On $\phi^3$ Theory Above Six Dimensions

We study $\phi^3$ theory above six dimensions. The beta function $\beta(g)=-\epsilon g-\frac{3}{4}g^3$ in $d=6-2\epsilon$ dimensions has a UV fixed point when $\epsilon<0$. Like for $O(N)$ vector models above four dimensions, such a fixed point observed perturbatively in fact corresponds to a pair of complex CFTs separated by a branch cut. Using both numerical bootstrap and Gliozzi's fusion rule truncation method, we argue that such a CFT exist.


Introduction
Consider scalar field theory with the following Lagrangian (1.1) The one loop beta function of theory in 6 − 2 dimensions has the following form Below six dimensions, the theory has a fixed point at purely imaginary coupling g * = i 2 . This fixed point has been used by Michael Fisher to study Lee-Yang edge singularity [1]. Formally, if one take to be negative, and hence to study the theory above six dimension, one can also find a fixed point. Notice g 2 is positive at the fixed point. One can also show that ∆ φ is greater that the unitarity bound. The fixed point is therefore perturbative unitary. The existence of such a non-renormalizable fixed point, at least perturbatively, was noticed in the 70's [2], and the leading correction to the scaling dimension ∆ φ was calculated by solving the Migdal-Polyakov bootstrap equation [3,4].
The idea behind modern bootstrap method also dates back to the 70's [5,6]. It was later applied to two dimensional conformal field theories in the famous work of Belavin, Polyakov and Zamolodchikov [7], where two dimensional minimal models were solved. It was until 2008 that some important progress was made in applying conformal bootstrap method to CFT in d > 2, with the help of a computer [8]. After that, numerical bootstrap has become an important method to study conformal theory in various dimensions, see [9] for a recent review. In some key examples, the critical exponents of certain models calculated using numerical bootstrap method were more precise that the Monte-Carlo simulations [10][11][12][13][14][15].
Since the φ 3 interaction in (1.1) is irrelevant in d > 6, the interacting fixed point in Ultra-Violet. The coupling constants of the irrelevant terms grow as one approach the interacting fixed point, higher weight terms such as φ 4 , φ 5 . . . shall also play a rule in renormalization.
We do not really know whether we should trust the naive expansion based on a single φ 3 interaction at large and negative . It will therefore be interesting to study such a fixed point using other methods such as numerical bootstrap. Another important remark is that scalar theory with cubic interaction is intrinsically non-unitary, due to the metastability of the φ 3 potential. We expect the scaling dimension ∆ φ and some OPE coefficients to have small imaginary part as d → 6 + . Perturbatively unitary fixed was shown to exist for O(N ) vector model in d = 4 + [16], and also in large N for 4 < d < 6 [17][18][19]. In both cases, using instanton method, it was shown that both the scaling dimension ∆ φ and some OPE coefficients receive small imaginary corrections [16,20]. For large enough N , the non-perturbative imaginary part could be neglected numerically, so that these fixed point appear in numerical bootstrap results [21][22][23]. Borrowing the intuition from these previous works, we expect that the ∆ φ and OPE of φ 3 theory above six dimensions to also develop a small imaginary part as d → 6 + .
In this work, we use two different methods to study φ 3 theory. Since conformal block can also be define in non-integer dimensions, one can use numerical bootstrap method to study CFT in factional dimensions [24]. We compare perturbative -expansion result in [19] with the result from numerical bootstrap. In close to six dimensions, the boundary of allowed region in (∆ φ , ∆ ) plane shows a sharp cliff precisely at the value of ∆ φ predicted by -expansion, see Fig. 1. Here ∆ mean the second primary operator appear in φ × φ OPE. As we increase the space-time dimension d, the cliff suddenly disappears. This leads us to the conjecture that one of the λ 2 φφO (square of operator product expansion coefficient), after neglecting the instantons effect, changes sign at this dimension. After the numerical bootstrap study, we then use Gliozzi's truncation bootstrap method [25,26] to study φ 3 theory by focusing on the fusion rule The calculation is precisely the same as in [25], except that we now set d > 6. We show that λ 2 φφT < 0 above certain fractional dimensions. This paper is organised as follows. We review the standard method in unitarity numerical bootstrap and give results in Λ = 23, 35. We then compare result with the anomalous dimension of φ calculated using perturbative calculation at four loop. After that, we use Gliozzi's fusion rule truncation bootstrap to study the same theory.

