Finite-temperature operator basis on $\mathbb{R}^3 \times S^1$ for SMEFT

We present the first complete non-redundant operator basis for the Standard Model Effective Field Theory (SMEFT) at finite temperature, using the imaginary-time formalism. By employing the Hilbert series method on the space-time manifold $\mathbb{R}^3 \times S^1$, we classify all effective operators up to dimension-six. In constructing the basis, we consistently impose integration-by-parts and equations-of-motion constraints along spatial directions. We further analyze the impact of additional constraints, including the vanishing of the curl of the electric and magnetic fields and gauge choices for the temporal components on an operator basis. We also express them in terms of static three-dimensional spatial and zero-temperature SMEFT operators. At dimension five and six, we identify intrinsically thermal operators that vanish in zero temperature. Our framework is fully general and extends to arbitrary mass dimension and compact connected internal symmetry groups.