Searching for gauge theories with the conformal bootstrap

Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for $SO(N)$ vector 4-point functions in general dimension $D$. In the large $N$ limit, upper bounds on the scaling dimensions of the lowest $SO(N)$ singlet and traceless symmetric scalars interpolate between two solutions at $\Delta =D/2-1$ and $\Delta =D-1$ via generalized free field theory. In 3D the critical $O(N)$ vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching $\Delta =1/2$ at large $N$. We show that the bootstrap bounds also admit another infinite family of kinks ${\cal T}_D$, which at large $N$ approach solutions containing free fermion bilinears at $\Delta=D-1$ from below. The kinks ${\cal T}_D$ appear in general dimensions with a $D$-dependent critical $N^*$ below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with $SO(N)$ vectors, $SU(N)$ fundamentals, and $SU(N)\times SU(N)$ bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of ${\cal T}_D$ are subgroups of $SO(N)$, and we speculate that the kinks ${\cal T}_D$ relate to the fixed points of gauge theories coupled to fermions.

The modern conformal bootstrap [1] provides a powerful nonperturbative approach to study higher dimensional conformal field theories (CFT). This method exploits general consistency conditions satisfied by all conformal theories to generate remarkably precise CFT data with rigorous control on the errors. This method is particularly useful for studying strongly-coupled conformal theories for which perturbative approaches are not applicable.
Following some remarkable successes in the 3D critical Ising and O(N ) vector models [2][3][4][5][6][7] (and more recently [8]), the conformal bootstrap has been used to tackle various types of CFTs in higher dimensions D > 2 (see [9] for a review). Nevertheless, most CFTs As a surprisingly powerful nonperturbative approach, the conformal bootstrap is expected to shed light on these profound strong coupling problems. In particular, the conformal bootstrap has been used to provide non-trivial constraints on the IR dynamics of QED 3 [19][20][21] and on those of 4D gauge theories [22][23][24][25][26]. These constraints are helpful for answering certain questions relevant to the dynamics of gauge interactions. Nevertheless, they are not as strong as the results of the 3D critical Ising model, which appear to saturate the bootstrap bounds at a kink-like discontinuity and provide extremal solutions to the bootstrap equations. More generally, a kink-like discontinuity suggests the existence of a non-trivial solution to the crossing equation which may potentially be promoted to a fullfledged theory. This can be further tested by exploiting the consistency conditions with mixed correlators under suitable assumptions on the theory, which may allow one to isolate the solution. Therefore, we can heuristically consider a kink-like discontinuity to be a precursor to identifying a theory that can be solved with the conformal bootstrap.
In particular, some promising evidence towards bootstrapping 3D gauged CFTs without supersymmetry was found recently in [21], which discovered a new family of kink-like discontinuities in the bootstrap bounds, with a possible relation to the infrared (IR) fixed points of QED 3 . In the present work, we will extend this analysis and identify a new infinite family of kinks in the bootstrap bounds in general dimensions, which we conjecture to be related to full-fledged non-supersymmetric CFTs with gauge interactions. We will particularly focus on the interpretation of these kinks as they appear in the 4D bootstrap applied to 4-point functions of fermion bilinears.
In search of an infinite family of CFTs, such as fixed points of QED 3 or QCD 4 , actually it is more illuminating to start with their large N limit, since in this limit the theory is significantly simplified (QCD 4 ) or even solvable (QED 3 ). This is counter to the history of the numerical conformal bootstrap, in which the first numerical solution was obtained for the critical Ising model [2,4,27] and then the critical O(N ) vector models [3,6]. In [21] an infinite family of kinks (T 3D ) beyond the well-known critical O(N ) vector model ones were discovered in 3D bootstrap bounds. Combining the results in [21] with the earlier bootstrap kinks connected to the 3D critical O(N ) vector model [3], it gives a rather interesting pattern of kinks in the 3D SO(N ) vector bootstrap: There are two infinite families of kinks in the 3D SO(N ) vector bootstrap, which respectively approach solutions to the crossing equation with a (scalar) SO(N ) vector at ∆ = 1/2 and ∆ = 2, both containing a series of conserved higher spin currents. The kinks approaching ∆ = 2 have an additional fine structure consisting of two nearby kinks at each value of N above a critical value N * 6.
The large N behavior of this new set of kinks is quite enlightening when considering the interpretation in terms of an underlying Lagrangian description. In higher dimensions (D > 2), a theory with conserved higher spin currents is essentially free [28,29]. In the large N limit the new family of kinks approach free fermion theory from below. 2 In 3D the structure at large N seems to cleanly resolve into two closely separated kinks and the the scaling dimensions of non-singlet fermion bilinears at finite N nicely agree with the 1/N corrections arising in QED 3 and QED 3 -GNY (Gross-Neveu-Yukawa) models. This leads to 2 One may wonder how a scalar SO(N ) vector appears in a free fermion theory. Actually there is a symmetry enhancement in the bootstrap results due to the bootstrap algorithm. We will discuss this phenomenon in section IV.
a conjecture that the kinks at all N relate to the IR fixed points of QED 3 and QED * 3 = QED 3 -GNY, and they merge at the critical flavor number N * ! It is natural to ask if we can find similar patterns in the bootstrap bounds beyond 3D. In 4D, there are no interacting IR fixed points in the O(N ) vector models and the corresponding kinks disappear in the 4D bootstrap results. 3 On the other hand, the IR fixed points can be realized in asymptotically free Yang-Mills theories coupled to massless fermions, known as Caswell-Banks-Zaks fixed points (CBZ) [31,32]. 4 To realize the CBZ fixed points, the number of massless fermions must be inside of an interval, namely the "conformal window". The upper limit of the conformal window is reached when asymptotic freedom is lost, while below the lower bound chiral symmetry breaking and confinement will be triggered in the low energy limit. The CBZ fixed points play important roles in possible scenarios of physics beyond the standard model, and provide classic examples of CFTs with strongly-coupled gauge interactions. They have also been extensively studied using lattice simulations. General bounds on the CFT data of CBZ fixed points can be obtained through the conformal bootstrap, though the bounds obtained so far are fairly weak [22][23][24][25][26].
