We derive model-independent, universal upper bounds on the Operator Product Expansion (OPE) coefficients in unitary 4-dimensional Conformal Field Theories. The method uses the conformal block decomposition and the crossing symmetry constraint of the 4-point function. In particular, the OPE coefficient of three identical dimension d scalar primaries is found to be bounded by 10(d − 1) for 1 < d < 1.7. This puts strong limits on unparticle self-interaction cross sections at the LHC.In this paper we will answer, in a particular well-defined context, the question: Is there an upper bound to the interaction strength in relativistic Quantum Field Theory (rQFT)?Intuitive reasons suggest that such a bound exists. Take QCD as a representative real-world example. At energies E above the scale Λ QCD ∼ 1 GeV, this is a perturbative theory of interacting quarks and gluons, and the interaction strength is measured by the dimensionless running coupling g s (E). The coupling starts small at very high energies E Λ QCD and grows at low energies, formally becoming infinite at E ∼ Λ QCD . However, perturbative expansion breaks down before this happens. L-loop diagrams are suppressed by factors ∼ (g 2 s /16π 2 ) L . As soon as g s ∼ 4π, all loop orders contribute equally. Thus in perturbation theory it is impossible to get couplings

We construct UV completions of bottom-up models with a pseudo Nambu-Goldstone Boson (NGB) composite Higgs and partial compositeness, admitting a weakly coupled description of the composite sector. This is identified as the low energy description of an SO(N ) supersymmetric gauge theory with matter fields in the fundamental of the group. The Higgs is a NGB associated to an SO(5)/SO(4) coset of a global symmetry group and is identified with certain components of matter fields in a Seiberg dual description of the theory. The Standard Model (SM) gauge fields are obtained by gauging a subgroup of the global group. The mass mixing between elementary SM and composite fermion fields advocated in partial compositeness arise from the flow in the IR of certain trilinear Yukawa couplings defined in the UV theory. We explicitly construct two models of this kind. Most qualitative properties of the bottom-up constructions are derived. The masses of gauge and fermion resonances in the composite sector are governed by different couplings and can naturally be separated. Accommodating all SM fermion masses within the partial compositeness paradigm remains the main open problem, since the SM gauge couplings develop Landau poles at unacceptably low energies.

We numerically study the crossing symmetry constraints in 4D CFTs, using previously introduced algorithms based on semidefinite programming. We study bounds on OPE coefficients of tensor operators as a function of their scaling dimension and extend previous studies of bounds on OPE coefficients of conserved vector currents to the product groups SO(N )×SO(M ). We also analyze the bounds on the OPE coefficients of the conserved vector currents associated with the groups SO(N ), SU(N ) and SO(N )×SO(M ) under the assumption that in the singlet channel no scalar operator has dimension less than four, namely that the CFT has no relevant deformations. This is motivated by applications in the context of composite Higgs models, where the strongly coupled sector is assumed to be a spontaneously broken CFT with a global symmetry.