Electroweak Baryon Number Violation Research Articles

Dynamical simulations of electroweak baryogenesis with fermions

We perform real-time numerical lattice simulations of a one-family version of the Standard Model. We model the quantum fermions using the ensemble method and treat the bosonic scalar and non-abelian gauge fields classically. Our main interest is electroweak baryogenesis, and we test the approach by considering Standard Model baryon number violation through the chiral anomaly, a truly quantum phenomenon. We find that the method correctly reproduces the anomaly, and perform the first dynamical simulations of electroweak baryon number violation including fermions.

Bödeker’s effective theory: From Langevin dynamics to Dyson–Schwinger equations

The dynamics of weakly coupled, non-abelian gauge fields at high temperature is non-perturbative if the characteristic momentum scale is of order | k | ∼ g 2 T . Such a situation is typical for the processes of electroweak baryon number violation in the early Universe. Bödeker has derived an effective theory that describes the dynamics of the soft field modes by means of a Langevin equation. This effective theory has been used for lattice calculations so far [G.D. Moore, Nucl. Phys. B568 (2000) 367. Available from: <hep-ph/9810313>; G.D. Moore, Phys. Rev. D62 (2000) 085011. Available from: <hep-ph/0001216>]. In this work we provide a complementary, more analytic approach based on Dyson–Schwinger equations. Using methods known from stochastic quantitation, we recast Bödeker’s Langevin equation in the form of a field theoretic path integral. We introduce gauge ghosts in order to help control possible gauge artefacts that might appear after truncation, and which leads to a BRST symmetric formulation and to corresponding Ward identities. A second set of Ward identities, reflecting the origin of the theory in a stochastic differential equation, is also obtained. Finally, Dyson–Schwinger equations are derived.

Anomalous commutators and electroweak baryogenesis

Electroweak vacuum transition processes (sphalerons) in the early Universe provide a possible explanation of the baryon asymmetry. Combining this physics with the anomalous commutators of Adler and Boulware and renormalization group invariance, we argue that electroweak baryon number violation also induces a “topological condensate” in the vacuum. QCD sphaleron processes act to distribute the baryon number violation between both left- and right-handed quarks and induce a spin independent component in this “condensate”.

An upper bound on the total cross-section for electroweak baryon number violation

An upper bound on the total cross-section of s-wave electroweak instanton/sphaleron induced baryon plus lepton number violating processes is presented. It is obtained by exploiting a recently reported lower bound on the corresponding tunneling suppression exponent and by estimating the pre-exponential factor. We find that the present knowledge about electroweak baryon plus lepton number violating processes still allows their eventual observability at the Very Large Hadron Collider, even as pure s-wave scattering. A possibly observable rate at cosmic ray facilities and neutrino telescopes is presently not excluded, but requires a substantial contribution from higher partial waves.

Spectral flow of chiral fermions in nondissipative Yang-Mills gauge field backgrounds

Real-time anomalous fermion number violation is investigated for massless chiral fermions in spherically symmetric SU(2) Yang-Mills gauge field backgrounds which can be weakly dissipative or even nondissipative. Restricting consideration to spherically symmetric fermion fields, the zero-eigenvalue equation of the time-dependent effective Dirac Hamiltonian is studied in detail. For generic spherically symmetric SU(2) gauge fields in Minkowski spacetime, a relation is presented between the spectral flow and two characteristics of the background gauge field. These characteristics are the well-known ``winding factor,'' which is defined to be the change of the Chern-Simons number of the associated vacuum sector of the background gauge field, and a new ``twist factor,'' which can be obtained from the zero-eigenvalue equation of the effective Dirac Hamiltonian but is entirely determined by the background gauge field. For a particular class of (weakly dissipative) Luscher-Schechter gauge field solutions, the level crossings are calculated directly and nontrivial contributions to the spectral flow from both the winding factor and the twist factor are observed. The general result for the spectral flow may be relevant to electroweak baryon number violation in the early universe.

High temperature color conductivity at next-to-leading log order

The non-Abelian analogue of electrical conductivity at high temperature has previously been known only at leading logarithmic order --- that is, neglecting effects suppressed only by an inverse logarithm of the gauge coupling. We calculate the first sub-leading correction. This has immediate application to improving, to next-to-leading log order, both effective theories of non-perturbative color dynamics, and calculations of the hot electroweak baryon number violation rate.

Nonperturbative dynamics of hot non-Abelian gauge fields: Beyond the leading log approximation

Many aspects of high-temperature gauge theories, such as the electroweak baryon number violation rate, color conductivity, and the hard gluon damping rate, have previously been understood only at leading logarithmic order (that is, neglecting effects suppressed only by an inverse logarithm of the gauge coupling). We discuss how to systematically go beyond leading logarithmic order in the analysis of physical quantities. Specifically, we extend to next-to-leading-log order (NLLO) the simple leading-log effective theory due to B\"odeker that describes non-perturbative color physics in hot non-Abelian plasmas. A suitable scaling analysis is used to show that no new operators enter the effective theory at next-to-leading-log order. However, a NLLO calculation of the color conductivity is required, and we report the resulting value. Our NLLO result for the color conductivity can be trivially combined with previous numerical work by Moore to yield a NLLO result for the hot electroweak baryon number violation rate.

