed as where the projectors are defined in (2.17). The summation runs over all conformal primary operators which appear in the OPE of φ i × φ j . The function g ∆, (u, v) is the so called conformal block, which depends on the cross ratios and is completely fixed by conformal symmetry. In D = 4, the conformal block was first obtained by Dolan and Osborn in [49,50]. In other dimensions, the construction can be found in [23].
Conformal bootstrap approach relies on the fact that operator algebra obeys associativity, hence the following two ways of computing four point function should lead to equivalent This equality is also known as crossing symmetry of four point functions. Notice that the right hand side of (3.3) is identical to the left hand side upon the replacement {i, Initially, the four index tensor d ijm d klm appearing in (3.1) and hence in (  where the ± refers to the parity under i ↔ k. The V (3.6) F and H are the shorthand notations for the convolved blocks defined as symmetry in D = 5.95 and D = 5. Different from the O(N ) conformal bootstrap, here the fundamental field φ i appears in its own OPE due to the cubic self-interaction.
We shall assume that the second lowest scalar primary in the 26 + channel has scaling dimension ∆ ≥ ∆ 2nd 26 . (The lowest scalar primary in this channel is just φ i .) To test this assumption, we search for a linear functional α with the following properties For a given choice of (∆ φ , ∆ 2nd 26 ), if such a linear functional α exists, then we have which contradicts (3.5). The contradiction simply implies that the second lowest scalar primary in the 26 + channel should have scaling dimension lower than ∆ 2nd 26 . We use "SDPB" [51]  plane exhibits a sharp peak at ∆ φ ≈ 1.978, which coincides precisely with the value of ∆ φ given by the Padé [2,1]  The D = 5 result is plotted in Figure 4. Unlike the D = 5.95 case, no peak seems to appear. However, a weak kink is observed at ∆ φ ≈ 1.6. The three loop result ∆ φ ≈ 1.55670 is indicated by the dashed line. The location of the week kink does not fit the 3-loop result may be expected, since in this case, = 0.5, and the D = 5 fixed point is highly nonperturbative in nature. We also notice that when increasing the derivative order, the weak kinks corresponding to different derivative orders tend to converge to a single weak kink.

Discussion
The weak kink observed on the D = 5 bootstrap curve indicates the possibility of the existence of a CFT with F 4 flavor symmetry. It would be interesting to further constrain the D = 5 fixed point using mixed correlators conformal bootstrap [22]. We leave this for future investigation.
The exceptional Lie group of F 4 belongs to the so called F 4 family of invariant groups, which is defined as the family of groups admitting a representation that satisfies the three conditions listed in Section 2.1. According to the Birdtrack classification, there are four choices, the groups and the dimensions of the relevant representations are listed in Table 3.
There exists a fixed point for each of these choices in 6 − 2 dimensions. In Appendix A, we list the dimensions of various operators computed for these theories at the fixed point.
They are obtained simply by substituting the value of n for each case to formulas given in Section 2.1 and Section 2.2. Whether some of these fixed points continue to exist in D = 5 or even D < 5 is also worth further study. Another interesting question is whether other resummation method would give us a better estimation of the operator dimension. Table 3: F 4 family of invariant groups and the dimensions of the relevant representations.
Here we use capital letter to label the compact real form of the Lie algebra.
It is also possible to consider more general φ 3 theory of the form (1. The four loop beta function for SU (N) invariant φ 3 theory has been computed in [15]. It would be interesting to carve out the possible IR fixed points in D < 6 using conformal bootstrap approach.
Besides the F 4 family, there is also the so called E 6 family of invariant groups [47]. The groups admits an invariant 2-tensor δ i j and an invariant symmetric 3-tensor d ijk (and its conjugate d ijk ) carrying indices in some representation and satisfying certain conditions similar to those listed in 2.1. The groups and dimensions of the relevant representations are summarized in Table 4. One can then write down the Lagrangian Here the capital letter labels the compact real form of the corresponding Lie algebra.
which is invariant under the E 6 -family of groups. The SU (3) × SU (3) invariant theory considered in [15,52,53] is just the special case with n = 9. It was argued in [15]  One can see that there exists a stable unitary fixed point with g 2 * = 4 . It would be interesting to study renormalization of the φ 3 theory invariant under the E 6 family at higher loop level using Birdtrack technique, and figure out whether the 1-loop fixed point continues to exist at larger value of using conformal bootstrap.
in the F 4 family can be obtained by simply substituting the value of n in each case to Eqs.
In summary, for F 4 group case, n = 26,