3 years ago

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, $a ( x ) F μ ν F ˜ μ ν$, reproducing the continuum limit to order $O ( d x μ 2 )$ 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 $K = F μ ν F ˜ μ ν$ that admits a lattice total derivative representation $K = Δ μ + K μ$, reproducing to order $O ( d x μ 2 )$ the continuum expression $K = ∂ μ K μ ∝ E → ⋅ B →$. 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 $O ( d x μ 2 )$ accuracy). We discuss an iterative scheme allowing to overcome this difficulty.

Publisher URL: www.sciencedirect.com/science

DOI: S0550321317303863

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