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-mu nu(F) over tilde (mu nu), reproducing the continuum limit to order O(dx(mu)(2)) 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-mu nu(F) over tilde (mu nu) that admits a lattice total derivative representation K = Delta(+)(mu) K-mu, reproducing to order O(dx(mu)(2)) the continuum expression K = partial derivative K-mu(mu) proportional to (E) over right arrow. (B) over right arrow. 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) over right arrow, t) is inhomogeneous, the set of lattice equations of motion do not admit however a simple explicit local solution (while preserving an O(dx(mu)(2)) accuracy). We discuss an iterative scheme allowing to overcome this difficulty. (C) 2017 The Authors. Published by Elsevier B.V.