An extremely interesting question is whether the CBZ fixed points can saturate bootstrap bounds at kink-like discontinuities, an indication that the theories could potentially be isolated and numerically solved using the conformal bootstrap. Surprisingly, we do find a family of kinks (T 4D ) in the 4D bootstrap bounds, as briefly sketched out in Figure 1, though we do not know their putative Lagrangian descriptions yet. In this work we will study the kink-like discontinuities T 4D based on the scenario mentioned before and discuss their possible relations with the CBZ fixed points.
This work is organized as follows. In section II we review results on the new kinks T 3D from the 3D SO(N ) vector bootstrap, their relation to the SU (N ) adjoint bootstrap, and their possible connections to the IR fixed points of QED 3 . In section III we move to the 4D SO(N ) vector bootstrap and study the behavior of the kinks both in the large N limit 3 Actually it has been suggested in [30] that no weakly coupled fixed point can be generated in 4D without gauge interactions. 4 In this paper, we use "CBZ fixed points" to denote the CFTs in whole conformal window. We note that in certain terminology "CBZ fixed points" refers to CFTs near the upper bound of the conformal window only. and near the apparent critical value N * . In section IV we study the relation between the 4D SO(N ) vector and SU (N f ) × SU (N f ) bi-fundamental bootstrap and give a proof of the coincidence between the the bootstrap bounds with different global symmetries. We further discuss the possible relation between the bootstrap results and the CBZ fixed points. In section V we describe a similar bootstrap study in 5D. We conclude in section VI and discuss future work towards bootstrapping the fixed points of gauge theories.   than that of the SO(N ) traceless symmetric scalar. On the other hand, they all become identical if these non-singlet scalars are restricted to have the same scaling dimension. In a physical theory, this would only hold in the large N limit when the composite operators appearing in the OPE are factorized. With finite N the assumption is true at leading order and is violated by 1/N corrections. Due to these coincidences of the bounds, it is subtle to determine the true global symmetry of a putative theory saturating the bounds. This problem will be studied further in section IV.
Here we primarily wish to highlight the new family of prominent kinks appearing in the bootstrap bounds, see Figure 2 for an example with N f = 4. Bounds on the scaling dimensions of the lowest scalars in the SU (4) singlet and (T,T ) 5 representations are shown in the figure. The kinks remain in the bounds at larger N f , and they approach ∆ adj = 2 from below in the limit N f → ∞. Meanwhile, the bound on the singlet scaling dimension becomes weaker and finally disappears when N f → ∞, while the scaling dimension of the SO(N ) traceless symmetric scalar has a scaling dimension ∆ = 4 near the kink.
Using the extremal functional method [4,22,41], we can obtain a picture of the spectrum near the kink. In the large N f limit, there appears both a series of conserved higher-spin currents as well as double-trace operators from generalized free field theory (see section III B for a similar analysis in 4D), suggesting that the large N f spectrum corresponds to a mixture between generalized free field theory and a free theory associated with a non-singlet scalar of scaling dimension 2, i.e., a free fermion theory.
Consequently, the kinks at finite N f , if they correspond to full-fledged theories, are expected to relate to interacting perturbations of free fermion theory! 6 A well-known example of such a deformation of free fermion theory is the Gross-Neveu model [42], which is typically realized as a UV fixed point containing a four-fermion interaction, or equivalently as an IR fixed point containing a Yukawa coupling (the Gross-Neveu-Yukawa model). However, in this non-gauged interacting theory the non-singlet fermion bilinears have positive anomalous dimension (see e.g. [43]). In the large N f limit, their scaling 5 Operators in this representation carry two fundamental and two anti-fundamental indices, both of which are symmetrized. 6 Like the result with N f → ∞, at large but finite N f , it is possible that the extremal solution at the kink still picks out a mixture between the underlying theory and a generalized free field theory. Therefore, without imposing a finite central charge, the kinks may relate to but perhaps cannot be directly identified with a (local) physical theory. dimension approaches ∆ adj = 2 from above. It turns out that the large N f behavior of ∆ adj shown in the numerical results is instead consistent with the large N f perturbative expansions of QED 3 and QED 3 -GNY [44][45][46][47][48][49], indicating that the underlying theories of the new kinks may be related to conformal QED 3 .
A particular advantage of the conformal bootstrap is that it works nicely no matter how strongly coupled the theory is. In QED 3 , it is believed that there is a critical flavor number N * f , below which the theory runs into a chiral symmetry breaking phase in the low-energy limit. Near the critical flavor number the theory is strongly coupled and the value of N * f is still under debate. According to the proposed connection between the new family of kinks and conformal QED 3 , its behavior at small N f could help us to estimate N * f . The results in [21] show that the kinks persist for N f 3, 7 giving evidence that N * f = 2.
Moreover, the results support the merger and annihilation mechanism [50][51][52][53][54], through which the IR fixed point of QED 3 merges with the QED 3 -GNY model and disappears near N * f . The merger and annihilation mechanism is suggested to be triggered when an SU (N f ) singlet four-fermion operator crosses marginality ∆ (ψψ) 2 = 3. 8 Below N * f , the relevant fourfermion interaction is expected to generate an RG flow to the phase with chiral symmetry breaking, while the physics in this region goes beyond the reach of the conformal bootstrap.
