Upper bound on the mass anomalous dimension in many-flavor gauge theories: a conformal bootstrap approach

We study four-dimensional conformal field theories with an $SU(N)$ global symmetry by employing the numerical conformal bootstrap. We consider the crossing relation associated with a four-point function of a spin~$0$ operator~$\phi_i^{\Bar{k}}$ which belongs to the adjoint representation of $SU(N)$. For~$N=12$ for example, we found that the theory contains a spin~$0$ $SU(12)$-breaking relevant operator when the scaling dimension of~$\phi_i^{\Bar{k}}$, $\Delta_{\phi_i^{\Bar{k}}}$, is smaller than~$1.71$. Considering the lattice simulation of many-flavor quantum chromodynamics with $12$~flavors on the basis of the staggered fermion, the above $SU(12)$-breaking relevant operator, if it exists, would be induced by the flavor-breaking effect of the staggered fermion and prevent an approach to an infrared fixed point. Actual lattice simulations do not show such signs. Thus, assuming the absence of the above $SU(12)$-breaking relevant operator, we have an upper bound on the mass anomalous dimension at the fixed point~$\gamma_m^*\leq1.29$ from the relation~$\gamma_m^*=3-\Delta_{\phi_i^{\Bar{k}}}$. Our upper bound is not so strong practically but it is strict within the numerical accuracy. We also find a kink-like behavior in the boundary curve for the scaling dimension of another $SU(12)$-breaking operator.

We study four-dimensional conformal field theories with an SU (N ) global symmetry by employing the numerical conformal bootstrap. We consider the crossing relation associated with a four-point function of a spin 0 operator φk i which belongs to the adjoint representation of SU (N ). For N = 12 for example, we found that the theory contains a spin 0 SU (12)-breaking relevant operator when the scaling dimension of φk i , ∆ φk i , is smaller than 1.71. Considering the lattice simulation of many-flavor quantum chromodynamics with 12 flavors on the basis of the staggered fermion, the above SU (12)breaking relevant operator, if it exists, would be induced by the flavor-breaking effect of the staggered fermion and prevent an approach to an infrared fixed point. Actual lattice simulations do not show such signs. Thus, assuming the absence of the above SU (12)breaking relevant operator, we have an upper bound on the mass anomalous dimension at the fixed point γ * m ≤ 1.29 from the relation γ * m = 3 − ∆ φk i . Our upper bound is not so strong practically but it is strict within the numerical accuracy. We also find a kink-like behavior in the boundary curve for the scaling dimension of another SU (12)-breaking operator.

Introduction and result
Four-dimensional conformal field theories that may be realized as a low-energy limit of a non-Abelian gauge theory with N flavor massless fermions [1] are of great interest phenomenologically because they can be a starting point for finding viable models of the walking technicolor [2][3][4][5][6][7]. Recognition that a non-perturbative study of such conformal theories is feasible with currently available lattice techniques [8] triggered many recent investigations; see a recent review [9] and the references cited therein. Here, one is particularly interested in the mass anomalous dimension of the fermion, γ m , which must be of order one in viable technicolor models.
It is always challenging, however, to determine something quantitative for a conformal field theory by lattice numerical simulations. This is natural because the conformal field theory has no specific length scale and consequently one ideally has to work with an infinite volume. 1 In fact, for example, although there seems to be a consensus that the SU (3) gauge theory with 12 fundamental massless fermions-12-flavor quantum chromodynamics (QCD)-has an infrared fixed point, there still exist large discrepancies among central values of the mass anomalous dimension at the fixed point, γ * m , depending on computational strategies; see Fig. 11 of Ref. [12] and Table 4 of Ref. [9].
Originally motivated by the above large discrepancies in γ * m , in this paper we apply the numerical conformal bootstrap-a powerful rigorous approach to higher-dimensional conformal field theories-to four-dimensional conformal field theories with an SU (N ) global symmetry. A partial list of references on the numerical conformal bootstrap is ; see also a most recent paper, Ref. [37], and the recent review [38] for a more complete list. Our formulation is valid for arbitrary N , but we will report our numerical results only for N = 12 in the main text (we present the results for N = 8 and N = 16 in Appendix A). As explained below, by combining a result from our numerical conformal bootstrap and the fact that lattice simulations of the 12-flavor QCD [12,[39][40][41][42][43][44][45][46][47] are consistent with the existence of an infrared fixed point, we obtain an upper bound on the mass anomalous dimension, γ * m ≤ 1.29, for N = 12. (1.1) Practically, this upper bound is not so strong, not being able to constrain values obtained by existing lattice simulations. 2 Nevertheless, it appears quite interesting that such a strict bound can be made from very general properties of a unitary conformal field theory, with additional information provided by lattice simulations. There even exists a possibility that this bound might become stronger if the level of approximations that we made in our numerical conformal bootstrap is increased. Now, in the context of the technicolor model, one is interested in the anomalous dimension of the flavor-singlet scalar density, 1 An intriguing possibility to evade this is to employ the conformal mapping from R 4 to R × S 3 and a lattice discretization of the latter space [10]. See also Ref. [11] for an alternative approach. 2 There exists a rigorous bound that follows from the unitarity [48], γ * m ≤ 2.
