Bounds in 4D conformal field theories with global symmetry
TLDR
In this article, a general analysis of crossing symmetry constraints in 4D conformal field theory with a continuous global symmetry group is given, where phi is a primary scalar operator in a given representation R and the coefficients in these sum rules are related to the Fierz transformation matrices for the R circle times R over bar circle times (R) over bar invariant tensors.Abstract:
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 , where phi 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 circle times R and R circle times (R) over bar. The coefficients in these sum rules are related to the Fierz transformation matrices for the R circle times R circle times (R) over bar circle times (R) over bar 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 phi x phi(dagger) OPE. We prove the existence of an upper bound on the dimension of this scalar. The bound depends on the conformal dimension of phi and approaches 2 in the limit dim(phi) -> 1. For several small groups, we compute the behavior of the bound at dim(phi) > 1. We discuss implications of our bound for the conformal technicolor scenario of electroweak symmetry breaking.read more
Citations
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Solving the 3D Ising Model with the Conformal Bootstrap
Sheer El-Showk,Miguel F. Paulos,David Poland,Slava Rychkov,David Simmons-Duffin,Alessandro Vichi +5 more
TL;DR: In this article, the constraints of crossing symmetry and unitarity in general 3D conformal field theories were studied, and it was shown that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space.
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The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
TL;DR: Conformal field theories have been long known to describe the universal physics of scale invariant critical points as discussed by the authors, and they describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory.
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The analytic bootstrap and AdS superhorizon locality
TL;DR: In this article, it was shown that every CFT with a scalar operator ϕ must contain infinite sequences of operators with twist approaching τ → 2Δ + 2n for each integer n as l → ∞.
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Spinning conformal correlators
TL;DR: The embedding formalism for conformal field theories is developed, aimed at doing computations with symmetric traceless operators of arbitrary spin, using an indexfree notation where tensors are encoded by polynomials in auxiliary polarization vectors.
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Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
Sheer El-Showk,Miguel F. Paulos,David Poland,Slava Rychkov,Slava Rychkov,David Simmons-Duffin,Alessandro Vichi +6 more
TL;DR: In this article, a conformal bootstrap was used to perform a precision study of the operator spectrum of the critical 3D Ising model, and it was shown that the 3d Ising spectrum minimizes the central charge c in the space of unitary solutions to crossing symmetry.
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