Bounding scalar operator dimensions in 4DCFT

Abstract

In an arbitrary unitary 4D CFT we consider a scalar operator φ, and the operator φ 2 defined as the lowest dimension scalar which appears in the OPE φ × φ with a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theory-independent inequality [φ 2 ] ≤ f ([φ]) for the dimensions of these two operators. The function f (d) entering this bound is computed numerically. For, which shows that the free theory limit is approached continuously. We perform some checks of our bound. We find that the bound is satisfied by all weakly coupled 4D conformal fixed points that we are able to construct. The Wilson-Fischer fixed points violate the bound by a constant O(1) factor, which must be due to the subtleties of extrapolating to 4 − ε dimensions. We use our method to derive an analogous bound in 2D, and check that the Minimal Models satisfy the bound, with the Ising model nearly-saturating it. Derivation of an analogous bound in 3D is currently not feasible because the explicit conformal blocks are not known in odd dimensions. We also discuss the main phenomenological motivation for studying this set of questions: constructing models of dynamical ElectroWeak Symmetry Breaking without flavor problems.

BootstrappingO(N)vector models in4<d<6

“…We start with N = 500 and using the large N expansion value for ∆ σ 2 and our most accurate numerics we find a kink at (∆ φ , ∆ σ ) = (1.500409, 2.027) that is very close to the large N expansion values (∆ φ , ∆ σ ) = (1.500414, 2.022) for the critical theory. 5 We then examine N ≤ 40 and find a kink that disappears around 15 < N crit < 22 for a reasonable assumption of ∆ σ 2 , which is consistent with the large N expansion prediction of N crit < 35. We check that our choice of ∆ σ 2 does not qualitatively affect our answers for 6 ≤ N ≤ 40.…”

“…We denote the maximum derivative order by Λ (as in [34]) and the maximum spin by J max . The truncated constraint problem can then be rephrased as a semidefinite programing problem using the method developed in [5]. This problem can be solved efficiently by freely available software such as sdpa gmp [62].…”

“…Introduced originally in the context of two-dimensional CFTs [1][2][3][4], this technique has been applied to higher-dimensional CFTs only recently starting with the work of [5]. The bootstrap uses the crossing symmetry of correlation functions of CFTs to put an infinite set of constraints on the CFT data.…”

Exploring T and S parameters in Vector Meson Dominance models of strong electroweak symmetry breaking

“…giving a quartically divergent contribution in S. 13 However, this contribution remains reasonably small if cut off at Λ ∼ 2M V , due to phase space suppression. In a more realistic setup, the AV formfactor would be softened giving an even smaller ∆S.…”

“…In this case, thinking in terms of gauge degrees of freedom may be misleading. One may try instead to use the language of Conformal Field Theory [13][14][15][16]. Conformal language is also indispensable in the context of strong EWSB models in warped extra dimensions [17].…”