Numerical bootstrap
Due to conformal symmetry, the four point function of four identical scalars are fixed to be where the function g(u, v) depends on the cross ration u = and v = . The functions g(u, v) admits the following conformal block expansion For a CFT, such series should converge, and the four point function shall be crossing symmetric: The lines show indicates how way operator product expansion is performed. (2.3) gives us the following crossing equation [8], In a bootstrap setup, we numerically search for a linear functional α such that These conditions look very much like the crossing equation for Ising model. The criti- being the only scalar primary operator whose dimension is lower that ∆ .
The problem of searching for such a linear functional can be translated into a semidefinite programming problem. In this work, we use the "SDPB" solver which was designed to study conformal bootstrap problems [36]. The maximal derivative order is chosen to be The crossing equation has an ubiquitous solution in any space-time dimension d, which is usually a conformal field theory called generalised free theory. The theory is equivalent to a free scalar propagating in AdS d+1 , see for example [27]. The operators appear in φ × φ OPE have scaling dimensions given by ∆ = 2∆ φ + l, with l ∈ even. (2.6) Strictly speaking speaking, this is not a full-fledged field theory since there is no conserved spin-2 current, hence no energy-momentum tensor. From the numerical bootstrap point of view, such a CFT satisfied both crossing symmetry and unitarity constrains, and therefore should fall into the allowed region. One may worry about the fact that the operator φ does not appear in φ × φ OPE for generalized free theory. Notice the conditions (2.5) allow φ to appear in the OPE. Its presence is however not guaranteed. Generalized free theory is therefore compatible with such conditions. It corresponds to the baseline ∆ = ∆ φ 2 = 2∆ φ in the numerical bootstrap curve. Above this baseline, we observe some interesting excess. When d is close to 6, one could observe a sharp cliff, which change into a much smoother mountain as d increase. The mountain gets even more flattened as d approaches 7 dimensions. Close to d = 6, we expect that perturbation calculation should give reasonable prediction for scaling dimension of operators. We quote here the result in [19] at four loop: The dashed line in Fig. 1 shows the prediction of -expansion, where we have used Padé [1,3] method to resume the series. Close to 6d, the location of the sharp cliff excess is precisely at the value predicted by -expansion. Such non-smoothness usually indicates that there exist a conformal field theory at the non-smooth point. A famous example is the kink observed in three dimensional numerical bootstrap bound curve, which corresponds three dimension Ising model [10]. The existence of such a cliff gives us confidence that indeed such a fixed point exist above six dimension. Close to six dimensions, the appearance of the cliff could also be understand from equation of motion of φ 3 theory, φ = 1 2 φ 2 . Clearly the operator φ 2 is now a conformal descendant of φ and the next to leading conformal primary scalar operator is therefore φ 3 , whose scaling dimension should be much higher that 2∆ φ .
The critical d c at which th cliff disappears depend on the number of derivatives Λ used in numerical bootstrap. At Λ = 23, 6.4 < d c < 6.5. At Λ = 35, 6.2 < d c < 6.3. The disappearance of the bootstrap cliff is affected by two factors. First, as mentioned in the introduction, φ 3 theory above six dimension is non-unitary due to instantons effect. As d increase, the imaginary part of ∆ φ and certain OPE coefficients might not be negligible anymore. Second, it is also possible that even if it is safe to neglect instantons effect, the OPE 2 of some low lying operators change sign at a certain space-time dimension. We will show in next section that λ 2 φφT does change sign at d ≈ 6.43, with T being the stress-tensor.

