Dietrich Bödeker

Effective dynamics of soft non-Abelian gauge fields at finite temperature

We consider time dependent correlation functions of non-Abelian gauge fields at finite temperature. An effective theory for the soft (p ∼ g 2 T) field modes is derived by integrating out the field modes with momenta of order T and of order gT in a leading logarithmic approximation. In this effective theory the time evolution of the soft fields is determined by a local Langevin-type equation. As an application, the rate for hot electroweak baryon number violation is estimated as Γ ∼ g 2 log(1/g)(g 2 T) 4. Furthermore , possible consequences for non-perturbative lattice computations of unequal time correlation functions are discussed.Abstract We consider time dependent correlation functions of non-Abelian gauge fields at finite temperature. An effective theory for the soft (p∼g2T) field modes is derived by integrating out the field modes with momenta of order T and of order gT in a leading logarithmic approximation. In this effective theory the time evolution of the soft fields is determined by a local Langevin-type equation. As an application, the rate for hot electroweak baryon number violation is estimated as Γ∼g2log(1/g)(g2T)4. Furthermore, possible consequences for non-perturbative lattice computations of unequal time correlation functions are discussed.

Chern-Simons number diffusion and hard thermal loops on the lattice

We develop a discrete lattice implementation of the hard thermal loop effective action by the method of added auxiliary fields. We use the resulting model to measure the sphaleron rate (topological susceptibility) of Yang-Mills theory at weak coupling. Our results give parametric behavior in accord with the arguments of Arnold, Son, and Yaffe, and are in quantitative agreement with the results of Moors, Hu, and Muller.

Magnetic fields produced by phase transition bubbles in the electroweak phase transition.

The electroweak phase transition, if proceeding through nucleation and growth of bubbles, should generate large scale turbulent flow, which in turn generates magnetic turbulence and hence magnetic fields on the scale of turbulence flow. We discuss the seeding of this turbulent field by the motion of the dipole charge layers in the phase transition bubble walls, and estimate the strength of the produced fields. {copyright} {ital 1996 The American Physical Society.}

The 2-loop MSSM finite temperature effective potential with stop condensation

We calculate the finite temperature 2-loop effective potential in the MSSM with stop condensation, using a three-dimensional effective theory. We find that in a part of the parameter space, a two-stage electroweak phase transition appears possible. The first stage would be the formation of a stop condensate, and the second stage is the transition to the standard electroweak minimum. The two-stage transition could significantly relax the baryon erasure bounds, but the parameter space allowing it (mH ⪅ 100 GeV, mtR ∼ 155–160 GeV) is not very large. We estimate the reliability of our results using renormalization scale and gauge dependence. Finally we discuss some real-time aspects relevant for the viability of the two-stage scenario.

Really computing nonperturbative real time correlation functions

It has been argued by Grigoriev and Rubakov that one can simulate real time processes involving baryon number nonconservation at high temperature using real time evolution of classical equations, and summing over initial conditions with a classical thermal weight. It is known that such a naive algorithm is plagued by ultraviolet divergences. In quantum theory the divergences are regularized, but the corresponding graphs involve the contributions from the hard momentum region and also the new scale {similar_to}{ital gT} comes into play. We propose a modified algorithm which involves solving the classical equations of motion for the effective hard thermal loop Hamiltonian with an ultraviolet cutoff {mu}{much_gt}{ital gT} and integrating over initial conditions with a proper thermal weight. Such an algorithm should provide a determination of the infrared behavior of the real time correlation function {l_angle}{ital Q}({ital t}){ital Q}(0){r_angle}{sub {ital T}} determining the baryon violation rate. Hopefully, the results obtained in this modified algorithm will be cutoff independent.

The Baryon asymmetry in the Standard Model with a low cut-off

We study the generation of the baryon asymmetry in a variant of the standard model, where the Higgs field is stabilized by a dimension-six interaction. Analyzing the one-loop potential, we find a strong first order electroweak phase transition for Higgs masses up to at least 170 GeV. Dimension-six operators induce also new sources of CP violation. We compute the baryon asymmetry in the WKB approximation. Novel source terms in the transport equations enhance the generated baryon asymmetry. For a wide range of parameters the model predicts a baryon asymmetry close to the observed value.

