Higgs Critical Exponents and Conformal Bootstrap in Four Dimensions

We investigate relevant properties of composite operators emerging in nonsupersymmetric, four-dimensional gauge-Yukawa theories with interacting conformal fixed points within a precise framework. The theories investigated in this work are structurally similar to the standard model of particle interactions, but differ by developing perturbative interacting fixed points. We investigate the physical properties of the singlet and the adjoint composite operators quadratic in the Higgs field, and discover that the singlet anomalous dimension is substantially larger than the adjoint one. The numerical bootstrap results are then compared to precise four dimensional conformal field theoretical results. To accomplish this, it was necessary to calculate explicitly the crossing symmetry relations for the global symmetry group SU($N$)$\times$SU($N$).


I. INTRODUCTION
The discovery of a Higgs particle at the Large Hadron Collider is a major leap forward towards the construction of a more complete theory of nature. If the discovered particle is the standard model Higgs, it is imperative to understand the gauge dynamics of nonsupersymmetric four-dimensional gauge-Yukawa theories.
Among all possible quantum field theories, the ones developing quantum conformal fixed points have a central role [1,2]. Quantum chromodynamics is a time-honored example [3,4], where the celebrated property of asymptotic freedom comes from a noninteracting ultraviolet fixed point [3,4]. One can also imagine the existence of ultraviolet fixed points that are interacting, and this scenario is referred to as asymptotic safety [5].
In this work, we therefore wish to press forward and investigate explicit conformal The existence of Banks-Zaks (BZ) [100] interacting fixed points in such a model has been established in [75,[101][102][103][104]. Furthermore, in [54] the reader will find an in depth study of the asymptotic safety scenario and crucial properties which are guaranteed to exist for some of these theories. In this case, the underlying gauge theory is fundamental even in the presence of elementary scalars [54].
Having nonsupersymmetric, interacting, four-dimensional conformal field theories (CFTs) at our disposal, we determine the physical properties of the singlet Tr[HH † ] and the adjoint Tr[T a HT a H † ] composite operators. Via an explicit computation, we discover that the singlet anomalous dimension is substantially larger than the adjoint one. We then construct the four-point correlations functions in which these operators play an important role, and explicitly check the crossing relations. Furthermore in the Veneziano limit, and at the maximum known order in perturbation theory, we argue that the singlet sector of the theory is nontrivial. We finally compare our precise results with the numerical bootstrap constraints [105][106][107][108][109].
The work is organized as follows. In Section II we briefly review the conformal bootstrap idea and the associated bounds [105][106][107]. We then move on to derive the conformal bootstrap sum rules in a CFT with non-Abelian global symmetry SU(N f ) × SU(N f ) in Section III. The four dimensional gauge-Yukawa theories used here are introduced in Section IV. In the same section we also demonstrate that the singlet sector decouples from the other operators. We then compare with the numerical bootstrap [105][106][107] constraints. In Section V, we offer our conclusions.