where k (k) denotes the index of the fundamental (anti-fundamental) representation of SU (N )-the flavor group-in a QCD-like theory. This is because the expectation value of S provides the technifermion condensate. Since the combination m 0 S is not renormalized, m 0 S = mS R , where m 0 is the bare mass parameter and the right-hand side is the product of the renormalized quantities, the anomalous dimension of S is given by the mass anomalous dimension γ m , defined by where the subscript 0 implies that bare quantities are kept fixed. We are interested in the value of γ m at the infrared fixed point, γ * m . In the above QCD-like theory, we assume that the SU (N ) flavor group is chiral in the sense that we actually have the chiral symmetry SU (N ) L × SU (N ) R . Then, applying the flavored chiral rotation to the scalar density (1.2), we have a pseudo-scalar density, which belongs to the adjoint representation of SU (N ). Since the flavor rotation and the scale transformation commute, the pseudo-scalar adjoint operator φk i possesses the same scaling dimension ∆ φk i as S (1.2). Then, the mass anomalous dimension γ * m and the scaling dimension ∆ φk i (at the fixed point) are related by This also directly follows from the partially conserved axial current (PCAC) relation. In Sect. 2, we consider a four-point function of a spin 0 adjoint operator φk i without specifying its actual microscopic structure such as Eq. (1.4). 3 We derive the crossing relation associated with the four-point function, 4 basically following the notational conventions of Ref. [18]. Then, in Sect. 3, we apply the numerical conformal bootstrap to the crossing relation. For this, we used a semidefinite programming code, the SDPB of Ref. [35].
In this way, among other things, we found that for N = 12 the system contains a spin 0 relevant operator in the representation [N − 1, N − 1, 1, 1] of SU (N ), 5 when Since this relevant operator in the [N − 1, N − 1, 1, 1] representation appears in the operator product expansion (OPE) of two φk i s, if the latter is identified with the pseudo-scalar density in Eq. (1.4), this is a scalar density. Such an SU (12) non-invariant operator is not 3 We do not assume the underlying gauge theory either; we assume only that the theory is conformal and possesses a global SU (N ) symmetry. 4 We learned that this crossing relation had already been derived in Ref. [25]. We would like to thank the referee for pointing out this fact. 5 We label representations of SU (N ) by a list of the (non-increasing) number of boxes in each column of the corresponding Young tableau. For example, the adjoint representation is denoted as [N − 1, 1]. For N = 12, we should say [11,11,1,1] rather than [N − 1, N − 1, 1, 1], but in this paper we use the latter notation even for N = 12. This remark applies also for other representations and for other values of N . radiatively induced, even if it is relevant, if our regularization preserves the SU (12) symmetry. We note, however, that in all existing lattice simulations of the 12-flavor QCD, the staggered fermion [49] is employed to prevent the fermion mass operator (which is believed to be a unique spin 0 SU (12)-invariant relevant operator associated with the infrared fixed point under consideration) from being radiatively induced. This is accomplished by the exact U (1) A symmetry [50] that the massless staggered fermion possesses. Still, however, the staggered fermion cannot preserve the full SU (12) flavor symmetry (the so-called taste breaking). Generally, when the regularization does not preserve a symmetry, relevant operators that are not invariant under the symmetry are radiatively induced and, to achieve the desired continuum or low-energy limit, one has to tune the coefficients of those non-invariant operators in the action. The fact that actual lattice simulations [12,[39][40][41][42][43][44][45][46][47] of the 12-flavor QCD are consistent with the existence of an infrared fixed point without such a fine-tuning strongly indicates that the theory does not contain the above SU (12) non-invariant relevant operator in the spectrum.
Thus, assuming the absence of the spin 0 relevant operator in the representation [N − 1, N − 1, 1, 1], we have the inequality ∆ φk i ≥ 1.71. Then the upper bound on the mass anomalous dimension (1.1) follows from the relation (1.5).
We stress that our upper bound (1.1) is a physical property of a conformal field theory at the infrared fixed point under consideration. The validity of our upper bound and whether one uses the staggered fermion in actual lattice simulations are completely independent issues. We have used the fact indicated by existing lattice simulations, just to support our assumption on the absence of the spin 0 relevant operator in the representation [N − 1, N − 1, 1, 1] around the fixed point. Whether there exists such a relevant operator in the RG flow near a fixed point or not is a property of the fixed point and this property should be independent of the way one studies the system.
