Rigorous limits on the interaction strength in quantum field theory

We derive model-independent, universal upper bounds on the operator product expansion coefficients in unitary 4-dimensional conformal field theories. The method uses the conformal block decomposition and the crossing symmetry constraint of the 4-point function. In particular, the operator product expansion coefficient of three identical dimension $d$ scalar primaries is found to be bounded by $\simeq 10(d-1)$ for $1<d<1.7$. This puts strong limits on unparticle self-interaction cross sections at the LHC.

Figure 1

Theoretical upper bound for $c_{\varphi \varphi O}$ as a function of the dimension $\overline{\Delta }$ of the scalar field $O$. The curves correspond to the $\varphi$’s dimension fixed at $d=1.005$, 1.02, 1.05, 1.1 (from the bottom up). The bound was computed for each of the shown points, with interpolation in between.

Figure 2

Same as Fig. 1 for the $\varphi$’s dimension fixed at $d=1.2$, 1.3, 1.4, 1.5, 1.6, 1.7 (from the bottom up).

Figure 3

Theoretical upper bound for the OPE coefficient $c_{\varphi \varphi \varphi }$ as a function of $\varphi$’s dimension $d$.