Bounds in 4D conformal field theories with global symmetry

By R. Rattazzi, S. Rychkov, A. Vichi. 2010

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We explore the constraining power of OPE associativity in 4D conformal field theory with a continuous global symmetry group. We give a general analysis of crossing symmetry constraints in the 4-point function ⟨ϕϕϕ†ϕ†⟩, where ϕ is a primary scalar operator in a given representation R. These constraints take the form of ‘vectorial sum rules’ for conformal blocks of operators whose representations appear in R⊗R and . The coefficients in these sum rules are related to the Fierz transformation matrices for the invariant tensors. We show that the number of equations is always equal to the number of symmetry channels to be constrained. We also analyze in detail two cases—the fundamental of SO(N) and the fundamental of SU(N). We derive the vectorial sum rules explicitly, and use them to study the dimension of the lowest singlet scalar in the ϕ × ϕ† OPE. We prove the existence of an upper bound on the dimension of this scalar. The bound depends on the conformal dimension of ϕ and approaches 2 in the limit For several small groups, we compute the behavior of the bound at . We discuss implications of our bound for the conformal technicolor scenario of electroweak symmetry breaking.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/05a4c81b6147631b3100e470df158b2ea011f8c1

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Citations in the corpus

Emergent spacetime and holographic CFTs (2011, methodology): “See [68], [69], [70], [71], [72], [73] for recent work were the unitarity of the CFT in d > 2 was used to derive certain constraints for the data ∆i, Cijk, in special cases.”
A universal inequality for CFT and quantum gravity (2009, unknown): “(Also see [16, 17, 18, 19, 20, 21, 22].)”
Spinning conformal blocks (2011, background): “The most general requirement in this case is that the whole matrix of N O × N O coeﬃcients be positive-deﬁnite; see an analogous discussion in section 2.5 of [5].”
Improved bounds for CFT’s with global symmetries (2011, methodology): “In this work we produced numerical bounds investigating the crossing symmetry constraints derived in [11] for Conformal Field Theories with global symmetries.”
Carving out OPE space and precise O(2) model critical exponents (2019, methodology): “In practice, this means we would like to study large systems of correlation functions involving multiple scalars [10–15], fermions [16–18], currents [19, 20], stress tensors [21], various global symmetry representations [22–42], etc.”
Projectors, shadows, and conformal blocks (2012, background): “Later, they were essential for improving these methods and deriving bounds with concrete phenomenological implications [16–18].”
On the CFT operator spectrum at large global charge (2015, methodology): “2 For progress on the simple 3D conformal theories using modern bootstrap methods, see for example [11–15].”
Fundamental composite dynamics: A review (2020, background): “The situation is however more complicated in cases with global symmetries [249], which are more relevant for composite Higgs models, as there are multiple symmetry channels appearing in the bootstrap calculation.”
The analytic bootstrap and AdS superhorizon locality (2012, background): “The last few years have seen a remarkable resurgence of interest in an old approach to Conformal Field Theory (CFT), the conformal bootstrap [1, 2], with a great deal of progress leading to new results of phenomenological [3–9] and theoretical [10–15] import.”
Carving out the space of 4D CFTs (2011, methodology): “An analysis of crossing relations in theories with SO(N) and SU(N) global symmetries was performed in [54], and improved bounds for SO(N) were presented in [55].”
Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$c-Minimization and Precise Critical Exponents (2012, background): “) This makes SDPA a good choice for studying theories with global symmetries, where we have different crossing relations for each tensor structure which can appear in a four-point function [55].”
Bootstrapping 3D fermions (2015, methodology): “A natural choice for the Gross-Neveu models would be to organize operators according to their O(N) representations and use the constraints of O(N) symmetry in the crossing equations, as in [9, 10, 61].”