Alessandro Vichi

Bounding scalar operator dimensions in 4D CFT

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 x phi with a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theory-independent inequality [phi(2)] 1 we have f(d) = 2 + O(root d - 1), 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 - epsilon 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.

Solving the 3D Ising Model with the Conformal Bootstrap

We study the constraints of crossing symmetry and unitarity in general 3D conformal field theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and product-expansion coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.

Carving out the space of 4D CFTs

A bstractWe introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and

Simplified models for dark matter searches at the LHC

\mathcal{N} = 1

Improved bounds for CFT's with global symmetries

superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In

Precision Islands in the Ising and

\mathcal{N} = 1