II. CONFORMAL BOOTSTRAP REVIEW
To set the stage, we provide a short, self-contained introduction to the idea of the conformal bootstrap and highlight its salient properties. We consider the set of correlation functions for all local operators of some quantum field theory. For this to constitute a conformal field theory, the set of correlation functions must obey a corresponding set of constrains, and presently, we set out to find it. A CFT consists of its conformal primary operators O i 1 , and their associated conformal dimensions ∆ i and spins l i . Because of conformality, the normalization is completely arbitrary, and we select a basis for the scalar operators such that the 2-point functions have the form In any CFT, it is possible to express the product of two local operators as a sum over all local operators in the theory which have a finite radius of convergence. This is called the operator product expansion (OPE), and we have where, as mentioned, the sum is over all (primary and non-primary) local operators M k and c k i j (x − y) are functions of the dimensions and spins (which we denote collectively by the index k = (∆ k , l k )) of the operators involved, and of the dynamics of the theory. Using equation (2) inside correlation functions, we can replace a product (like the LHS) by a sum (like the RHS), as long as there are no other operators at smaller distances from y than |x − y|.
The OPE above is quite general, and by also imposing conformal invariance it can be shown [110] that the kinematics of the primary operators uniquely determines the coefficients c k i j (x − y) belonging to their descendant operators 2 . Thus, all dynamical information in the OPE is encoded in the coefficients for the primary operators where C k i j are now translation invariant constants. The complete OPE (with both primary and descendant contributions) is then where L k (x − y, ∂ y ) are differential operators that only depend on the kinematics, that is the dimensions and spins of the primary operators O k . They do not depend on the dynamics of the CFT. By using the OPE on the two operators that are closest together, it is now a straightforward matter to reduce an n-point function to an infinite sum over (n − 1)-point functions, which in turn can be reduced to an infinite 3 sum of (n − 2)-point functions, and so on down to the 2-point functions, which have the simple structure seen in (1). Thus, if we know the conformal dimensions ∆ i , the spins l i and the 3-point coefficients C k i j of the primary operators, we know the entire CFT. 2 The descendant operators are obtained by acting on the primaries with the translation operator i.e. taking derivatives of the primaries. 3 Due to the peculiar nature of infinities and the fact that the number of operators in the CFT is countably infinite, successive applications of the OPE does not actually increase the cardinality of the infinity.
If we have multiple operators, there are several ways of using the OPE to reduce an n-point function. However, this obviously cannot change the result, and thus we must insist that regardless of the order in which multiple OPE contractions are used, the end results must be equal. This leads to non-trivial constraints on the possible values of ∆ i and C k i j that can make up a consistent CFT. As an instructive example, we consider the 4-point function We can evaluate this using the OPE between the operators at x 1 and x 2 and simultaneously at x 3 and x 4 , or alternatively by performing the OPE between the operators at x 1 and x 4 and simultaneously at x 2 and x 3 . This corresponds to the s-channel (12) → (34) and t-channel (14) → (23) respectively. 4 The contraction in the s-channel yields In this expression, only the OPE coefficients C k 12 and C k 34 depend on the dynamics of the CFT. It is therefore convenient to define the conformal blocks which contain every contribution from the local operator O k and its many descendants.
As mentioned above, these conformal blocks are dependent only on the kinematics of the conformal group, and explicit expressions for them are given in [111,112].
The above evaluation was done in the s-channel (12) → (34), but we could equally well have performed it in the t-channel (14) → (23). This would have given us a similar, but distinct, expression with 2 and 4 interchanged. Imposing that these two procedures give equal expressions is what yields the non-trivial conformal bootstrap equation which, together with (6), tells us how the dimensions, spins and OPE coefficients must relate to each other in order for the theory in question to be conformal.
In addition, conformal symmetry allows us to further constrain the coordinate dependence of the 4-point function and the most general conformally invariant expression is where g(u, v) is an arbitrary function 5 of the conformally-invariant cross-ratios: In [105], the bootstrap equation for the 4-point function of four identical scalar operators φφφφ was considered. Starting from the OPE: and using (8) with all ∆ i = d equal, we obtain: where we explicitly separated the contribution of the identity operator. The explicit expression for the conformal blocks g k (u, v) reads: where 2 F 1 is Gauss's hypergeometric function.
The 4-point function on the left-hand side of Eq. (11) is obviously symmetric under the interchange of any two x i , and its conformal block decomposition (12) must therefore also respect this symmetry. Invariance with respect to x 1 ↔ x 2 or x 3 ↔ x 4 implies that only operators of even spin are exchanged. The non-trivial constraint comes from the symmetry with respect to x 1 ↔ x 3 and gives the following condition (see Fig. 1 for an 5 Note that we absorbed the OPE coefficients C k 12 and C k 34 into the definition of g(u, v). which is not automatically satisfied for g(u, v) as given in equation (12). 6 Following [105], it is useful to rewrite (14) by separating the unit operator contribution, which gives where the index k covers the conformal dimension ∆ and the spin l, as in (13). The LHS of this equation is the imbalance created by the presence of the unit operator in the OPE.
This imbalance has to be compensated by contributions of the other fields on the RHS.
In practice, it is convenient to normalize (15) by dividing both sides by u d − v d . The resulting sum rule takes the form: For a given spectrum of operator dimensions and spins {∆, l} the sum rule (16) can be viewed as an equation for the coefficients p ∆,l ≥ 0. If there are no solutions to this equation, the corresponding CFT would be ruled out.
To achieve a concrete realization of this idea, it is necessary to have a practical recipe to show that the solution does not exist. For a simple example of such recipe, imagine that a certain derivative, e.g. ∂ x , when applied to every F d,∆,l and evaluated at a certain point, is strictly positive. Since the same derivative applied to the LHS of (16) gives identically zero, a solution where all coefficients p ∆,l are non-negative would clearly be impossible.
Using this logic, a first model-independent bound on the dimension of the operator φ 2 was numerically found in [105,106] by using linear programming methods: 6 The appearance of the (u/v) d factor in this relation is due to a nontrivial transformation of the prefactor where d is the conformal dimension of the scalar φ, d ≡ ∆ φ , and ∆ is the dimension of the operator φ 2 , ∆ ≡ ∆ φ 2 . In [107] a semidefinite programming algorithm was used and the bound was improved further to the current strongest limit: There does not seem to be any known 4D unitary CFT saturating this bound.

