Abstract
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 densityK=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.
Highlights
Real-time evolution of classical fields has many applications in different areas of high energy physics and cosmology
It does not seem possible to find a set of discrete equations reproducing to order O(dx2) the dynamics of an Abelian gauge theory in the presence of a general axion-like field a(x) = a(t, x), and at the same time being solvable by an explicit scheme
As promised, a lattice expression for the Pontryagin density KL Eq (66), that admits a total derivative representation as in Eq (73), with KL0 and KLi given by Eqs. (67), (72), respectively
Summary
Real-time evolution of classical fields has many applications in different areas of high energy physics and cosmology. The statement that an implementation of K respecting the topological property Eq (2) can be done exactly in Abelian lattice gauge theories goes back to the works of Guy Moore [52,53], where the motion of Chern–Simons number in SU(2) and SU(3) gauge at high temperatures under a chemical potential was investigated, significantly improving earlier studies [12]. Let us note that due to the topological nature of FμνFμν , action Eq (3) is (‘topologically’) invariant under a(x) → a(x) + C, with C an arbitrary constant This is reflected in the fact that the linear coupling of a(x) to the Pontryagin density, d4x a(x) FμνFμν , represents a derivative coupling: the total derivative nature of FμνFμν = ∂μKμ, after integration by parts, gives d4x Kμ∂μa(x).
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