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)FμνF˜μν , reproducing the continuum limit to order O(dxμ2) 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=FμνF˜μν that admits a lattice total derivative representation K=Δμ+Kμ , reproducing to order O(dxμ2) the continuum expression K=∂μKμ∝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(dxμ2) accuracy). We discuss an iterative scheme allowing to overcome this difficulty.