To really claim that the SU (12) non-invariant operator in the [N − 1, N − 1, 1, 1] representation is induced with the staggered fermion, we still have to show that it is not prohibited by exact symmetries of the staggered fermion [51,52]. This group-theoretical question can be studied with the help of Ref. [53], which provides a complete list of SU (12) non-invariant 6 operators up to the canonical mass dimension 6; these are consistent with (i.e., not prohibited by) exact symmetries of the staggered fermion. The authors of Ref. [53] show that, for example, the following four-Fermi scalar operator is consistent with exact symmetries of the staggered fermion: where γ µ is the conventional Dirac matrix and ξ 5 is a flavor-space counterpart of the where ( ) stands for the symmetrization of the indices enclosed, and consider the two-point function in the system of free fermions. If this two-point function is non-zero, then the operator X contains the component of the [N − 1, N − 1, 1, 1] representation. Assuming a particular representation of ξ 5 in which the component (ξ 5 )1 1 is non-zero, it is easy to see that XO (11) (11) ∝ −32(1 − 2/N + 4/N 2 ). This shows the above assertion: Exact symmetries of the staggered fermion cannot exclude the relevant operator in the [N − 1, N − 1, 1, 1] representation of SU (12) from being radiatively induced.

SU (N ) crossing relation
As noted in the previous section, we consider a four-point correlation function of a spin 0 operator in the adjoint representation of the global symmetry SU (N ), where the lower (upper) indices stand for indices of the fundamental (anti-fundamental) representation of SU (N ). In what follows, the scaling dimension of φk i , ∆ φk i , is also denoted as d: In the conformal field theory, four-point functions such as Eq. (2.1) can be computed by applying the OPE to pairs of operators. The OPE between two operators in the adjoint representation of SU (N ) is decomposed into the sum over operators in various irreducible representations of SU (N ) (the Clebsch-Gordon decomposition) as Then, we have [cd] (ab) g ∆,ℓ (u, v) [cd] [ab] g ∆,ℓ (u, v) In deriving this, we have used the tensorial structure of the two-point function of the adjoint operator, . (2.7) g ∆,ℓ (u, v) is the so-called conformal block and its explicit form in four dimensions is given by [54] [ij] δ [cd] [ab] , (2.14) and δ (cd) The index structure of these symbols is fixed by the symmetry. The signs are fixed by requiring positiveness for i =d, j =c,k = b, andl = a (see Sect. 2.2 of Ref. [17], for example). Noting the identities which must result in an identical expression. This requirement imposes a strong consistency condition called the crossing relation. In our case, this is obtained from the invariance of Eq. (2.5) under the exchange (x 1 , i,k) ↔ (x 3 , a,c). Noting that u ↔ v under this exchange, we have, for example, as the coefficient of δk i δl j δc a δd b , We will also use the combination In a similar way, we have 4! = 24 relations as the coefficients of various combinations of Kronecker deltas. However, not all the relations are linearly independent. We find that the linearly independent relations are summarized as for a primary operator with the spin ℓ (except the identity operator, for which ∆ = ℓ = 0).

Numerical conformal bootstrap
We now apply the numerical conformal bootstrap to the crossing relation (2.25). We assume that the spin 0 adjoint operator φk i possesses the smallest scaling dimension d = ∆ φk i among all spin 0 operators appearing in Eq. (2.25), except the identity operator for which ∆ = 0. First, we investigate a possible bound on the smallest scaling dimension of a spin 0 operator in the [N − 1, N − 1, 1, 1] representation. For this, for a fixed d, we take an appropriate number ∆ trial ≥ d. Then we seek a linear differential operator Λ, which acts on a 6-component vector V as where coefficients λ i m,n are real, and which fulfills the following conditions: • As a condition for the identity operator for which ∆ = ℓ = 0, Λ(V 1 + d,0,0 ) = 1. If we can find a Λ which fulfills the above conditions, Λ acting on the crossing relation (2.25) yields a contradiction, a strictly positive number = 0. Thus, we can conclude that, if the system is a unitary conformal field theory, there must exist a spin 0 operator in the [N − 1, N − 1, 1, 1] representation which possesses the scaling dimension smaller than the assumed ∆ trial . Changing ∆ trial , we can find a restriction on the scaling dimension of the spin 0 operator in the [N − 1, N − 1, 1, 1] representation.
