Lattice implementation of Abelian gauge theories with Chern-Simons number and an axion field

Real time evolution of classical gauge fields is relevant for a number of applications in particle physics and cosmology, ranging from the early Universe to dynamics of quark-gluon plasma. We present an explicit non-compact lattice formulation of the interaction between a shift-symmetric field and some U (1) gauge sector, a (x)F_μνF^∼μν, reproducing the continuum limit to order O (dx_μ²) and obeying the following properties: (i) the system is gauge invariant and (ii) shift symmetry is exact on the lattice. For this end we construct a definition of the topological number density K =F_μνF^∼μν that admits a lattice total derivative representation K = ∆_μ⁺ K^μ, reproducing to order O (dx_μ²) the continuum expression K =∂_μK^μ ∝ E → ⋅ B → . If we consider a homogeneous field a (x) = a (t), the system can be mapped into an Abelian gauge theory with Hamiltonian containing a Chern-Simons term for the gauge fields. This allow us to study in an accompanying paper the real time dynamics of fermion number non-conservation (or chirality breaking) in Abelian gauge theories at finite temperature. When a (x) = a (x → , t) is inhomogeneous, the set of lattice equations of motion do not admit however a simple explicit local solution (while preserving an O (dx_μ²) accuracy). We discuss an iterative scheme allowing to overcome this difficulty.