For the QED 3 -GNY model with flavor number N f = 2, there is evidence supporting an SO(5) symmetry enhancement in the IR phase [55,56], while it is questionable if it relates to a unitary CFT based on previous bootstrap studies [9,19,57,58]. Bootstrap results in [21] suggest that the putative CFT with enhanced SO(5) symmetry is likely to have an N f just below the conformal window in 3D. 9 7 QED 3 with an odd flavor number of two-component Dirac fermions has a parity anomaly. Here we interpret odd N f as an analytical continuation of the CFT data while ignoring the parity anomaly. It will be interesting in the future to explore the implications of imposing parity symmetry in the mixed correlator bootstrap. 8 Note that the IR fixed point does not necessarily merge with another UV fixed point and disappear when a four-fermion operator crosses marginality. It is possible that two lines of fixed points cross instead of merge. In conformal QED 3 , we observe a relevant four-fermion operator in a non-singlet representation of the flavor symmetry SU (N f ), while the bound still shows a prominent kink. In the future we hope to provide a more detailed study on loss of conformality of fermionic gauge theories using the conformal bootstrap, both in 3D and higher dimensions. We thank S. Rychkov for insightful discussions on the mechanisms by which conformality can be lost. 9 A dimensional continuation of this theory in the context of a D = 2+ dimensional nonlinear sigma model The 3D bootstrap results show promising evidence that the bounds can access CFTs perturbed from free fermion theory through U (1) gauge interactions. It is tempting to ask if we can get similar results in 4D and even higher dimensions. Although gauge dynamics in 4D are quite different from those of 3D, on the conformal bootstrap side the spacetime dimension D is just a parameter in the implementation, and it is straightforward to apply a similar analysis to CFTs with D 4.
where (S, T, A) denote singlet, traceless symmetric and anti-symmetric representations of SO(N ) symmetry. The superscript signs in X ± denote the even/odd spins that can appear was studied in [59,60], suggesting that conformality is lost at D 2.77. In the numerical bootstrap one can also study the dimensional continuation of the SO(5) [61] or SO(4 + ) [62] vector bounds. In these cases the sharp kink seems to disappear near D 2.7 − 2.8.
Before proceeding, let us note that in this correlator the bootstrap implementations for potentially suggesting a critical number N * in between these values. While suggestive, it is hard to determine the precise N * based on the smoothness of the bounds with the current numerical precision. Note that the bounds are still not well converged even at Λ ∼ 31. In 3D, a similar family of kinks disappear when the singlet scaling dimension approaches 3 and becomes marginal [21]. However, in Figure 3 the singlet scaling dimension of the prominent kink with N = 32 is above 7. For smaller N we expect that the optimal bounds with Λ → ∞ should be notably lower than this value. It is not clear if the scaling dimension of the singlet will approach 4 near N * . Note that in 3D, the singlet bounds usually show two nearby kinks at sufficient high numerical precision, while in 4D, we do not yet see another adjacent kink at the current numerical precision. It would be interesting to see if an adjacent kink also appears in the 4D bound at higher numerical precision.
At small N the bounds on the scaling dimensions of the lowest scalars in the traceless symmetric sector (T ) seem to be featureless, and they do not change much for different N ∼ 20. However, there is surprising information hidden in the smooth bound. Let us compare the bounds on the scaling dimensions of the singlet and traceless symmetric scalars.
The series of kinks seem to disappear at a certain N * below N = 18, and for N = 18, there is a mild kink and the scaling dimension of the SO(N ) vector is roughly estimated in the range ∆ φ ∈ (1.7, 1.8). In the bound on the traceless symmetric scalar, near ∆ φ ∼ 1.7 the upper bound approaches ∆ T = 4, i.e., the lowest traceless symmetric scalar is close to being marginal. Thus, our preliminary results suggest that the disappearance of the kink coincides with a non-singlet scalar crossing marginality! This is different from the 3D bootstrap results, in which the kink disappears as the singlet bound crosses marginality. If the kinks do relate to full-fledged CFTs, this could provide strong evidence on the mechanism by which conformality is lost. We will give additional discussion on this point after clarifying several aspects of the 4D results.
B. Bounds on the scaling dimensions at large N The kinks shown in Figure 3 persist with larger N , and approach the position ∆ φ = 3 in the large N limit. The upper bound on the scaling dimension of the singlet scalar becomes weaker at larger N and disappears as N → ∞. In contrast, the upper bound on the scaling dimension of the traceless symmetric scalar gets stronger at larger N . In the large N limit the bound in the region ∆ φ < 3 is saturated by generalized free field theory. This was previously conjectured in [25]. On the other hand, at precisely ∆ φ = 3 there is another solution to   Figure 9). In Figure 4, we can observe an interesting property of these kink/jump locations: the anomalous dimension of the SO(N ) vector at large N seems to scale as For example, if we try to estimate the position at which the jump occurs at each N , we find , where the precise value obtained depends on the chosen points, the details of the fit procedure, the inclusion of subleading corrections, etc. 10 By contrast, we find that assuming a 1/N scaling generally leads to much poorer fits (R 2 < 95%). It will be interesting in future work to compute this coefficient more precisely. In typical known theories with a proper SO(N ) global symmetry, like the critical O(N ) vector models, the anomalous dimensions of SO(N ) vectors scale as 1/N in the large N expansion. Thus, the above scaling behavior seems to be exotic if the SO(N ) symmetry is the proper global symmetry of the underlying theory.
In the large N limit, we see that the non-singlet sectors play an important role in the analysis. From the bootstrap point of view, bounds in the non-singlet sectors are typically stronger (lower) at larger N [3,68]. When this is the case they are guaranteed to be finite in the large N limit. The difference between the singlet and non-singlet sectors can be clearly explained in the large N extremal solutions which we have observed to coincide with generalized free field theory.
Generalized free field theories are non-local CFTs that describe the leading behavior of general large N CFTs. In these theories, the 4-point correlator where x ij = x i −x j and (u, v) are the standard conformal invariant cross ratios u = . The three terms in the right hand side of (5) give contributions from singlet, traceless symmetric and anti-symmetric representations of SO(N ) symmetry. In the N → ∞ limit the singlet sector becomes trivial, and there is no non-unit singlet operator that can appear in the conformal partial wave decomposition of 4-point correlator At ∆ φ = 2.2, there are no operators in the singlet sector, while in the traceless symmetric and anti-symmetric sectors only double-trace operators appear. This is consistent with the fact that in the region ∆ φ < 3, the extremal solution is given by generalized free field theory.
