Causality, Analyticity and an IR Obstruction to UV Completion

TL;DR

This paper identifies positivity constraints on leading irrelevant operators as a diagnostic for UV completion within local QFTs and perturbative string theories. It connects these UV analyticity requirements to IR phenomena, showing that wrong signs induce macroscopic nonlocality and superluminal propagation on nontrivial backgrounds, which undermines a Lorentz-invariant causal evolution. The authors apply the framework to the DGP brane-world model and the electroweak chiral Lagrangian, arguing that both cannot arise from standard UV-complete theories if positivity is violated. They further explore string theory examples (Little String Theory, noncommutative theories) and discuss gravitational extensions, highlighting that any experimental evidence of positivity violation would signal physics beyond local QFT and perturbative string theory, with far-reaching implications for causality and locality.

Abstract

We argue that certain apparently consistent low-energy effective field theories described by local, Lorentz-invariant Lagrangians, secretly exhibit macroscopic non-locality and cannot be embedded in any UV theory whose S-matrix satisfies canonical analyticity constraints. The obstruction involves the signs of a set of leading irrelevant operators, which must be strictly positive to ensure UV analyticity. An IR manifestation of this restriction is that the "wrong" signs lead to superluminal fluctuations around non-trivial backgrounds, making it impossible to define local, causal evolution, and implying a surprising IR breakdown of the effective theory. Such effective theories can not arise in quantum field theories or weakly coupled string theories, whose S-matrices satisfy the usual analyticity properties. This conclusion applies to the DGP brane-world model modifying gravity in the IR, giving a simple explanation for the difficulty of embedding this model into controlled stringy backgrounds, and to models of electroweak symmetry breaking that predict negative anomalous quartic couplings for the W and Z. Conversely, any experimental support for the DGP model, or measured negative signs for anomalous quartic gauge boson couplings at future accelerators, would constitute direct evidence for the existence of superluminality and macroscopic non-locality unlike anything previously seen in physics, and almost incidentally falsify both local quantum field theory and perturbative string theory.

Figures (10)

• Figure 1: Bubbles of non-trivial vacua, $\pi = C_\mu x^\mu$, in our Goldstone model with $c_{3}<0$. (a) In the rest frame of the bubble, $C_{\mu}=(C,0,0,0)$. The solid lines denote the causal cone inside of which small fluctuations are constrained to propagate. (b) The same system in a boosted frame in which the bubble moves with a large velocity in the positive $x'$ direction. For sufficiently large boosts, the causal cone dips below horizontal, and small fluctuations are only seen to propagate to the left with a different temporal ordering than in the unboosted frame.
• Figure 2: Two finite bubbles moving with large opposite velocities in the $x$ direction and separated by a finite distance in the $y$ direction. The open cones indicate the local causal cones of $\pi$-fluctuations, and the red line the closed trajectory of a series of small fluctuations along these cones. Such closed time-like trajectories make it clear that no notion of causality or locality survives in a theory which violates positivity.
• Figure 3: The field between the plates of a charged capacitor coaxial with a current-carrying solenoid is of the form $\vec{E}=\frac{A}{r}\hat{r}$ and $\vec{B}=B\hat{z}$. When $c_{1}<0$, small fluctuations at fixed $r$ propagate superluminally. For sufficiently large fieldstrengths, but still within the regime of validity of the effective field theory, the "causal cone" of small fluctuations dips below horizontal, allowing for purely spacelike evolution all the way around the capacitor at fixed $t$, a dramatic violation of locality and causality.
• Figure 4: Propagation of a small fluctuation around a background represented as a sequence of scattering events.
• Figure 5: Analytic structure of the forward $2\to2$ scattering amplitude at tree level, in the theory of a Goldstone boson UV completed into a linear sigma model with Higgs mass $M_h^2$. The poles arise from tree-level Higgs exchange