III. CONFORMAL BOOTSTRAP SUM RULES IN CFT WITH
We will construct explicit examples of CFTs stemming from four-dimensional, nonsupersymmetric gauge-Yukawa theories possessing the global, non-Abelian symmetry For this reason, we will proceed to generalize the conformal block decomposition to this particular case since it has not, to our knowledge, previously been studied in the literature. Similar analyses have been carried out for the SO(N) and SU(N) cases in [113]. The relevant gauge singlet complex scalar degrees of freedom are bi-fundamental with respect to the SU(N f ) L × SU(N f ) R global symmetry and can be mathematically represented as: where all indices i, j, α, β = 1, 2, . . . , N f . Latin indices are for SU(N f ) L and Greek indices for SU(N f ) R respectively. It is convenient to introduce the following matrix notation We start with the OPE analysis for the following composite operator: where, in the free theory, Tr[HH † ]δ i j δ αβ and d H is the conformal dimension of the H field. The group-theoretical content of the OPE above is: The crossing symmetry constraints are derived by equating the (12)→ (34) and (14)→ (23) s-and t-channel conformal block decompositions of the following 4-point function There are four basic invariants contained in [H( we see that the overall singlet terms contributing are: where 1 AA means that we have to extract the singlet from the tensor product of the two adjoint representations.
We now derive the constraint stemming from crossing symmetry in terms of these four basic invariants. For the s-and t-channel conformal block decompositions we obtain: where H i = H(x i ), d H is the quantum physical dimension of the H field, G ≡ G u↔v and we used a graphical notation for the tensor contractions. The squaring of the contractions (. . .) 2 means that we have to perform the same contraction for both SU(N f ) factors. Every line means that the corresponding indices are contracted with the δ−tensor: Now, equating the s-and t-channel decompositions and demanding that the coefficients multiplying the corresponding tensors match, we deduce: which yields four equations with four unknowns. These equations generalize (14) to the theories possessing the non-abelian symmetry SU(N f ) L × SU(N f ) R and can be solved numerically.

IV. A FOUR-DIMENSIONAL CALCULABLE GAUGE-YUKAWA CFT
We where T A iα are the usual generalized Gell-Mann matrices. The fields H and H † can be contracted to form a singlet or an adjoint with respect to the right or left handed groups: while (Adj,Adj) can be formed as a tensor product .
The Lagrangian of the theory is Here Throughout this section we will work with the rescaled couplings which enable a finite Veneziano limit of the theory at fixed . That is, we let both N c , N f → ∞ while keeping x ≡ N f /N c fixed. The appropriately rescaled couplings are This model was introduced in [75, 101] to investigate near-conformal dynamics and its impact on the spectrum of the theory. Special attention was paid to the appearance of a dilaton, the Goldstone boson associated with the breaking of conformal symmetry, and its properties. The model was further investigated at higher orders in [102], and the properties related to the a-theorem were considered in [103].