However, at ∆ φ = 2.9999, we observe an interesting mixing in the spectrum. Both doubletrace operators and a series of conserved higher spin currents (but not a spin 0 current) appear in the spectrum. 11 As there is a scalar with ∆ = 3, the higher spin currents are likely to be constructed with fermion bilinears. In a free fermion theory, the spin 0 current j 0 =ψψ can not appear in the j 0 × j 0 OPE due to parity symmetry. This explains the absence of a scalar current in L = 0 singlet sector. In the large N limit, the fermion bilinear 4-point correlator contains two parts: the disconnected part given by generalized free field theory and a connected part containing contributions from higher spin currents. Besides the generalized free field theory, the connected part of the 4-point correlator also provides a solution to the crossing equation [69]. 11 The readers should be reminded that spurious operators could appear in the extremal spectra and not all the zeros in the extremal function are necessarily identified with physical operators. For instance, in the third graph of Figure 5 there is a spurious spin 1 conserved current. These can be distinguished by the fact that their OPE coefficients in the extremal solution are vanishingly small.
The mixing in the spectrum suggests that the extremal solution at the top of the jump (∆ φ = 3, ∆ T = 8) likely corresponds to a linear combination of the generalized free field theory solution and the free fermion solution, such that the ∆ T = 6 operator is absent. In fact, by explicit construction one can establish this and also show that the extremal solution at the top of the jump contains a series of higher-spin conserved currents. We will not dwell on this construction or its spectrum, as the detailed solution to the crossing equation and its relation to the free fermion bilinear 4-point correlators [28,63,[69][70][71][72] has been presented in the parallel work [62]. 12 However, we expect that similar mixing phenomena could be seen numerically at large but finite N . In consequence, the original kinks may correspond to certain underlying theories mixed with a generalized free field theory. The mixing problem can be solved by imposing more constraints in the bootstrap implementation, for example by imposing a finite c central charge or conserved current central charge. Both of the central charges are significantly different between a physical theory at finite N and a generalized free field theory, and therefore they can be used to separate the underlying physical theory from unphysical solutions [74].
At large N the kink locations show an interesting behavior which suggests that the putative underlying theory of the kinks may be a deformation from free fermion theory. If they correspond to physical theories, one hopes that they could be studied perturbatively through a 1/N expansion and then one may try to compare the bootstrap results with perturbative predictions of certain known Lagrangian theories. This has been done in 3D where the x-positions of the kinks are close to the large N results of QED 3 [21]. However, in 4D non-supersymmetric CFTs, like the CBZ fixed points with gauge group SU (N c ), there are two control parameters: the flavor number N f and the degree of the color group N c . In this two dimensional parameter space, only a very special line of (N f , N c ) could possibly saturate the bootstrap bounds. One simple possibility is that the kinks pick out a theory at or near the top of the conformal window at a given N f . 13 Establishing this or some other scenario 12 The analytical solution was first described in the talk [73]. 13 In the Veneziano limit, a theory with N f = 11 2 N c − n has an anomalous dimension γ m ∼ 22 25 n N f (see e.g. [75]), where in physical theories n is half-integer. The coefficient 22 25 n = 0.88n can be compared with our initial estimate a √ 2 ∼ 1.4 − 3.5 after matching N = 2N 2 f . It will be interesting to do this comparison with higher precision data. using the bootstrap results will require quite high precision. 14 More CFT data from both the bootstrap and perturbative sides will likely be needed to extract a firm conclusion about the large N behavior. However in the next section we will see that interesting comparisons can still be made at small N after inputting the full flavor group of the CBZ fixed points.

IV. BOOTSTRAPPING FERMION BILINEARS IN 4D
In this section we will further explore the connection between the bootstrap kinks and the CBZ fixed points. We'll also study coincidences between bootstrap bounds with different global symmetries, making a connection between the SO(N ) vector bounds described above It is straightforward to obtain the crossing equation following the general procedure provided in [77].  (7) then includes a symmetrized double copy of the above SU (N f ) representations. An explicit formula for the crossing equation has been given in [25], in which a bootstrap study aimed at the CBZ fixed points was performed. This work resulted in a lower bound on the scaling dimension ∆ψ ψ of the fermion bilinear O¯i i under the assumption that the presumed IR fixed point can be realized within a given lattice regularization.
The crossing equation can be written in a compact form [25] O∈O×O † where V ± X are 9-component vectors. Details on the vectors are presented in Appendix A. Contributions of the sectors V S,Adj and V A,T are suppressed in (8) as they will not give new constraints. We also suppressed their complex conjugate representations in the above crossing equation. (For brevity, these representations will all be implicitly assumed in the OPE, crossing equations, and branching rules which we will discuss later.) We are interested in bounds on the scaling dimensions of the scalars and will mainly focus on the scalars in the (S, S) and (T, T ) sectors.
Surprisingly, the bound on the scaling dimension of the lowest scalar in the singlet sector (8) is exactly the same (up to the precision of our binary search) as that from the SO(N * ) vector bootstrap, given N * = 2N 2 f ! On the other hand, the bound on scaling dimension of the lowest scalar in the (T, T ) sector is weaker than that from SO(N * ) vector bootstrap. As we will discuss later, it can be made identical to the latter by imposing some additional conditions.
In the next section we will study the general relations between bootstrap bounds with different global symmetries, which will be important to giving a proper interpretation of the bootstrap results.