Gliozzi's Fusion Rule Truncation Bootstrap
In this section, we use the method developed in [25] to study φ 3 theory above d = 6, which allow us to extract information about low dimension operators in φ×φ OPE. The calculation is precisely the same as in [25], except we now set d > 6. One advantage of Gliozzi's method works also for non-unitary CFTs as long as the theory can be approximated by a truncated fusion rule. We follow the procedure in [25] and define CFTs are called "truncable" [26].
As in [25], we truncate the fusion rule of φ 3 theory to be 2) as in [25]. In this case we have 4! 3! = 4 possible minors. The determinants of the 3 × 3 minors at d = 7 are shown in Fig. 2. One can see that they intersect at an unique point, which corresponds to the φ 3 theory. Setting ∆ φ and ∆ 4 to be at the point where the minors intersect, we can now solve (3.2) with a non-zero vector, plugging in (3.3) fixes for us the normalization. In this way, we obtain the OPE coefficients.
It is important to notice that the solutions we have found have purely real scaling dimensions and OPE 2 , this means that we are neglecting the imaginary part due to instantons.
It is not clear why such a truncated fusion rule should also work in our case, except for the fact that it can be used to calculate the critical exponents for Lee-Yang edge singularities below 6 dimension [25]. We will compare the result obtained though this truncation with numerical bootstrap and -expansion. The solution between 6 and 7 dimensions is summarised in Fig. 3. We have also marked the scaling dimension ∆ φ in Fig. 1 using solid lines. For all these solutions, we have checked that the three primary operators (and φφT change sign at d c ≈ 6.43, which is possibly related the disappearance of the cliff in the bootstrap curve. Also, comparing to -expansion result, the fusion rule truncation method is less accurate at d → 6. This is due to that the fusion rule become more like free theory fusion rule, and the truncation to three primaries is not valid. Also notice when doing numerical bootstrap with smaller number of derivatives (Λ = 23), at d = 7, we do observe a bump of the bootstrap curve. This seems to suggest that non-unitary solution to crossing equation can affect bootstrap curve in low derivatives.
The solutions of truncated bootstrap equations could be continued to even higher spacetime dimensions. We have checked that they exist at d = 8, 9 and 10, which are summarised in Table 1. We however emphasis here that there is no reason to assume the instantons effects to be small. We are not sure whether they give a good approximation of the fixed points of φ 3 theory in the corresponding space-time dimensions.   CFTs with complex spectrum was named "complex CFTs" and was shown to be related to the "walking" of RG flow recently [28]. It would be interesting to look into the details of how this pair are created. Let's start with the λφ 4 theory, hence a free scalar theory with L int = λ 4! φ 4 interaction. The leading terms of the beta function in D = 4 − is

Discussion
The coupling at the Wilson-Fisher fixed point is g * = 3 . As we vary space time dimension from d < 4 to d > 4. The Wilson-Fisher fixed point (as indicated by the blue dots in Figure   4) hit the free theory fixed point (as indicated by the black dots in Figure 4) at d = 4. Notice that due to the instability of the potential at negative coupling, the free theory in fact lives at the tip of a branch cut. Such a branch cut was argued to exist in QED by Dyson [29], and was made rigorous for one dimension φ 4 theory (quantum mechanics) in [30]. It may be safe to expect such a cut in higher dimensions. In [16], the critical exponent of λφ 4 theory at d = 4 + was shown to have small imaginary part, and the cut plays an essential role in the calculation. After the Wilson-Fisher fixed point collide with the free theory fixed point at d = 4, it bifurcates into a pair of complex CFTs with conjugate coupling λ.
The creation of the complex CFTs in gφ 3 theory could be understood in a similar manner. In one dimension field theory (hence quantum mechanics), it was shown in [31] that such a theory has a branch cut along the real and positive g 2 axis. We can borrow the one dimensional intuition to think about the higher dimensional gφ 3 theory. As we increase space time dimension, the Lee-Yang edge singularity which lives on the negative g 2 axis hit the free theory at d = 6. Due to the branch cut, it then bifurcate into a pair of complex CFTs. As we further increase d, they might go far away from the real axis, it is however hard to imagine that they suddenly disappear. So we expect such a pair of complex CFTs to exist even in higher dimensions.
Such complex CFTs should also exist in gauge theories. The conformal window 3N c > N f > 3 2 N c of supersymmetric QCD can also be understood as an interacting fixed point hitting the free theory [28,34]. The upper end of the conformal window is determined by asymptotic freedom. According to Seiberg duality, we know that the theory has a dual description in terms of magnetic gauge theory. Even though the electric theory becomes strongly coupled as N f → 3 2 N c , the dual magnetic gauge theory is weakly coupled. We expect a branch cut to exist when the magnetic gauge theory has negative g 2 m . As we vary N f , the Banks-Zaks fixed point hit the magnetic free gauge theory and then bifurcate into a pair of complex CFTs. There exist another ubiquitous phenomenon by which CFTs could go into the complex plane [17,28,[32][33][34][35]. In that case, two interacting CFTs collide and then becomes a pair of conjugate complex CFTs. This scenario is different form the scenario when an interacting CFT hit a branch point CFT and bifurcate. One difference, for example, is that in the later case, after the collision, the branch point CFT is still present. We have three fixed points in the complex plane. As mentioned, the lower end of the conformal window of super QCD terminates because the Banks-Zaks fixed point hit the free theory. The lower end of the conformal window of normal QCD, however, can be described by interacting CFTs' merging. This difference was made clear in [28,34].
Notice in all the examples mentioned above, the fixed point at the tip of the branch cut is a free theory. It would be interesting to understand whether a branch point CFT could be interacting.