A. Beta functions and Weyl consistency conditions
In order to perform a four-dimensional comparison with the bootstrap bound, we start by providing a calculable CFT at the highest known perturbative order. Following previous studies [103,104] the beta functions of the theory are Here we have already assumed the Veneziano limit and is the number of SU(N c ) adjoint Weyl fermions of the theory 7 . We used the results of [114][115][116][117] to determine the beta functions and anomalous dimensions of the gauge-Yukawa theories investigated here.
The perturbative gauge beta function is considered up to and including the three loop order, the Yukawa to two and the scalar quartic couplings to the first order. This is the proper way of organizing perturbation theory for a multiple coupling theory as shown in [103,104]. In fact this counting can be mathematically related to the Weyl consistency conditions unveiled in the pioneering work by Osborn [118] and demonstrated to be relevant also for the standard model in [103,104]. These conditions require the different beta functions to be related across different loop orders. Mathematically these conditions read: with where g i ≡ (a g , α H , z 1 , z 2 ) refers to the couplings collectively. To help the reader identify the related terms, according to the Weyl conditions, across the different couplings, we 7 In Table I we assumed = 1.
color-coded them directly in the beta functions. It is clear that these conditions relate the two-loop coefficients in the gauge beta function with one-loop coefficients in the Yukawa beta function (red color) and the two-loop coefficients in the Yukawa beta function with the one-loop coefficients in the quartic beta function (blue and brown colors). Our perturbative interacting CFTs live at the fixed point (FP) identified by the simultaneous zeros of the previous beta functions, i.e. we need to solve for β a g = β a H = β z 1 = β z 2 = 0.
The study of the beta functions above allowing to establish the existence of perturbative CFTs has been performed in [54,103]. We will investigate the explicit physical results stemming from the analysis of these beta functions in IV C.

