Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fermion Conformal Bootstrap in 4d

Denis Karateev, Petr Kravchuk, Marco Serone, Alessandro Vichi

Abstract

We apply numerical conformal bootstrap techniques to the four-point function of a Weyl spinor in 4d non-supersymmetric CFTs. We find universal bounds on operator dimensions and OPE coefficients, including bounds on operators in mixed symmetry representations of the Lorentz group, which were inaccessible in previous bootstrap studies. We find discontinuities in some of the bounds on operator dimensions, and we show that they arise due to a generic yet previously unobserved fake primary effect, which is related to the existence of poles in conformal blocks. We show that this effect is also responsible for similar discontinuities found in four-fermion bootstrap in 3d, as well as in the mixed-correlator analysis of the 3d Ising CFT. As an important byproduct of our work, we develop a practical technology for numerical approximation of general 4d conformal blocks.

Fermion Conformal Bootstrap in 4d

Abstract

We apply numerical conformal bootstrap techniques to the four-point function of a Weyl spinor in 4d non-supersymmetric CFTs. We find universal bounds on operator dimensions and OPE coefficients, including bounds on operators in mixed symmetry representations of the Lorentz group, which were inaccessible in previous bootstrap studies. We find discontinuities in some of the bounds on operator dimensions, and we show that they arise due to a generic yet previously unobserved fake primary effect, which is related to the existence of poles in conformal blocks. We show that this effect is also responsible for similar discontinuities found in four-fermion bootstrap in 3d, as well as in the mixed-correlator analysis of the 3d Ising CFT. As an important byproduct of our work, we develop a practical technology for numerical approximation of general 4d conformal blocks.

Paper Structure

This paper contains 51 sections, 224 equations, 24 figures, 1 table.

Table of Contents

  1. Introduction and summary of results
  2. Setup
  3. Two- and three-point functions
  4. Neutral channel
  5. Charged channel
  6. Four-point tensor structures
  7. Ordering $\langle\overline{\psi}\psi\overline{\psi}\psi\rangle$
  8. Orderings $\langle\overline{\psi}\psi\psi\overline{\psi}\rangle$ and $\langle\psi\psi\overline{\psi}\overline{\psi}\rangle$
  9. Crossing equations
  10. Decomposition into conformal blocks
  11. Orderings $\langle\overline{\psi}\psi\overline{\psi}\psi\rangle$ and $\langle\overline{\psi}\psi\psi\overline{\psi}\rangle$.
  12. Ordering $\langle\psi\psi\overline{\psi}\overline{\psi}\rangle$.
  13. Semidefinite problems
  14. Bounds on the spectrum
  15. Bounds on OPE coefficients
  16. ...and 36 more sections

Figures (24)

  • Figure 1: Naive topology of $\Sigma$ in charge $q=0$ sector.
  • Figure 2: Merging of scalar and vector lines due to a pole in vector blocks. Scaling dimensions indicate dimensions of the scalar blocks.
  • Figure 3: Topology of $\Sigma$ in neutral sector after taking into account poles. The dimensions shown near intersections correspond to the block which appears as the residue.
  • Figure 4: Topology of $\Sigma$ in charged sector after taking into account poles. The dimensions shown near intersections correspond to the block which appears as the residue.
  • Figure 5: Topology of $\Sigma$ near charged scalar sector with a gap. Left: the gap in scalar sector is less than $4$. Right: the gap in scalar sector is greater than $4$.
  • ...and 19 more figures