Bounding scalar operator dimensions in 4D CFT
Abstract
In an arbitrary unitary 4D CFT we consider a scalar operator phi, and the operator phi^{2} defined as the lowest dimension scalar which appears in the OPE phi × phi with a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theoryindependent inequality [phi^{2}] <= f([phi]) for the dimensions of these two operators. The function f(d) entering this bound is computed numerically. For d→1 we have f(d) = 2+O((d1)^{1/2}), 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 WilsonFischer fixed points violate the bound by a constant O(1) factor, which must be due to the subtleties of extrapolating to 4∊ 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 nearlysaturating 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.
 Publication:

Journal of High Energy Physics
 Pub Date:
 December 2008
 DOI:
 10.1088/11266708/2008/12/031
 arXiv:
 arXiv:0807.0004
 Bibcode:
 2008JHEP...12..031R
 Keywords:

 High Energy Physics  Theory;
 High Energy Physics  Phenomenology
 EPrint:
 v2: 48 pp, the published version (a simple explanation of why the bound is at all possible added, linear functional coefficients in Table 1 rescaled to correct the previously wrong normalization, notation improved, 2 refs added)