B. Higgs anomalous dimensions and bootstrap
The existence of a perturbative CFT permits us to determine the conformal dimensions of the (1, 1) singlet ∆ S ≡ 2+γ S and of the (Adj,Adj) adjoint ∆ A ≡ 2+γ A composite operators.
For reader's convenience, we recall how these dimensions enters the OPE (21): To compute these anomalous dimensions, we add to the Lagrangian (35) 8 and use [119] to specialize the formulae given in [114] to the present case.
We know from the Weyl consistency conditions [103,104] that the order to which beta functions are computed in a gauge-Yukawa theory is distinctly non-trivial, and we must therefore also take care to compute the anomalous dimensions of the composite operators, as well as the Higgs field, to the proper order. To find this, we consider that if two of the four external legs on a Feynman digram that contributes to the quartic beta function are joined together, the resulting diagram is a constituent of the anomalous dimension of the composite operators to one higher order in the loop expansion. We therefore conclude that the anomalous dimensions should be computed to two loop order.
Thus, for the Higgs field H, we have that the anomalous dimension is and for the singlet and the adjoint composite operators: Interestingly, in the Veneziano limit, the conformal dimension of the (Adj, Adj) operator factorizes ∆ A = 2d H (γ A = 2γ H ), suggesting that, to two-loop order and in the Veneziano limit, we can identify M i jαβ (0) in (43) with the operator where we define the normal-ordered product : : of two operators as the non-singular part of the OPE in the limit where the two space-time points are brought together. Because the anomalous dimension of the adjoint is twice that of the H field, this sector of the theory enjoys properties resembling those of a generalized free scalar H(x) with conformal dimension d H = 1 + γ H . Therefore the correlation functions involving the composite adjoint operator are disconnected and can be written as products of 2-point function. For example, using (46): we can compute the 4-point function using the basic 2-point function: Moreover, since the 3-point function, defining the OPE coefficient c A in (43), is fixed by comparing with (47) and using (48) we see that c A = 1.
The factorization property of the (Adj, Adj) operators allows us to compute G A and G A to this order in perturbation theory and in the Veneziano limit. Indeed, to compute G A , for example, we start with the general expressions (8) (with all ∆ i = d H ) and using notation of (25) write: where we showed explicitly only the contributions from the conformal block G A . As indicated by index contractions, we have to consider the correlator with external indices (i = m, j = k) and (α = σ, β = δ). Using the factorization property of (Adj,Adj), we calculate the G A contribution as follows: and therefore by comparing with (50) we deduce that : Similarly, for G A we obtain G A = (v/u) d H . In terms of Feynman diagrams, factorization implies that the conformal block G A contributes only to the disconnected diagrams to this order in perturbation theory and in the Veneziano limit. These disconnected contributions provide the leading-N f dependence of the correlators which is known as large-N f factorization [120]. In fact, using the standard 't Hooft counting, it is easy to show that the disconnected contribution to our 4-point function in (51) where we formally divided the connected (conn) and disconnected (disc) contributions [121] to the conformal blocks G S and G A . We also used the fact that G L and G R appear at the order O(1/N f ) because they are disconnected with respect to just one of the two At the leading O(1) in the large-N f expansion, from the bootstrap equations (28)- (29) we have:  (53) By matching the 1/N 2 f terms we have: The last equation is obtained by subtracting (31) from (30). Using (59) in the equation obtained by subtracting (58) from (57) we arrive at: Let us now consider the contributions to the conformal blocks G conn S,A and G conn S,A more carefully. We will be using the work of [121] where the 4-point function of the singlet operators O(x) was considered. In this case, the lowest dimensional operator, aside from the unit operator, in the O × O OPE is the double trace operator O 2 whose dimension f ) factorises at the lowest order in 1/N f expansion. In our model, from (45), the dimensions of the adjoint operator satisfy the same factorization property and therefore the analysis of [121] applies. There it was shown that the conformal block G conn A (and G conn A ) receives the contributions from the sum of two terms: • O(1/N 2 f ) correction to the OPE coefficients p A ∆,l . We will denote this contribution by (G conn A ) OPE • O(1/N 2 f ) corrections to the anomalous dimension ∆ A which enter the functions g A ∆,l (u, v). We will denote this contribution by (G conn A ) AD .
The connected contribution to the conformal block can thus be expanded [121]: Furthermore, in our model the anomalous dimension for the singlet operator from (45) equals the anomalous dimension of the adjoint plus an additional non-factoriziable contri- f not present in the anomalous dimensions for the adjoint (45). The nonfactorizable contribution is not present in the analysis of [121] and it will be taken into account.
Based on the above discussion it seems reasonable, but should still be proven, that G conn A matches the factorazible part of G conn S . Assuming that this is true, the contributions due to the conformal blocks ((G conn S,A ) AD ,( G conn S,A ) AD ) and ((G conn S,A ) OPE ,( G conn S,A ) OPE ) cancel out in (60): The non-factorizable contribution to the singlet anomalous dimension quantifies the departure from the Gaussian limit and stems from an additional part of the singlet conformal block (G conn S ) non−fact . This part will not be balanced by an appropriate term associated with the adjoint composite operator in (60). This leads to a suggestive bootstrap equation for the non-factorizable part of the singlet which has precisely the form of (15), and we can even expand the conformal block in functions of the kinematics of the CFT (G conn S ) non−fact = p S,n f ∆,l g S,n f ∆,l (u, v). Just as in (15), the right-hand side of (63) is the contribution from the Gaussian part of the theory which is balanced by the left-hand side. Therefore it would be tempting to interpret this result as a bound for ∆ S , similar to the bound on the lowest dimensional operators of the theory coming from Eq. (15), though holding only to the next-to-leading order in the couplings and in the Veneziano limit. This would mean that for low values of the couplings and high values of N f , ∆ S should obey the bound given by Eq. (18) with d = d H . However, there are several caveats to this suggestive statement that require further investigation. The most pressing is that our expression holds only for a part of G S , and it is not clear how (or even if) a consistency equation on such a part would translate into a bound on ∆ S . Another concern is that, as pointed out in [121], unitarity only implies positivity of p ∆,l to leading order in 1/N f , and if the expansion parameters p S,n f ∆,l are allowed to take either sign, the large N f analogue of the proof provided in [105] would be affected.
Fortunately the bound of [107] applies to the adjoint composite operator, the lowest dimensional operator, without any caveats. And as we shall see in the specific examples provided below, the bound is well satisfied.
In these examples, we will also see that the anomalous dimension of the singlet composite operator can be substantially larger than that of the adjoint composite operator.
Given that the bound on the singlet is unknown, this is a welcome feature which has been long sought after for nonperturbative models of near conformal dynamics used to describe composite Higgs scenarios, see [55] for a recent review.

