Abstract
The time-evolution of the magnetic field in hot homogeneous nuclear matter has two qualitatively different stages separated by the sphaleron transition time ${\ensuremath{\tau}}_{c}$. At early times, when the chiral conductivity ${\ensuremath{\sigma}}_{\ensuremath{\chi}}$ is a slow function of time, the soft chiral modes $k<{\ensuremath{\sigma}}_{\ensuremath{\chi}}$ of the magnetic field grow exponentially with time, which is known as the chiral instability. At later times ${\ensuremath{\sigma}}_{\ensuremath{\chi}}$ fluctuates due to the sphaleron transitions and can be regarded as a random process. It is argued that the average magnetic field is exponentially damped at later times. The time-evolution of the average field energy is more complicated and depends on the electrical conductivity of the chiral matter but does not depend on chirality. It exhibits instability only if the matter is a poor electrical conductor, such as the quark-gluon plasma near the critical temperature. The precise conditions for the instability and the growth rate of the unstable modes are derived.
Highlights
The topological configurations of gluon fields form P and T -odd domains in hot nuclear matter
The dynamics of the electromagnetic field in the spatially uniform chiral matter with electrical conductivity σ can be described by the vector potential A which satisfies the following equation in the radiation gauge
This paper extends analysis of the chiral instability to systems whose topological charge fluctuations are stochastic in the long time limit
Summary
The topological configurations of gluon fields form P and T -odd domains in hot nuclear matter. The second and crucial assumption is the slow variation of the chiral conductivity It holds while t < τc where τc is the sphaleron transition time, i.e. the transition time between the gauge field configurations of different topological charge. In order to study the magnetic field instability in the hot nuclear matter, one is required to examine the opposite limit of t τc, when many sphaleron transition can occur causing fluctuations of the topological charge of the chiral domains. This paper extends analysis of the chiral instability to systems whose topological charge fluctuations are stochastic in the long time limit This approach is adequate for hot nuclear matter since the topological charge is determined mostly by the strong interactions, but may not universally hold in other chiral systems.
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