Hard thermal loops and the sphaleron rate on the lattice

Abstract

We measure the sphaleron rate (topological susceptibility) of hot SU(2) gauge theory, using a lattice implementation of the hard thermal loop (HTL) effective action. The HTL degrees of freedom are implemented by an expansion in spherical harmonics and truncation. Our results for the sphaleron rate agree with the parametric prediction of Arnold, Son and Yaffe: Γ ∝ α 5 T 4 . MOTIVATIONBaryon number is not a conserved quantity in the Standard Model: due to the anomaly, the violation is related to the (Minkowski time) topological susceptibility of the SU(2) weak group. While at low temperatures the violation is totally negligible [1], at temperatures above the electroweak symmetry restoration temperature (∼ 100 GeV) the rate of the baryon number violation (sphaleron rate) Γ is large. This can have significant repercussions for baryon number generation in the early Universe, and it opens the avenue for purely electroweak baryogenesis.Even though the weak coupling constant is small, at high temperatures the sphaleron processes are dominated by IR momenta k ∼ g 2 T and are thus inherently non-perturbative. Moreover, the IR modes behave essentially classically, which is signalled, for example, by the large occupation numbers of the Bose fields:. This has motivated the much utilized method of using the classical equations of motion to calculate Γ in hot SU(2) theories [2] (the Higgs and fermionic degrees of freedom effectively decouple in the hot EW phase). For recent reviews, see [3], [4].The success of the classical method hinges on the efficient decoupling of the almost-classical IR modes relevant for the sphaleron processes and * Presented by K. Rummukainen at the conference LAT-TICE '99, Pisa, Italy, July 1999.the strongly non-classical UV modes. However, as argued by Arnold, Son and Yaffe [5], this decoupling is not complete. A step beyond the classical approximation is the hard thermal loop (HTL) effective theory [6], which incorporates the leading order effects of the UV modes. The HTL theory can be cast in various forms; most practical for lattice computations is the one where the the hard modes are described by including a large number of classical massless particles with adjoint charge moving on the background of IR fields. This field + particles system can be put on a lattice as such, and it has been succesfully used in simulations [7]. In this work we use an alternative Boltzmann-Vlasov approach, where the particles are described with local density functions n(t, x, k). For full description, see [8]. HTL THEORY ON THE LATTICELet us consider a system consisting of the HTL particles moving on the background of IR gauge fields. The particle density functions n(t, x, k) obey the Vlasov equationwhere n 0 = (e k/T − 1) −1 , n = n 0 + δn a , and the Lorentz-force ∝ v × B has been neglected. The IR gauge fields evolve according to the Yang-Mills

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“…The time scale necessary to achieve a sphaleron transition turns out not to be 1 g 2 T as previously argued [64], but a longer time scale 1 g 4 T , which accounts for the "Landau damping" of the transition time due to the interaction of the soft magnetic modes with hard modes on the scale of the temperature. This parametric scaling has also been confirmed independently by numerical simulations [15,[71][72][73] which extract [15]…”

Chiral anomaly in SU(2)R-axion inflation and the new prediction for particle cosmology

“…4π . Using lattice simulations the transition rate for the SU(2) L weak sphalerons has been found in [72] as…”