C. Physical results
Here we review the salient points behind the existence of perturbative interacting CFTs [54,103] and then determine the physical dimensions of the composite operators at the FPs of the theory.
The two-loop gauge beta function has a perturbative Banks-Zaks FP if the one-loop coefficient b 0 of the gauge beta function is small and the signs of the one-loop b 0 and two-loop b 1 coefficients are opposite. Therefore, our first task is to find a region in the parameter space of the model where the BZ FP exists. Solving (38) to one-loop and substituting into (37) we obtain From the asymptotic freedom (AF) boundary condition b 0 = 0, we obtain x = (11 − 2 )/2.
After substituting this value of we observe that for the unphysical value * ≈ 0.37 the coefficient b 1AF vanishes. For = 1 we have that b 1AF is negative and for = 0 it is positive. Therefore in the first case we have an infrared BZ FP, and in the second we have an ultraviolet BZ FP. Note also [103] that in the absence of the Yukawa interactions the coefficient b 1AF in (65) is always negative and therefore the physical BZ FP can lead only to an infrared FP.
We are now ready to present our results for the = 0 and = 1 cases. Our strategy is the following • For a given FP in all the couplings (a * g , a * H , z * 1 , z * 2 ) at a given value of x ≡ N f /N c , we determine the anomalous dimensions for the composite operators γ S and γ A . We also determine the associated anomalous dimension of the scalar field γ H . These results were obtained by means of the equations (44) and (45).
• We then insert d = d H = 1 + γ H in the right hand-side of (17) and (18) to determine ∆ max − 2. Finally, we compare the result with γ A , which turns out to be the operator for which the bootstrap bound applies in all cases under consideration, and display γ S , which is a more interesting quantity for phenomenology.
Because of the factorization property for the adjoint composite operator, i.e. γ A = 2γ H , it is guaranteed that the physical dimension of the (Adj,Adj) operator satisfies the bound (18) [106]. The situation, as we shall see, is much more intriguing for the scalar operator.
Furthermore, because the more recent numerically determined functional form, given in (18) [107], has a different dependence of γ H in the very perturbative region than the previous bound (17) [106], we also display this for a comparison.

The = 0 case
The asymptotic freedom boundary, where the first coefficient of the gauge beta function vanishes, b 0 = 0, occurs at x AF = (11 − 2 )/2 = 5.5. Increasing x > x AF slightly results in the appearance of an ultraviolet BZ FP, see Fig. 3.a. An in depth analysis of the FP structure and its theoretical and phenomenological consequences for the asymptotic safety scenario has just appeared in [54].
When the three-loop gauge beta function is considered, an infrared FP emergers along with the ultraviolet BZ FP in the range x ≤ x * ≈ 5.617. At x * the ultraviolet BZ FP and  the infrared FP collide and both fixed points disappear. Perturbation theory is, of course, valid only for values of (x − x AF )/x AF 1. As shown in [54] perturbation theory is valid The comparison with the bootstrap bound is shown in Fig. 3.b. We first note that as expected, the bound is clearly respected by the anomalous dimension of the adjoint operator. More interestingly, we discover that the anomalous dimension of the singlet composite operator γ S is substantially larger than the anomalous dimension of the adjoint operator. If this also holds in the non-perturbative regime, this has important and welcome implications for model building.
To better understand the behavior of the anomalous dimension of the singlet, it would be interesting to do a full bootstrap analysis of this operator, and compute a bound for it like the ones that have been found for theories obeying an SU(N) or SO(N) global symmetry [107]. In particular, this would shed light on the question of whether the observed behavior for the singlet anomalous dimension holds in the nonperturbative limit.

