Abstract: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.

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 deriving 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.

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weightshifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and "seed" operators in 4d, and MFT for currents and stress-tensors in 3d.Mean Field Theory (MFT) provides some of the simplest examples of crossing-symmetric, conformally-invariant correlation functions. Correlators in MFT are simply sums of products of two-point functions. In theories exhibiting large-N factorization, MFT is the leading contribution at large-N . For example, in AdS/CFT, MFT is the leading contribution to correlators in bulk perturbation theory [1][2][3]. In the analytic conformal bootstrap, MFT is the leading contribution to correlators at large spin [4][5][6][7][8][9][10]. MFT provides crucial example data for the numerical bootstrap [11], especially for spinning operators [12][13][14][15][16]. Furthermore, MFT OPE coefficients form the "ladder kernel" in SYK-like models [17][18][19][20][21][22][23]. Consequently, the OPE data of MFT (i.e. the scaling dimensions and OPE coefficients) is the starting point for many computations.Although correlators in MFT are simple, the OPE data can be nontrivial. OPE coefficients for a four-point function of fundamental scalars in MFT in 2-and 4-dimensions were guessed in [24]. 1 They were subsequently generalized to d-dimensions in [25] using a technique dubbed "conglomeration." In this work, we point out that conglomeration is part of a general toolkit of harmonic analysis for the Euclidean conformal group SO(d + 1, 1) [26]. Although harmonic analysis was first applied to CFTs in the 70's, it has played an especially important role in recent developments [17][18][19][20][21][22][23][27][28][29][30][31]. In section 2, we give an introduction to harmonic analysis for (Euclidean) CFTs.The calculation of [25] can be rephrased in terms of simple ingredients: the Plancherel measure, three-point pairings, and the "shadow transform" [32,33]. In particular, the computation of MFT OPE coefficients factorizes into two independent shadow transforms of threepoint functions, which are essentially generalizations of the famous "star-triangle" integral [34,35]. Using these observations, we write a general formula for OPE data of fundamental MFT fields in arbitrary Lorentz representations in section 2.8. Along the way, we derive orthogonality relations for conformal partial waves with arbitrary (internal and external) Lorentz representations.Our derivation essentially uses a "Euclidean inversion formula" -a formula that expresses OPE data as an integral of a four-point function over Euclidean space. MFT OPE data can also in principle be computed by applying the Lorentzian i...

We compute in closed analytical form the minimal set of "seed" conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation ( ,¯ ) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0, | −¯ |) and one (| −¯ |, 0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any ( ,¯ ), by using an educated ansatz and reducing the problem to an algebraic linear system. Various details on the form of the ansatz have been deduced by using the so called shadow formalism. The complexity of the conformal blocks depends on the value of p = | −¯ | and grows with p, in analogy to what happens to scalar conformal blocks in d even space-time dimensions as d increases. These results open the way to bootstrap 4-point functions involving arbitrary spinor/tensor operators in four dimensional conformal field theories.

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.