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

Abstract

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.

Bootstrap bound for conformal multi-flavor QCD on lattice

“…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.…”

“…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 of the scalar operators in the adjoint representation of the SU(N f ) V symmetry.…”

“…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 multi-flavor 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.…”

The effective bootstrap

“…refs. [4,10,[17][18][19][20][21][22]33], where bounds of this kind (and others) have been determined with the derivative method using both linear and semi-definite programming.…”

“…[25] might be useful in this respect. 21 An important line of development in the numerical bootstrap is the analysis of mixed correlators which so far are numerically accessible only using semi-definite programming [15]. It would be very interesting to implement mixed correlators in the multipoint bootstrap, either by adapting the semi-definite programming techniques or by extending the linear programming techniques.…”