Unitarity and microscopic acausality in a nonlocal theory

We consider unitarity and causality in a higher-derivative theory of infinite order, where propagators fall off more quickly in the ultraviolet due to the presence of a transcendental entire function of the momentum. Like Lee-Wick theories, these field theories might provide new avenues for addressing the hierarchy problem; unlike Lee-Wick theories, tree-level propagators do not have additional poles corresponding to unobserved particles with unusual properties. We consider microscopic acausality in these nonlocal theories. The acausal ordering of production and decay vertices for ordinary resonant particles may provide a phenomenologically distinct signature for these models.

Figure 1

Imaginary part of the self-energy functions $\mathrm{\Sigma }(s)$ as a function of $\sqrt{s}$. The open circular points indicate the results of the direct numerical evaluation of Eq. (2.9), for the cases where $\eta =0$ and $\eta =0.3$. The long dashed line gives the $\eta =0$ expectation, proportional to $(1-4m^2/s{)}^{1/2}$ for $s>4m^2$. The solid points are the result of Eqs. (2.29), after a numerical evaluation of the second term, as discussed in the text.

Figure 2

The commutator $C(\rho )=⟨0|[\varphi (x),\varphi (y)]|0⟩$, as a function of $\rho =(x^0-y^0{)}^2-|\overrightarrow{x}-\overrightarrow{y}{|}^2$ at spacelike separation $\rho <0$, for $m=0$ and $\eta =1$.

Figure 3

Time dependence of the smeared source function at the origin $j(t{)}_s\equiv j(y^0=t,\overrightarrow{y}=\overrightarrow{0}{)}_s$ for $\eta =1$ and $c_0=1$.

Figure 4

Contours discussed in the Appendix for studying the reality properties of $\mathrm{\Sigma }(p^0)$ for real $p^0<2m$ and $\overrightarrow{p}=0$. The contour for $I_C$ terminates on the $I_1$ line.