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Bounding scalar operator dimensions in 4D CFT

Riccardo Rattazzi, Vyacheslav S. Rychkov, Erik Tonni, Alessandro Vichi

TL;DR

The paper derives a model-independent bound on the first scalar in the $\phi\times\phi$ OPE, showing $[\phi^2]\le f([\phi])$ with $f$ computed numerically and satisfying $f(1)=2$. Using crossing symmetry, OPE structure, and conformal blocks, the authors formulate a sum rule whose positivity constraints define a convex cone; by a contradiction argument they bound the scalar content of the OPE and extract $f(d)$. The bound is tested against known 4D fixed points (where it holds) and 2D CFTs (where the Ising model nearly saturates it), while Wilson–Fisher in $4-\varepsilon$ shows deviations likely due to extrapolation subtleties. The results illuminate how conformal bootstrap-like constraints can inform phenomenology, particularly naturalness and flavor in dynamical EWSB scenarios, and lay out a concrete program to tighten the bounds, extend to other dimensions, and incorporate global symmetries.

Abstract

In an arbitrary unitary 4D CFT we consider a scalar operator φ, and the operator φ^2 defined as the lowest dimension scalar which appears in the OPE φ\timesφwith a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theory-independent inequality [φ^2] \leq f([φ]) for the dimensions of these two operators. The function f(d) entering this bound is computed numerically. For d->1 we have f(d)=2+O(\sqrt{d-1}), which shows that the free theory limit is approached continuously. We perform some checks of our bound. We find that the bound is satisfied by all weakly coupled 4D conformal fixed points that we are able to construct. The Wilson-Fischer fixed points violate the bound by a constant O(1) factor, which must be due to the subtleties of extrapolating to 4-εdimensions. We use our method to derive an analogous bound in 2D, and check that the Minimal Models satisfy the bound, with the Ising model nearly-saturating it. Derivation of an analogous bound in 3D is currently not feasible because the explicit conformal blocks are not known in odd dimensions. We also discuss the main phenomenological motivation for studying this set of questions: constructing models of dynamical ElectroWeak Symmetry Breaking without flavor problems.

Bounding scalar operator dimensions in 4D CFT

TL;DR

The paper derives a model-independent bound on the first scalar in the OPE, showing with computed numerically and satisfying . Using crossing symmetry, OPE structure, and conformal blocks, the authors formulate a sum rule whose positivity constraints define a convex cone; by a contradiction argument they bound the scalar content of the OPE and extract . The bound is tested against known 4D fixed points (where it holds) and 2D CFTs (where the Ising model nearly saturates it), while Wilson–Fisher in shows deviations likely due to extrapolation subtleties. The results illuminate how conformal bootstrap-like constraints can inform phenomenology, particularly naturalness and flavor in dynamical EWSB scenarios, and lay out a concrete program to tighten the bounds, extend to other dimensions, and incorporate global symmetries.

Abstract

In an arbitrary unitary 4D CFT we consider a scalar operator φ, and the operator φ^2 defined as the lowest dimension scalar which appears in the OPE φ\timesφwith a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theory-independent inequality [φ^2] \leq f([φ]) for the dimensions of these two operators. The function f(d) entering this bound is computed numerically. For d->1 we have f(d)=2+O(\sqrt{d-1}), which shows that the free theory limit is approached continuously. We perform some checks of our bound. We find that the bound is satisfied by all weakly coupled 4D conformal fixed points that we are able to construct. The Wilson-Fischer fixed points violate the bound by a constant O(1) factor, which must be due to the subtleties of extrapolating to 4-εdimensions. We use our method to derive an analogous bound in 2D, and check that the Minimal Models satisfy the bound, with the Ising model nearly-saturating it. Derivation of an analogous bound in 3D is currently not feasible because the explicit conformal blocks are not known in odd dimensions. We also discuss the main phenomenological motivation for studying this set of questions: constructing models of dynamical ElectroWeak Symmetry Breaking without flavor problems.

Paper Structure

This paper contains 25 sections, 132 equations, 17 figures, 1 table.

Table of Contents

  1. The problem and the result
  2. Phenomenological motivation
  3. Quantitative analysis
  4. Necessary CFT techniques
  5. Primary fields and unitarity bounds
  6. Correlation functions
  7. Operator Product Expansion
  8. Conformal blocks
  9. Crossing symmetry and the sum rule
  10. The sum rule in the free scalar theory
  11. Main results
  12. Why is the bound at all possible?
  13. Geometry of the sum rule
  14. Warmup example: $d=1$
  15. Simplest bound satisfying $f(1)=2$
  16. ...and 10 more sections

Figures (17)

  • Figure 1: The best current bound (\ref{['main']}), obtained by the method described in Section \ref{['results']}. The subscript in $f_{6}$ refers to the order of derivatives used to compute this bound.
  • Figure 2: The “ spacelike diamond" (\ref{['eq:diamond']}) in which the conformal blocks are real and regular, see the text.
  • Figure 3: The RHS of the sum rule in the free scalar theory, summed over $l\leq0,2,4,8,16$ (from below up) and plotted for $0\leq z=\bar{z}\leq1.$ The asymptotic approach to $1$ (dashed line) is evident. Notice the symmetry with respect to $z=1/2$, a consequence of (\ref{['inv']}).
  • Figure 4: The shape of $F_{d,\Delta,l}$ for $d=1$, $l=2,4$ and several values of $\Delta$ satisfying the unitarity bound.
  • Figure 5: Same as Fig. \ref{['fig:expl']}, for $l=0.$
  • ...and 12 more figures