The parameter N max in Eq. (3.1) parametrizes the search space of Λ. When N max is increased, the possible form of Λ has more varieties and it becomes easier to find the Λ which fulfills the above conditions. As a consequence, the restriction on the scaling dimension on the operator becomes stronger when N max is increased. In our present problem, the upper bound on the mass anomalous dimension becomes lower when N max is increased.
The above search for Λ can effectively be carried out by using the semidefinite programming, as emphasized in Ref. [18]. For this, we used a semidefinite programming code, SDPB of Ref. [35]. There are two parameters characterizing the level of approximation in this approach. One is the maximal spin in the above search of Λ, Lmax. Another is the order of the rational approximation of the conformal block, keptPoleOrder. Our most strict bound below was obtained by setting parameters as (derivativeOrder = N max , keptPoleOrder, Lmax) = (16,20,24). We confirmed that the boundary curves in Figs. 1 and 2 do not change, even if we change the parameters (derivativeOrder, keptPoleOrder, Lmax) to, for example, (10,11,22) for the N max = 10 case and to (16,18,22) (this is only for Fig. 1) and (16,18,24) for the N max = 16 case. 7 Figure 1 is our result obtained by the above procedure. The horizontal axis is the scaling dimension of the spin 0 adjoint operator φk i , d = ∆ φk i . The shaded region is the smallest scaling dimension of a spin 0 operator in the [N − 1, N − 1, 1, 1] representation of SU (N ) with N = 12 in a unitary conformal field theory. We stress again that to have a unitary conformal field theory, there must exist at least one spin 0 operator in the [N − 1, N − 1, 1, 1] representation in the shaded region. In particular, we see that, when d = ∆ φk i < 1.71,  (12,14,24), (14,16,24), and (16,20,24), respectively. We see that the operator becomes relevant, i.e., the scaling dimension becomes smaller than 4, when d = ∆ φk i < 1.71. Although our analysis on the representation [N − 2, 2] does not provide a useful upper bound on γ * m , quite interestingly, we see a kink-like behavior in the boundary curves in Fig. 2 around d = ∆ φk i ∼ 1.5. Recalling the fact that in the numerical conformal bootstrap quite often one finds a known conformal field theory at a kink point on the boundary curve, the behavior in Fig. 2 is quite suggestive. It would be interesting to study this kink-like behavior in more detail and seek a possible conformal field theory with a global SU (12) symmetry that corresponds to the (possible) kink in Fig. 2.
Among other representations in Eqs. (2.3) and (2.25), [N − 2, 1, 1] and its conjugate possess only odd spin operators, and spin 0 operators which can correspond to a term in the action are not included. The representations [N − 1, 1] and 1 are somewhat special because, depending on the underlying field theory (e.g., 12-flavor QCD), by using the flavored chiral rotation it is possible to construct spin 0 operators in these representations whose scaling dimension  (12,14,24), (14,16,24), and (16,20,26), respectively. We see that the operator becomes relevant when d = ∆ φk i < 1.41.
is degenerate with d = ∆ φk i . For such a case, to draw a non-trivial conclusion one has to consider the second operator in these representations that has the scaling dimension greater than or equal to d. Although we carried out such an analysis for the representations [N − 1, 1] and 1, we do not present those results here, because the conclusion on the mass anomalous dimension seems quite dependent on the detail of the underlying theory. Figure A1 is our result on the smallest scaling dimension of a spin 0 operator in the [N − 1, N − 1, 1, 1] representation of SU (N ) with N = 8, N = 12, and N = 16 (from left to right). Boundary curves are obtained by setting (derivativeOrder = N max , keptPoleOrder, Lmax) = (14,16,24). As for N = 12 in the main text, we see that when d < 1.67 for N = 8, and when d < 1.71 for N = 16, there emerges an SU (N )-breaking relevant operator in the system. Thus, by assuming the absence of such an operator, we have an upper bound on the mass anomalous dimension as γ * m ≤ 1. 33 for N = 8, and γ * m ≤ 1.29 for N = 16.
Although the latter bound is numerically the same as Eq. (1.1), which is for N = 12, there is no contradiction because here we are using a somewhat narrower search space for the linear operator Λ (N max = 14) than that in the main text (N max = 16); the bound on γ * m here is thus somewhat weaker than would be obtained from the setting in the main text.  The parameters (derivativeOrder = N max , keptPoleOrder, Lmax) are the same as above. As for the N = 12 case in the main text, although the consideration of the operator in the [N − 2, 2] representation does not provide a useful bound on γ * m , we also observe a kinklike behavior for N = 8 and N = 16. Again, it would be interesting to study this kink-like point, if any, cannot be realized by using the staggered fermion formulation without fine tuning, but may be realized by the other regularization. behavior in more detail and seek a possible conformal field theory that corresponds to these (possible) kinks.