The = 1 case
When the model is expanded to include adjoint fermions, the infrared BZ FP originates just below the asymptotic freedom boundary x AF = (11 − 2 )/2 = 4.5 as shown in Fig. 4.a.
The comparison of the composite operator anomalous dimensions with the two numerical bootstrap bounds is shown in Fig. 4.b. As explained above, γ A is consistently below the bound in the perturbative regime. As for the = 0 case we determine the relevant quantity γ S and show that it is, also in this case, substantially larger than the adjoint operator.

The = 2 case
For completeness we also consider the case of large values of , and compare the anomalous dimensions to the bounds. At = 2 the infrared BZ FP originates at x AF = (11 − 2 )/2 = 3.5 as shown in Fig. 5.a. In this case, we note a new feature; at x * ≈ 2.4 the infrared BZ FP and the ultraviolet FP merge within the asymptotically free region. 9 The 9 We have previously investigated such a merger for the same model with = 1 when using the beta functions to two loops in all the couplings [102]. Though using the beta functions to this loop order is unphysical, the analysis, given the merger, is sound, and we expect an analoguous analysis to be valid here.
comparison of the composite operator anomalous dimensions with the two numerical bootstrap bounds is shown in Fig. 5.b. We notice that in this case, in a perturbatevely trustable regime, γ S obeys with the bound, and comes suggestively close to saturating it.

The = 5 case
As we continue to increase , γ S continues to decrease relative to the bound. For the biggest possible integer value where a BZ fixed point can occur, = 5, the infrared BZ FP originates at x AF = (11 − 2 )/2 = 0.5 as shown in Fig. 6.a. The comparison of the composite operator anomalous dimensions with the two numerical bootstrap bounds is shown in Fig. 6.b. We notice that in this case, both γ A and γ S are consistently below the bound, and that γ S is only slightly above γ A .
In summary, our numerical analysis revealed that the singlet anomalous dimension decreases with increasing value of . The main reason for this behavior is an increasing FP value of the Yukawa coupling a * H , relative to the other couplings, which is shown in Figs. 3.a-6.a with the black dotted line. We have also performed a thorough numerical study away from the Veneziano limit with finite N f and N c values where we found the same correlation. In Appendix A we review, for completeness, the leading finite N f corrections to the beta functions and anomalous dimensions of the theory [102].
The correlation between high and low γ S is interesting, as the Yukawa coupling a * H provides the link between the gauge and the scalar sector of the theory and controls the strength of the attractive Yukawa interaction. Our results show that as we increase (and, as a consequence, decrease x) while keeping the singlet anomalous dimension γ S fixed, the anomalous dimension γ H of the field H is an increasing function of .

V. CONCLUSIONS
We provided a systematic investigation of interesting properties of relevant composite operators stemming from gauge-Yukawa theories developing conformal fixed points in four dimensions. These theories are structurally similar to the standard model of particle interactions and have already been employed for interesting model building [40]. Having We showed that in the Veneziano limit, and at the maximum known order in perturbation theory, adjoint composite operator is Gaussian and automatically obeys the bootstrap bounds on the anomalous dimension. We also discovered that the singlet composite operator anomalous dimension at the interacting FP is substantially larger than the one for the adjoint composite operator. This is an interesting observation for phenomenologically driven questions regarding the possibility of large anomalous dimensions for singlet operators needed, for example, in theories of composite Higgs dynamics [55]. It would be interesting to analyze more generally the full bootstrap equations for these patterns of chiral symmetry.
Our results demonstrate the relevance of constructing conformal nonsupersymmetric four dimensional gauge-Yukawa theories that can be used for demonstrating the existence of four dimensional asymptotically safe theories [54], for interesting model building [40], probing the a-theorem [103], but also to either accurately test numerical solutions of the bootstrap constraints or determine novel anomalous dimensions of relevant composite operators. Following the pioneering work of Seiberg [122] it would be interesting to explore whether the weakly coupled four dimensional gauge-Yukawa theories investigated here have strongly coupled duals [123,124].
Remarkably all the leading 1/N f corrections emerge only at the order 1/N 2 f order.