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Carving out the space of 4D CFTs

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Abstract

We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and \( \mathcal{N} = 1 \) superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In \( \mathcal{N} = 1 \) superconformal theories, we place strong bounds on dim(Φ†Φ), where Φ is a chiral operator. These bounds asymptote to the line dim(Φ†Φ) ≤ 2 dim(Φ) near dim(Φ) ≃ 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Φ × Φ OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of \( \mathcal{N} = 1 \) theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.

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Authors and Affiliations

  1. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, U.S.A.

    David Poland & David Simmons-Duffin

  2. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, U.S.A.

    David Poland

  3. Institut de Théorie des Phénoménes Physiques, EPFL, Lausanne, Switzerland

    Alessandro Vichi

  4. Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA, 94720, U.S.A.

    Alessandro Vichi

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  1. David Poland
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  2. David Simmons-Duffin
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Correspondence to David Poland.

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ArXiv ePrint: 1109.5176

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Poland, D., Simmons-Duffin, D. & Vichi, A. Carving out the space of 4D CFTs. J. High Energ. Phys. 2012, 110 (2012). https://doi.org/10.1007/JHEP05(2012)110

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