A. Coincidences between bootstrap bounds with different global symmetries

Coincidence of singlet bounds
Coincidences between bootstrap bounds on the scaling dimensions of the singlet scalars seem to be quite general. One example is the coincidence between the singlet bounds from the bootstrap with SO(2N ) vector and SU (N ) fundamental scalars [22]. Bounds arising from 4-point functions of SO(N 2 − 1) vector and SU (N ) adjoint scalars [21,40] are also known to coincide with each other. By comparing the symmetries of external scalars involved in the bound coincidences, one may notice that the representations of different groups that lead to the same bounds actually have the same dimension (or number of components), among which the SO(N ) vector realizes the largest symmetry group with a representation of the given dimension. One may expect a more general statement: 15 Given a scalar which forms an N -dimensional representation R of a group G, the bootstrap bound on the lowest singlet scalar obtained from the 4-point correlator RRRR will coincide with the singlet bound from the SO(N ) vector bootstrap.
Besides coincidences between the singlet bounds, we can further ask if the whole spectra of the extremal solutions to the crossing equations, after decomposing the SO(N ) 15 We have so far tested three examples for this conjecture. It would be interesting to check this statement with other continuous symmetry groups. In this work we do not discuss discrete symmetry groups, but one may also wonder if there are similar coincidences for discrete symmetry groups as well. An example is given in [78], which shows that bounds on the singlet scaling dimension from the bootstrap with SO(N ) symmetry are the same as bounds assuming a discrete symmetry C N = S N Z N 2 .
representations into the representations of its subgroup G, are also identical with each other.
If this is true, then it means the extremal solutions to the crossing equation of RRRR at the boundary are actually enhanced to SO(N ) global symmetry. We'll use the SO(2N 2 f ) vector and SU (N f ) × SU (N f ) bi-fundamental bootstrap as an example for this study.
In Figure 6 we present the extremal functions of the SO(32) vector bootstrap compared to the SU (4) × SU (4) bi-fundamental bootstrap at ∆ φ /∆ bf = 2, which is slightly to the left of the kink shown in Figure 3. The upper bound locates in the range ∆ S ∈ (7.04916, 7.04917) at Λ = 25, slightly weaker than the upper bound ∼ 6.9 computed in Figure 3  Following the notation used in the SU (4) × SU (4) bi-fundamental crossing equation (8), the mapping between the SO(32) and SU (4) × SU (4) extremal functions is given by 16 In Figure 6 only the extremal functions of the first three lowest spins in each representation are shown, while similar agreement also appears in the extremal functions with higher spins.
The conclusion of the above analysis is that the upper bound on the scaling dimension of the lowest singlet scalar obtained from the SU (N f ) × SU (N f ) bi-fundamental bootstrap is given by an extremal solution with fully enhanced SO(2N 2 f ) global symmetry.
We have also checked that there is a similar mapping between the spectra of extremal 16 As before, we suppress the extra sectors V The SU (N ) crossing equation (12) was previously presented in [77]. We'll show the form of the explicit vectors appearing in this crossing equation later in (19). Like the SU (N f ) × SU (N f ) bi-fundamental bootstrap, the extremal spectra coincide across different sectors. Specifically we see the mapping Apparently the mappings (9-15) are nothing else but the SO(N ) → G branching rules for SO(N ) representations. It is quite amazing that although the crossing equations, such as equations (1), (8), and (12), are endowed with different forms, the numerical bootstrap can figure out precise branching rules just from general consistency conditions. Note that a necessary condition for the above branching rules is that the external scalar in the representation R of group G should have the same number of degrees of freedom (or dimension of R) as the SO(N ) vector, otherwise the extremal solutions from the two different crossing equations cannot contain the same information and a one-to-one mapping between the extremal solutions is not possible. We will come back to this point when discussing a possible approach to avoid such symmetry enhancement.

Coincidence of non-singlet bounds
According to the above analysis, the coincidence of the singlet bounds follows the branching rules (9)(10)(11)(12)(13)(14)(15). In these branching rules the singlet sector is on a similar footing as the non-singlet representations -the only relevant property is that it relates to a one-to-one mapping. One may expect a similar coincidence between bounds of non-singlet operators as long as the representations appearing in the branching rules are treated carefully.
Let us take the SO(2N We address these two questions in the following subsections.

B. A proof of the coincidences between bootstrap bounds
The "symmetry enhancement" phenomena was first observed in [22], which implies the relation (16). 18 This condition is important, otherwise the definition of the bootstrap problem could be implicitly modified. Thus, the SO(2N ) singlet bound cannot be weaker (higher) than the SU (N ) singlet bound. This conclusion was also obtained in [22]. The coincidence of the SO(2N ) and

SU (N ) bounds further suggests that the extremal solution of SO(2N ) is also extremal for
SU (N ). Alternatively, any point (∆ φ , ∆ S ) that is excluded by the SO(2N ) vector bootstrap is also excluded by the SU (N ) fundamental bootstrap.
We will study the structure of the crossing equation. For this purpose, it is helpful to write the crossing equations (1) and (12) into matrix forms: where the columns are given by the vectors which appear in the crossing equations (1) and (12) respectively.
In the SO(2N ) vector bootstrap, the bootstrap problem can be rewritten in the form The bootstrap algorithm is then to test if such linear functionals α i exist for a given (∆ φ , ∆ S ).
Similarly, we can rewrite the SU (N ) fundamental bootstrap problem as In particular, we expect a mapping of the following form between the functions in each sector of the crossing equation: or in a vector form: Here we set the coefficient of the first component α S to be 1 as a normalization condition. 19 Since the functions β V ± X and α S/T /A are obtained from linear functionals α i /β i applied to two independent bases of functions F (u, v)/H(u, v), the above equation essentially amounts to a reconstruction of the linear functionals β i for the SU (N ) fundamental bootstrap from the linear functionals α i for the SO(2N ) vector bootstrap. 19 In the numerical bootstrap, we typically choose an operator, like the unit operator, to set the overall normalization of our functional. In the extremal solution, the positive value used in this normalization is numerically exponentially small compared to the action of the functional on other operators. Our choice β V + S = α S stipulates that if the functional is normalized to 1 on the SO(2N ) unit operator, then it will also be normalized to 1 on the SU (N ) unit operator.
