How to compute the anomalous baryon number violation at high temperature

We present an effective classical theory for the soft modes of a high temperature non-Abelian plasma. This theory is formulated in local Hamiltonian form, and is well suited for numerical non-perturbative studies of the real-time dynamics. It includes the short-range dynamics in the hard thermal loop approximation, with appropriate corrections which take care of the cutoff effects in a gauge-invariant way. By using this theory for lattice calculations, one should be able to compute the rate of anomalous baryon number violation, thus solving a recent controversy. ∗Laboratoire de la Direction des Sciences de la Matière du Commissariat à l’Energie Atomique Motivated by several physical problems — chiefly among them, the calculation of the rate of hot anomalous baryon number violation [1] — there is currently a large interest in the non-perturbative real-time dynamics of soft fields in hot gauge theories. The non-perturbative phenomena are generally associated with long wavelength (λ >∼ 1/g T ) quasistatic (ω <∼ g T ) magnetic field configurations, the only ones which are not screened at the scale gT by collective phenomena. (T denotes the temperature, assumed to be large enough for the coupling constant g(T ) to be small: g ≪ 1.) Since Euclidean lattice gauge theory, which is the standard non-perturbative tool, is not well-suited for real-time calculations, much effort has been spent on numerical simulations of the classical Yang-Mills theory [2, 3, 4]. The justification for the classical approximation is that the relevant soft modes have large thermal occupation numbers and should therefore exhibit a classical behaviour. However, it has been recently recognized [5, 6] that the high-temperature baryon violation rate Γ could be sensitive to the hard thermal modes (with momenta ∼ T ), for which the classical description is not appropriate (recall, e.g., the Rayleigh-Jeans ultraviolet divergence). The hard modes cause the damping of the soft field configurations, an effect which is predicted to reduce Γ by a factor of g as compared with its classical estimate. In order to verify this prediction, and eventually compute Γ, one has to treat properly the hard, quantum, modes. Since the latter are perturbative, they can be explicitly integrated over to yield an effective theory for the soft modes, to be eventually used in classical numerical simulations. The effective theory obtained by integrating out the hard modes to leading order in g involves non-local corrections to the soft fields propagators and vertices, known as “hard thermal loops” (HTL) [7, 8, 9]. They encompass the damping phenomena alluded to before. It has been already proposed in Ref. [10] to use the HTL effective theory as a classical lattice theory for computing Γ (see also Refs. [6, 11]), but this raised some technical problems which remained unsolved up to now. The first problem refers to the gauge-