Does scale invariance imply conformal invariance in four dimensions
In plain words
A theory that looks the same at all magnifications usually has a larger symmetry, conformal symmetry. Whether this always holds in four dimensions is not proven.
Precise statement
For a unitary, Poincare-invariant 4D QFT with a well-defined stress tensor, determine whether scale invariance implies conformal invariance nonperturbatively. It is proven in 2D (Polchinski 1988), shown to all orders of perturbation theory in 4D (Luty, Polchinski, Rattazzi; Fortin, Grinstein, Stergiou, 2012), and argued nonperturbatively under assumptions (Dymarsky, Komargodski, Schwimmer, Theisen 2013).
What would settle it
A nonperturbative argument without extra assumptions, or a unitary scale-invariant but non-conformal 4D theory.
Status in the literature
Proven to all orders in perturbation theory (2012) and nonperturbatively under extra assumptions (2013); no general proof.