Fermionic UV completions of Composite Higgs models

We classify the four-dimensional purely fermionic gauge theories that give a UV completion of composite Higgs models. Our analysis is at the group theoretical level, addressing the necessary (but not sufficient) conditions for the viability of these models, such as the existence of top partners and custodial symmetry. The minimal cosets arising are those of type SU(5)/SO(5) and SU(4)/Sp(4). We list all the possible "hyper-color" groups allowed and point out the simplest and most promising ones.Comment: 15 pages, 4 tables; V2 Comments and references added. To appear in JHEP. V3 Coset of type $SU(4)\times SU(4)'/SU(4)_D$ added to the classificatio

Kinematics of 4D Conformal Field Theories

In this thesis we develop a framework for performing bootstrap computations in 4-dimensional conformal field theories. We use the conformal symmetry to construct generic 2-, 3- and 4-point functions and in turn generic bootstrap equations. An emphasis is made on the unification of all the obtained theoretical results and on their implementation into a Mathematica package \u201cCFTs4D\u201d for an easy and convenient use. The two main conceptual problems one faces are the construction of generic n-point tensor structures and the construction of generic conformal blocks. We address the first problem using 2 alternative methods: the covariant (embedding space) formalism and the non-covariant (conformal frame) formalism. Both have their advantages and disadvantages. We establish a precise connection between them which allows their interchangeable use depending on the situation. We address the second problem by reducing generic conformal blocks to an (infinite) set of seed conformal blocks. This is done using the so called spinning differential operators. We first construct explicitly a suitable (finite) set of such operators. We then introduce a new formalism which provides an (infinite) set of conformally covariant differential operators. The spinning operators are obtained as their invariant products. This heavily enlarges the original list of spinning differential operators. Finally we compute the seed conformal blocks in two different ways: by directly solving the Casimir equation and by using the shadow formalism augmented with group-theoretic properties of our new covariant differential operators

Weight Shifting Operators and Conformal Blocks

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for "seed conformal blocks" in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including derivation of recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/"crossing kernels") of the conformal group.Comment: 84 page

General Bootstrap Equations in 4D CFTs

We provide a framework for generic 4D conformal bootstrap computations. It is based on the unification of two independent approaches, the covariant (embedding) formalism and the non-covariant (conformal frame) formalism. We construct their main ingredients (tensor structures and differential operators) and establish a precise connection between them. We supplement the discussion by additional details like classification of tensor structures of n-point functions, normalization of 2-point functions and seed conformal blocks, Casimir differential operators and treatment of conserved operators and permutation symmetries. Finally, we implement our framework in a Mathematica package and make it freely available.Comment: 57 page

Bootstrapping the aa-anomaly in 4d4d QFTs

We study gapped 4d quantum field theories (QFTs) obtained from a relevant deformation of a UV conformal field theory (CFT). For simplicity, we assume the existence of a $\mathbb{Z}_2$ symmetry and a single $\mathbb{Z}_2$-odd stable particle and no $\mathbb{Z}_2$-even particles at low energies. Using unitarity, crossing and the assumption of maximal analyticity we compute numerically a lower bound on the value of the $a$-anomaly of the UV CFT as a function of various non-perturbative parameters describing the two-to-two scattering amplitude of the particle.Comment: 41 pages + appendices, 20 figure

General Three-Point Functions in 4D CFT

We classify and compute, by means of the six-dimensional embedding formalism in twistor space, all possible three-point functions in four dimensional conformal field theories involving bosonic or fermionic operators in irreducible representations of the Lorentz group. We show how to impose in this formalism constraints due to conservation of bosonic or fermionic currents. The number of independent tensor structures appearing in any three-point function is obtained by a simple counting. Using the Operator Product Expansion (OPE), we can then determine the number of structures appearing in 4-point functions with arbitrary operators. This procedure is independent of the way we take the OPE between pairs of operators, namely it is consistent with crossing symmetry, as it should be. An analytic formula for the number of tensor structures for three-point correlators with two symmetric and an arbitrary bosonic (non-conserved) operators is found, which in turn allows to analytically determine the number of structures in 4-point functions of symmetric traceless tensors