An obstruction to positivity bounds with spontaneously broken Lorentz invariance

Positivity bounds are a set of consistency conditions that EFTs shall obey in order to admit a sensible UV completion. Generically, they arise as inequalities for the Wilson coefficients of the low-energy theory and, in the Lorentz invariant framework, they are extracted exploiting the analyticity properties of the S-Matrix. 
I will show how the analytic structure of the S-Matrix gets deeply modified by the presence of a spontaneous breaking of Lorentz invariance. I will discuss the results of (JHEP 06 (2024) 201), where we studied the S-matrix of Goldstones in the renormalizable theory of a U(1) complex scalar at finite charge, i.e. in a state that breaks Lorentz invariance. The theory is weakly coupled so that this S-matrix exists at all energies. 
Unlike the Lorentz invariant case, the resulting S-matrix is not analytic in the exchanged (complexified) four-momentum. The non-analyticities stem from the LSZ reduction formula, as a consequence of the energy-dependent mixing between the radial and Goldstone modes.  
I will comment on the implications for the formulation of positivity bounds in a cosmological/condensed matter setting, including some possible future directions.