Using the matrix form of the SO(2N ) crossing equation (20), the RHS of (25) is Then combined with (21) we obtain the conditions We will now solve for the linear functionals β i in terms of the α i using the above equations.
We then propose the following ansatz for the solution of (27): where T is a 3 × 6 matrix with 5 unknown parameters x i . The general formula for the transformation matrix T can be obtained by multiplying M −1 SU (N ) on both sides of equation (27). Note that for the single correlator bootstrap, as proved in [77], the number of bootstrap Both α i and β i in (29) are linear functionals which act on the functions F or H. As F and H have different parity symmetry under the transformation x 2 ↔ x 4 , we expect the linear functionals acting on F or H do not mix with each other. Specifically, β 1 , β 3 , β 5 , and α 1 , α 2 are applied to F , and so β 1,3,5 should depend on α 1,2 only while being independent of α 3 . Similarly β 2,4,6 act on H and should only depend on α 3 . Then we have the following constraints on the linear transformation T : Surprisingly, the above 9 equations can be solved for the 5 variables x i : and the transformation matrix T has a unique solution (with the normalization adopted in (25)): Combining this with the relation (16) coming from symmetry arguments, 20 we find The conserved current J A SO(N ) of SO(N ) symmetry decomposes into several representations of the group G in which the J R G are also conserved currents in addition to J adj G . Such currents are expected to be absent in a G-symmetric theory unless there is a real symmetry enhancement G → SO(N ).
In the crossing equations of the correlators (35), we fix the global symmetry to be G and only J adj G appears as an external operator in the crossing equation. Therefore comparing with the SO(N ) symmetric mixed correlators (containing a scalar φ i and current J A SO(N ) ), One can further probe this question by asking how the kinks behave when one relaxes the symmetry assumptions: if the SO(2N 2 f ) global symmetry is intrinsic to the theories at the kinks, then by breaking the SO(2N 2 f ) symmetry explicitly in the bootstrap setup, the kinks will likely become weaker or even disappear; otherwise it is more likely that the kinks are essentially endowed with a global symmetry G ⊂ SO(2N 2 f ). In the following we'll show that the second possibility is favored by our numerical results.
Our strategy is to impose gaps δ X in the bootstrap conditions. We require the lowest operators O in certain sectors V X of the crossing equation (8) have scaling dimensions ∆ O : where ∆ * is the unitary bound of operators in V X . Without imposing any additional constraints in the bootstrap setup (δ X = 0), the singlet upper bound shows exact SO(2N 2 f ) symmetry, and the spectra are replicated in different sectors following the branching rules (9)(10)(11). Such symmetry enhancement can not be realized if δ X break the SO(2N 2 f ) symmetry explicitly.
Let us consider the SU (4) × SU (4) bi-fundamental bootstrap as an example. The results are shown in Figure 7. We test two sets of gaps in the scaling dimensions of the lowest scalars The SO(32) symmetry is broken by the gaps (38) and (39) sector and a mild gap is expected. Nevertheless, we are not aware of evidence on how large the physical gap could be. We adopt the gaps given in (38) and (39) for numerical tests.
Actually by imposing the gaps (38) or (39), the singlet upper bound near the kink becomes slightly stronger, however, the change is not easy to detect in Figure 7. One can also check that outside of the sectors where the gaps are imposed, spectra in the extremal functions do not change significantly. The most significant effects of the gaps are to create the sharp cuts in the left region of the graph. The position of the cut depends on the gaps that are imposed.
The behavior of the bounds with different gaps is reminiscent of the bounds obtained from the 3D U (1) T monopole bootstrap [20]. The results suggest that near the kink, most of the operators appearing in the OPE of SU (N f ) × SU (N f ) bi-fundamental scalars are irrelevant, which is consistent with the general expectation of gauged fermionic theories. It would be interesting to know which gaps will cause the cut to approach the kink, as well as whether there exist any gaps that could cause a cut on the right and help to isolate the kink into a closed region. We leave this problem for future study.
In the region with larger dimensions, the bound, including the kink (though its position changes slightly), is quite insensitive to the gaps that break SO(2N 2 f ) symmetry! This can be viewed as a signal that SO(32) is not the intrinsic symmetry of the theory underlying the kink, even though it appears in the SO(32) bound. One may doubt that the reason the kink is not sensitive to the gaps (39) might be simply because these operators do not It is also possible that the kink remains by breaking the SO(2N 2 f ) symmetry to other subgroups like SU (N 2 f ) or SU (N f ) with an adjoint representation as the external operator. It is hard to uniquely fix the proper global symmetry of the theory at the kink using the current setup. One may try to constrain the bootstrap results to specifically relate to SU (N f ) × SU (N f ) symmetry by using mixed correlators consisting of fermion bilinears and SU (N f ) × SU (N f ) conserved currents (along with gaps forbidding additional symmetry enhancement). By bootstrapping these mixed correlators we may obtain bootstrap results without ambiguities in the symmetry. However, the crossing equations of these mixed correlators involve complicated flavor and spinning indices and a bootstrap study would be quite intricate. We expect that lessons learned in previous work on the conserved current bootstrap [79][80][81], as well as recent improvements in numerical bootstrap algorithms [8,39,82,83], will all be helpful for this study.  21 Based on the current numerical precision, we cannot make a reliable conclusion if the bound has two nearby kinks (as seems to occur in the analogous 3D bounds [21]). It would be interesting to check whether this is the case at higher numerical precision.
As shown in Figure 3, the bootstrap results suggest that the critical flavor number N * is slightly below N * ∼ 18. Due to the "fake" symmetry enhancement effect from the bootstrap algorithm, the value N * should be considered as describing the dimension of the representation of the external scalar for the symmetry G ⊂ SO(N * ).
