Bootstrap bound for conformal multi-flavor QCD on lattice

The recent work by Iha et al shows an upper bound on mass anomalous dimension $\gamma_m$ of multi-flavor massless QCD at the renormalization group fixed point from the conformal bootstrap in $SU(N_F)_V$ symmetric conformal field theories under the assumption that the fixed point is realizable with the lattice regularization based on staggered fermions. We show that the almost identical but slightly stronger bound applies to the regularization based on Wilson fermions (or domain wall fermions) by studying the conformal bootstrap in $SU(N_f)_L \times SU(N_f)_R$ symmetric conformal field theories. For $N_f=8$, our bound implies $\gamma_m<1.31$ to avoid dangerously irrelevant operators that are not compatible with the lattice symmetry.


Introduction
Realizing strongly coupled conformal gauge theories on a lattice has been one of the active research arenas in theoretical physics. It would give non-perturbative information on various critical exponents such as mass anomalous dimension γ m = 3 − ∆ψ ψ . Finding a model with larger mass anomalous dimension is important for its possible application to particle physics (see e.g. [1] and reference therein). We note that in four dimensions, as long as we work in Lagrangian quantum field theories, it is necessary to consider gauge theories to realize any non-trivial renormalization group fixed point, and the Monte-Carlo simulation to probe such non-trivial renormalization group fixed points is thus much more difficult than in the lower dimensions.

One of the simplest examples of possible conformal gauge theories in four dimensions
is multi-flavor massless QCD [2]. While we have a better theoretical control in the weakly coupled regime [3] (so called Banks-Zaks fixed point in the literature), we have two unsolved issues in the strongly coupled regime: (1) what is the critical number of flavors with a given gauge group? (2) what is the maximal value of the mass anomalous dimension?
We concentrate on the second issue here. There is an absolute bound on the mass anomalous dimension γ m ≤ 2 from the unitarity bound of conformal field theories [4]. 1 There is a proposed more stringent bound γ m ≤ 1 from the analysis of the Schwinger-Dyson equation [7]. Intuitively, the latter corresponds to the bound on the coupling constant where the SU (N f ) L × SU (N f ) R symmetric four-Fermi terms become relevant, inducing the Nambu-Jona-Lasinio mechanism of chiral condensation. Note, however, that unless in the large N f limit, in which the mass anomalous dimension and the relevance of the four-Fermi interaction may be related, there is no simple relation such as ∆ four−fermi = 2∆ψ ψ , relating the mass anomalous dimension and the instability due to the four-Fermi terms. Indeed, the approximation used in the Schwinger-Dyson equation may be only justifiable in the large N f limit. Whether γ m = 1 or γ m = 2 is when the conformality will be lost in multi-flavor massless QCD is still under debate [8] [9].
Along the line of this reasoning, the conformal bootstrap gives more quantitative bound on the mass anomalous dimension in relation to the (ir)relevance of the SU (N f ) L × 1 Throughout the work, we assume that the renormalization group fixed point shows conformal invariance rather than mere scale invariance, which is true in the weakly coupled regime [5]. See e.g. [6] for the validity of this assumption.
1 SU (N f ) R symmetric four-Fermi terms. While being rigorous, the bound is weak as we will see in the next section and does not exclude the possibility of having larger mass anomalous dimension, in particular in the large N f limit.
On the other hand, in the recent paper [10], Iha et al addressed a similar but slightly different question: what is the largest mass anomalous dimension of conformal multiflavor QCD under the assumption that it is realizable with the regularization based on staggered fermions. The action for the staggered fermion has a smaller symmetry of U (N f /4) L × U (N f /4) R than massless QCD in the continuum limit, and we obtain a more constraint from the condition that there is no relevant deformations that cannot be forbidden by the lattice symmetry. It should be noted that their results do not exclude the possibility for multi-flavor massless QCD to realize a fixed point violating their bound.
Rather, it means that such a fixed point cannot be approached with the lattice action based on the staggered fermions without further fine-tuning.
In this paper, we address the similar question for the other regularization based on

