Central Charge Bounds in 4D Conformal Field Theory

TL;DR

The work derives universal, model-independent lower bounds on the 4D CFT stress-tensor central charge $C_T$ as a function of the dimensions of the lightest scalar operators. It uses crossing symmetry of the four-point function, conformal blocks, and the linear-functional (bootstrap) method to bound the stress-tensor OPE coefficient $c_{4,2}$, which translates into a bound on $C_T$. The bounds show $C_T\to 4/3$ as the lowest scalar dimension $d\to1$ and become stronger with a larger scalar gap $\Delta_* - d$, mapping how operator content constrains degrees of freedom. This provides a quantitative framework for constraining the landscape of 4D CFTs and demonstrates the power of bootstrap techniques for central-charge bounds.

Abstract

We derive model-independent lower bounds on the stress tensor central charge C_T in terms of the operator content of a 4-dimensional Conformal Field Theory. More precisely, C_T is bounded from below by a universal function of the dimensions of the lowest and second-lowest scalars present in the CFT. The method uses the crossing symmetry constraint of the 4-point function, analyzed by means of the conformal block decomposition.

Figures (6)

• Figure 1: Using conformal freedom, any configuration of 4 points can be mapped into the one shown in this figure, in which 3 points are fixed and one ($x_{2}$) is moving in a two-plane passing through $x_{1}$ and $x_{3}$. The complex coordinate of $x_{2}$ in this plane is precisely the $z$ of (\ref{['eq:confblock']}), while $\bar{z}=z^{\ast}$. The conformal blocks are smooth everywhere in the plane except for $z=0$ and the shown $(1,+\infty)$ cut along the real axis.
• Figure 2: This configuration, with 4 points at the vertices of a square, is conformally equivalent to the one in Fig. \ref{['fig:fig1']} with $z=1/2$.
• Figure 3: Geometric interpretation of the sum rule: (a) the sum rule has a solution $\Leftrightarrow$$f\equiv1$ belongs to the cone; (b) the assumed spectrum is such that the sum rule does not allow for a solution $\Leftrightarrow$$f\equiv1$ does not belong to the cone; (c) in the latter situation, a hyperplane (the zero set of a linear functional) can be found separating $f\equiv1$ from the cone.
• Figure 4: Geometric interpretation of Eq. (\ref{['eq:sumrule1']}). As $t$ increases, the vector $1-t\,F_{d,4,2}$ eventually leaves the cone.
• Figure 5: The lower bound on the central charge $C_{T}$ in terms of the dimension $d$ of the lowest-dimension scalar primary. The stronger bound (upper blue curve) is obtained with $N=16$. For comparison we give a weaker bound obtained with $N=12$ (lower red curve), which corresponds to the horizontal axis $\Delta_{*}=d$ in the following Fig. \ref{['fig:ct(d,delta)']}. The horizontal dashed line $C_{T}=4/3$ shows where our bound stays above the free scalar central charge.