The critical flavor number N * ∼ 18 corresponds to a small value of N f when interpreted in terms of an SU (N f ) × SU (N f ) chiral symmetry. For example, the singlet bound from 21 The kink becomes weaker near the critical flavor number N * and one can only identify a transition region in the bootstrap bound. A better estimation on ∆ψ ψ could likely be obtained with higher numerical precision or the addition of gaps that create a sharp feature. such as N f = 6, 8 [84], is significantly larger than 18. The conclusion is that the kinks near N * are unlikely to be realized by SU (N c ) gauge theories coupled to fundamental fermions.
Instead, theories possessing a conformal window with small N f can potentially be obtained using fermions carrying multiple color indices [85]. In [75,86] with the ± respectively corresponding to symmetric and anti-symmetric representations.
On the other hand, the lower bounds on the conformal windows are difficult to compute.
Perturbatively, they can be estimated from a loop expansion of the beta function, where the leading terms are [86] N f,min ∼ 17 The above perturbative results on N f,min need to be treated carefully. However, it is clear Nc , the anomalous mass dimension is expected to be γ m = 1. In this limit the loss of conformality at the lower end of the conformal window x = x * is usually ascribed to an irrelevant four-fermion operator crossing marginality ∆ (ψψ) 2 = 4. In the Veneziano limit we have the factorization ∆ (ψψ) 2 = 2∆ψ ψ , which gives γ m = 1 at x * . A similar bound on the anomalous mass dimension can also be obtained from the Schwinger-Dyson equation [87]. The anomalous dimension γ m necessarily receives corrections at finite N f , and our results near N * may provide evidence of a large fermion anomalous mass dimension γ m > 1.
Remarkably, near the kink with ∆ψ ψ ∼ (1.7, 1. This is nicely consistent with the scenario of merger and annihilation of fixed points [50][51][52][53][54], which was proposed as a scenario explaining how conformality is lost in the CBZ fixed points near N * . In [53] the authors suggested that near the lower bound of the conformal window, the CBZ fixed points approach another UV fixed point, QCD * , generated by a certain irrelevant operator (e.g. a scalar built out of four fermions). At the critical flavor number N * , the irrelevant operator becomes marginal, which further drives the two fixed points to merge with each other and triggers chiral symmetry breaking. The two fixed points disappear for N < N * . A more comprehensive study on how conformality could be lost is provided in [54], which analyzed additional scenarios for the behavior of fixed points near N * . In general, the loss of conformality could possibly be triggered by a non-singlet scalar reaching marginality and the two lines of IR and UV fixed points may cross instead of merge with each other, depending on the symmetry of the marginal operator. The evolution of fixed points can be quantitatively described in the context of bifurcation theory [88].
A solid prediction of this mechanism is that an irrelevant operator should become marginal at N * . The results in Figure 8 provide direct evidence that there is a scalar close to 22 Bounds on the scaling dimensions of scalars in other sectors are higher. However, in certain sectors they are close to the bound of V (+) T,T and they may also play an important role near N * . We leave a more comprehensive analysis of these sectors for a future study. marginality near the critical N * . On the other hand, it suggests the operator that becomes marginal at N * is the leading scalar in the V (+) T,T sector. This is however different from the original expectation in [53]: the authors of [53] expected the loss of conformality is triggered by a marginal singlet scalar, while our bootstrap results suggest that, in the particular extremal theory described by the bootstrap kink with flavor number N * , the scalar in the symmetric-symmetric representation of chiral symmetry approaches marginality faster than the singlet.
According to the merger and annihilation mechanism, regardless of the details of the deformation, there should be a nearby UV fixed point generated by the approximately marginal operator. On the other hand, we do not see another obvious adjacent kink in the bound. This is not surprising since the approximately marginal operator is in the symmetric-symmetric representation, which breaks the SU (N f ) × SU (N f ) chiral symmetry to its diagonal subgroup SU (N f ) V . Therefore the presumed UV fixed point has a smaller symmetry group and it corresponds to quantitatively different bootstrap bound. This is also different from the 3D bootstrap results, in which the singlet bound shows two nearby kinks above the critical flavor number N * [21]. We find it quite inspiring that such a qualitative difference between the 3D and 4D bootstrap results near the critical flavor number N * may be able to shed light on the mechanism by which conformality is lost.
The kinks appearing in the SU (N f ) × SU (N f ) singlet bounds correspond to a series of non-trivial solutions to the crossing equations, which may relate to full-fledged CFTs.
The bootstrap results from the crossing equation (8)   where {α, β, γ} ({i, j, k}) are the color (flavor) indices. The operator O ijk is not gauge invariant in SU (N c ) gauge theories with N c = 3 and could help to distinguish this particular theory from many other candidates. This operator is fermionic and lives in a nontrivial representation of the flavor symmetry. The 4D fermion bootstrap has been studied in [26] and the techniques developed in the work can be directly employed for this study. 23 The baryon operator has UV scaling dimension ∆ U V B = 9/2, while it receives significant corrections due to strong coupling, and its IR scaling dimension ∆ IR B could possibly be much lower, see [108] for a scheme-independent perturbative study on the anomalous dimensions of baryon operators. The magnitude of ∆ IR B will likely directly affect whether the numerical bootstrap can generate strong bounds towards a physical theory. The main technical obstacle is due to the global symmetry and spinor indices, which lead to quite cumbersome crossing equations.
Recent developments in numerical bootstrap techniques [8,39,82,83] will be helpful for pursuing this study. Notable kinks in the singlet bound do appear for large N , e.g., N = 500, though we do not know if the kinks remain in the bounds with higher numerical precision. The kinks in 5D seem to have a notably larger critical number N * than their lower dimension analogs.