Bootstrap program and results
Our numerical conformal bootstrap program is basically same as the seminal studies in [11][12] [13][14] [15] (but with more complicated symmetry group of SU (N f ) L × SU (N f ) R than SO(N ) or SU (N )) except that we use the refined implementation of the numerical semidefinite programming by SDPB [16]. We consider the four-point functions

meson operator in QCD)
with the scaling dimension ∆ Φ and their complex conjugatesΦ i L i R . In the following equa-tions, we sometimes use the abbreviation of symmetric traceless T , anti-symmetric A, and singlet S representation in each SU (N f ).
The crossing symmetry of the four-point functions gives the following OPE sum rule 2 : where (±) denotes the even (+) or odd (−) spin contributions. By using the convention with the conformal block g ∆ O ,l being normalized as in [17], whose explicit expression can 2 This sum rule was originally obtained for the purpose of bootstrapping the finite temperature QCD.
The computation is straightforward but tedious group theoretic combinatorics, which we have implemented with Mathematica based on the general algorithm presented in [12]. Eventually, we have shifted our focus on the case with N f = 2 and O(n) × O(m) models in [20] [21]. The author would like to thank T. Ohtsuki for the collaboration.
3 be found in [18], the each representation contributes to the sum rule as Here, we have symmetrized the two SU (N f )s because in the sum rule, one cannot distinguish the SU (N f ) L and SU (N f ) R . We also note that for this four-point function, the sum rule is the same with or without extra U (1) A . In other words, the U (1) A anomaly 4 cannot be seen in the bootstrap bound we study. 3 Our main focus in this paper is N f = 8 which is under debate whether the theory possesses a fixed point for the SU (3) gauge group, but we may study the other flavor groups with no difficulty. In this section, we will set N f = 8 in the following, but we will comment on the case N f = 16 in section 3. A reader who is interested in the other N f may contact the author.
The actual implementation of the numerical conformal bootstrap is based on cboot [19], which uses the SDPB as a core part of the semi-definite programming. For the truncation of the search space by number of derivatives, our numerical results in this section are based on Λ(= N max ) = 17, which is slightly better than the one used in [10].
The other parameters such as the number of included spin are chosen appropriately so that the numerical optimization is stable.
Our first result is the bound on the scaling dimensions of the singlet operator in SU (N f ) L × SU (N f ) R symmetric conformal field theories that appears in the OPE of the scalar operators Φ andΦ. When the scaling dimension of this operator becomes less than four, it is regarded as a dangerously irrelevant operator (because in our explicit models in mind, there is no such relevant operators in the UV continuum limit), and we have to tune the corresponding coupling to realize the fixed point in any regularization in use.
We again emphasize that this does not mean that such a fixed point cannot exist, but it only means that the realization on the lattice requires further fine-tuning due to the dangerously irrelevant operator.
A plot in Fig 1 shows  Let us compare our bound with the one that has been (implicitly) studied in [10].
They studied the bound on the scaling dimensions of operators that appear in the OPE 3 A possible way to avoid U (1) A symmetry is to introduce a gap assumption in spin one sector in the singlet representation, but we a priori do not know how much gap should be introduced. The resulting bound on the scaling dimensions can be found in Fig 2 and Fig 3. Our result shows that in order to avoid these dangerously irrelevant operators, we need ∆ Φ > 1.69 or γ m < 1.31 from the bound for the symmetric traceless × symmetric traceless representation in Fig 2, which is stronger than the one from the anti-symmetric × antisymmetric representation.  Our bound can be compared with the one studied in SU (N f ) V symmetric conformal field theories from the four-point functions of scalar operators in the adjoint representation [10]. At first sight, it seems that our bound in Fig 2 coincides with their bound in Fig A1, and our bound in Fig 3 coincides with their bound in Fig A2. However, a more delicate comparison shows that they are very close but not identical if we use the same search space dimension Λ for the comparison. It turns out that our bound on ∆ Φ is slightly stronger (i.e. the bound on mass anomalous dimension γ m is stronger). As for the bound on the mass anomalous dimension for the possible phenomenological applications, however, the difference is minor in practice. 4 We also note that unlike in three dimensions near the kink, the convergence of the numerical conformal bootstrap bound with respect to the search space dimension Λ is not dramatically fast in four dimensions, so one may expect a further improvement on the bound by increasing Λ. Indeed, we see certain degrees of improvement by increasing Λ in Fig 4, but we believe that the possible improvement on the bound must be much less than 0.1 and does not reach γ m = 1.0. We will discuss the Λ → ∞ limit further from a slightly different viewpoint in the next section.

Discussions
In this work, we have obtained an upper bound on the mass anomalous dimension in conformal multi-flavor QCD, assuming that the fixed point is realizable by regularization The bound is numerically very similar but not identical to the one studied in [10]. For N f = 8, our bound implies γ m < 1.31 to avoid dangerously irrelevant operators that are not compatible with the lattice symmetry. To conclude the paper, we would like to discuss the large N f limit of our bound. In