Thermal production of relativistic Majorana neutrinos: strong enhancement by multiple soft scattering

The production rate of heavy Majorana neutrinos is relevant for models of thermal leptogenesis in the early Universe. In the high temperature limit the production can proceed via the 1↔2 (inverse) decays which are allowed by the thermal masses. We consider new production mechanisms which are obtained by including additional soft gauge interactions with the plasma. We show that an arbitrary number of such interactions gives leading order contributions, and we sum all of them. The rate turns out to be smooth in the region where the 1↔2 processes are kinematically forbidden. At higher temperature it is enhanced by a factor 3 compared to the 1↔2 rate.

Thermal production of ultrarelativistic right-handed neutrinos: Complete leading-order results

The thermal production of relativistic right-handed Majorana neutrinos is of importance for models of thermal leptogenesis in the early Universe. Right-handed neutrinos can be produced both by 1↔2 decay or inverse decay and by 2 → 2 scattering processes. In a previous publication we have studied the production via 1↔2 (inverse) decay processes. There we have shown that multiple scattering mediated by soft gauge boson exchange also contributes to the production rate at leading order and gives a strong enhancement. Here we complete the leading order calculation by adding 2 → 2 scattering processes involving either electroweak gauge bosons or third-generation quarks. We find that processes with gauge interactions give the most important contributions. We also obtain a new sum rule for the Hard Thermal Loop resummed fermion propagator.

FROM HARD THERMAL LOOPS TO LANGEVIN DYNAMICS

In hot non-Abelian gauge theories, processes characterized by the momentum scale g2T (such as electroweak baryon number violation in the very early universe) are non-perturbative. An effective theory for the soft (|p| ∼ g2T ) field modes is obtained by integrating out momenta larger than g2T . Starting from the hard thermal loop effective theory, which is the result of integrating out the scale T , it is shown how to integrate out the scale gT in an expansion in the gauge coupling g. At leading order in g, one obtains Vlasov-Boltzmann equations for the soft field modes, which contain a Gaussian noise and a collision term. The 2-point function of the noise and the collision term are explicitly calculated in a leading logarithmic approximation. In this approximation the Boltzmann equation is solved. The resulting effective theory for the soft field modes is described by a Langevin equation. It determines the parametric form of the hot baryon number violation rate as Γ = κg10 log(1/g)T 4, and it allows for a calculation of κ on the lattice.In hot non-Abelian gauge theories, processes characterized by the momentum scale g2T (such as electroweak baryon number violation in the very early universe) are non-perturbative. An effective theory for the soft (|p| ∼ g2T) field modes is obtained by integrating out momenta larger than than g2T. Starting from the hard thermal loop effective theory, which is the result of integrating out the scale T, it is shown how to integrate out the scale gT in an expansion in the gauge coupling g. At leading order in g, one obtains Vlasov-Boltzmann equations for the soft field modes, which contain a Gaussian noise and a collision term. The 2-point function of the noise and the collision term are explicitly calculated in a leading logarithmic approximation. In this approximation the Boltzmann equation is solved. The resulting effective theory for the soft field modes is described by a Langevin equation. It determines the parametric form of the hot baryon number violation rate as Γ = κg10 log(1/g)gT4, and it allows for a calculation for κ on the lattice.

Non-abelian plasma instabilities for strong anisotropy

We numerically investigate gauge field instabilities in anisotropic SU(2) plasmas using weak field initial conditions. The growth of unstable modes is stopped by non-abelian effects for moderate anisotropy. If we increase the anisotropy the growth continues beyond the non-abelian saturation bound. We find strong indications that the continued growth is not due to over-saturation of infrared field modes, but instead due to very rapid growth of high momentum modes which are not unstable in the weak field limit. The saturation amplitude strongly depends on the initial conditions. For strong initial fields we do not observe the sustained growth.