It is interesting to compare our results with previous studies on the 5D "cubic" model [110,111] with bosonic interactions L ∼ σφ i φ i , which, in the large N limit, approaches another theory with conserved higher spin currents: the (N = ∞) critical boson theory. 23 The 3D fermion bootstrap has been explored in [43,68] which show kinks corresponding to the Gross-Neveu-Yukawa models. The cubic model provides a UV-complete version of the 5D dimensional continuation of the critical O(N ) vector models in 2 < D < 4. Perturbatively the theory is stable and unitary for sufficiently large N above a critical value N > N * . Remarkably, the bootstrap approach can provide sharp results on the critical 5D cubic models which are consistent with the large N or (6 − )-expansion perturbative predictions [112][113][114][115][116]. 24 On the other 24 Note that the critical boson has scaling dimension ∆ = 2 + O(1/N ), which is actually lower than that of the 5D free boson bilinear operators ∆ = 3. Therefore we cannot the kinks corresponding to these theories in Figure 10 unless we impose gaps in certain sectors which carve out regions below the free boson theory.
hand, it turns out to be quite subtle to determine the critical number N * . Especially in the mixed correlator bootstrap [115], for relatively small N * 100 the physically allowed CFT data can be isolated into a small island with suitable assumptions, while after increasing numerical precision the island disappears. An explanation of this phenomena is that there are small non-unitary effects in the CFT data which can only be seen using the conformal bootstrap with sufficiently high precision. Remarkably, such non-unitary effects have been found to be generated by instantons and can be computed analytically [117].
One of the motivations of the cubic model [110,111] is to construct 5D theories with slightly broken higher spin symmetry. The cubic model, in the (6− )-expansion description, corresponds to a free boson theory perturbed by cubic interactions. Theories with exact higher spin symmetry can also be realized with free fermions, and one may alternatively construct theories with slightly broken higher spin symmetry by perturbing free fermion theory. In 3D, these two different types of theories have been partially revealed in bootstrap results: we have two families of prominent kinks which respectively approach the critical boson or free fermions in the large N limit [21].
Inspired by the 5D cubic model and bootstrap results, we may ask this question: can we construct 5D theories as deformations of free fermion theory which admit IR fixed points?
The theory could be unitary and stable perturbatively for sufficiently large flavor number.
One might expect that the deformations correspond to adding gauge interactions, as is suggested by the lower-dimension bootstrap results. This is also evident for a simple reason that in a non-gauged fermionic theory, local interactions with fermions are strongly UV irrelevant in higher dimensions D 5. A natural approach to realize 5D UV fixed points by perturbing free fermion theory is to take the D = 4 + dimensional continuation of 4D gauge theories, e.g., QED coupled to fermions.
In the (4 + )-expansion, at the one-loop level QED in D > 4 admits a UV fixed point. We also studied the coincidences between bootstrap bounds on scaling dimensions assuming different global symmetries. Generically, for the single correlator bootstrap with an external scalar in a representation R of a group G with dimensionality N , the bound on the scaling dimension of the lowest singlet scalar appearing in the R ×R OPE is expected to coincide with the singlet bound from the SO(N ) vector bootstrap. Moreover, we found a coincidence to also appear in the whole spectra of extremal solutions following the branching rules of SO(N ) → G. We provided an explicit proof for the bound coincidence between the SO(2N ) vector and SU (N ) fundamental bootstrap, by finding a transformation between the 25 Fixed points of 6D QCD have been studied in [119] using a perturbative approach, which suggests a conformal window analogous to 4D QCD. We also observed kinks in the bounds from the 6D SO(N ) vector bootstrap, but it requires a careful study before any solid conclusion can be reached.
linear functionals for the SO(2N ) vector and SU (N ) fundamental crossing equations, which satisfy the required positivity conditions. The proof can be straightforwardly generalized to different global symmetries and it reveals an interesting connection between the structure of group representations and the bootstrap constraints.
We particularly focused on the 4D bootstrap results and their possible connections with the well known 4D CBZ fixed points of Yang-Mills theories coupled to massless fermions.
These theories could have interesting applications to physics beyond the Standard Model or nonperturbative approaches to ordinary QCD (such as Hamiltonian truncation). Since the new family of kinks seem to correspond to deformations of free fermion theory, also inspired by their 3D analogs, we speculate these kinks correspond to fermionic theories with gauge interactions, of which the CBZ fixed points are known candidates. We discussed possible field realizations of the kinks near the apparent critical flavor number, which corresponds to a small flavor number and a large anomalous mass dimension.
We showed that the bootstrap results may help us to understand how conformality is lost near the lower bound of the conformal window, where we noticed a non-singlet operator The new kinks that appear to correspond to deformations of free fermion theories are welcome surprises for the conformal bootstrap. This is especially encouraging for the possibility to bootstrap fermionic gauge theories with strong interactions, which play critical roles in many applications but require nonperturbative treatments. One obstacle towards a precise estimation of the CFT data of the underlying theories is that the singlet bound converges rather slowly and is not yet close to its optimal solution. In the limit N → ∞, the singlet upper bound also tends to disappear. At finite but large N , the OPE coefficients in the singlet sector are 1/N suppressed so it becomes challenging for the numerical bootstrap to capture these small factors. 26 The problem remains but is less severe at the more interesting (for applications) case of small N or flavor number, where the theories are expected to be more strongly coupled. Bounds on the scaling dimensions of certain non-singlet scalars perform better in this aspect. In comparison with the 4D CBZ fixed points, conformal QED 3 provides a relatively simple laboratory for understanding these issues. We expect that a precision bootstrap study of mixed correlators in QED 3 can help to illustrate an effective approach for evaluating the CFT data of fermionic gauged CFTs. We hope to report developments on this problem in the near future [74].
where the V (±/+/−) X represent the following 9-component vectors: (A3) Here the conformal blocks F/H are respectively defined through (2) and (3), and the · · · represent symmetrized/conjugate representations whose vectors take the same form as above.