From hard thermal loops to Langevin dynamics

In hot non-Abelian gauge theories, processes characterized by the momentum scale g 2 T (such as electroweak baryon number violation in the very early universe) are non-perturbative. An effective theory for the soft (| p| ∼ g 2T) field modes is obtained by integrating out momenta larger than than g 2 T. 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 = κg 10 log(1/ g) gT 4, and it allows for a calculation for κ on the lattice.

Longitudinal subtleties in diffusive Langevin equations for non-Abelian plasmas

B\"odeker has recently argued that non-perturbative processes in very high temperature non-Abelian plasmas (such as electroweak baryon number violation in the very hot early Universe) can be quantitatively described, to leading logarithmic accuracy, by a simple diffusive effective theory. B\"odeker's effective theory is intended to describe the long-distance transverse electric and magnetic fields which are responsible for non-perturbative dynamics. His effective theory, however, also contains long-wavelength longitudinal electric fields. We discuss several subtleties in the treatment of longitudinal dynamics which were not closely examined in B\"odeker's original treatment. Somewhat to our surprise, we find that within its domain of validity B\"odeker's effective theory does correctly describe both longitudinal and transverse fluctuations. We also show that, as far as the transverse dynamics of interest is concerned, B\"odeker's effective theory could be replaced by a transverse-only theory that removes the longitudinal dynamics altogether. In the process, we discuss several interesting aspects of stochastic field theories.

Effective dynamics of hot, soft non-Abelian gauge fields: Color conductivity and log(1/α)effects

B\"odeker has recently argued that non-perturbative processes in very high temperature non-Abelian plasmas (such as electroweak baryon number violation in the very hot early Universe) are logarithmically enhanced over previous estimates and take place at a rate per unit volume of order ${\ensuremath{\alpha}}^{5}{T}^{4}\mathrm{ln}(1/\ensuremath{\alpha})$ for small coupling. We give a simple physical interpretation of B\"odeker's qualitative and quantitative results in terms of Lenz's Law---the fact that conducting media resist changes in the magnetic field---and earlier authors' calculations of the color conductivity of such plasmas. In the process, we resolve some confusion in the literature about the value of the color conductivity and present an independent calculation. We also discuss the issue of whether the classical effective theory proposed by B\"odeker has a good continuum limit.

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∼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.

The hot baryon violation rate isO(αw5T4)

The rate per unit volume for anomalous electroweak baryon number violation at high temperatures, in the symmetric phase, has been estimated in the literature to be $O({\ensuremath{\alpha}}_{\mathrm{w}}^{4}{T}^{4})$ based on simple scaling arguments. We argue that damping effects in the plasma suppress the rate by an extra power of ${\ensuremath{\alpha}}_{\mathrm{w}}$ to give $O({\ensuremath{\alpha}}_{\mathrm{w}}^{5}{T}^{4})$. We show how to understand this effect in a variety of ways ranging from an effective description of the long-distance modes responsible for baryon number violation to a microscopic picture of the short-distance modes responsible for damping. In particular, we resolve an old controversy as to whether damping effects are relevant. Finally, we argue that similar damping effects should occur in numerical simulations of the rate in classical thermal field theory on a spatial lattice, and we point out a potential problem with simulations in the literature that have not found such an effect.

Total cross section for anomalous fermion-number violation via dispersion relation

We investigate the behaviour of the total cross section for electroweak baryon and lepton number violation (in association with multi-W and multi-Higgs production) at low energies (√ s < O(10)TeV) by means of a dispersion relation. The imaginary part of the relevant forward elastic scattering amplitude is calculated in a saddle-point approximation, systematically introduced by the so-called valley method. Taking into account the first correction to the basic instanton result (and neglecting the higher order corrections) we find that the total cross section does not get exponentially large. The question whether the cross section becomes of observable size remains still open, however.

Sphaleron production in high energy two particle collisions

I estimate the amplitude for producing a classical vector (W) field configuration with non-zero Chern-Simons number in a high energy q q collision. The calculation is performed within a Minkowski framework, with use made of amplitudes derived for multigluon processes in unbroken gauge theories. The degree of suppression (or non-suppression) of the process depends on the detailed structure of the field configuration. Thus, although it is difficult to assess whether the amplitude is sizeable in the 10 TeV region, there seems to be no a priori reason why electroweak baryon number violation (which, in a Minkowski space formulation, requires the generation of the field configurations considered here) should be suppressed in high energy collisions.

Electroweak baryon number violation at high energies

The electroweak baryon number violation is expected to be rapid at high temperatures. This implies that at least some multiparticle processes with B-violation are rapid at high energies. We discuss recent attempts to study the two-particle cross sections of instanton-like processes, which show that at energies below 10 TeV, the cross sections increase exponentially, and might even become large at about 10 TeV.

Electroweak baryon number violation at finite temperature

We consider baryon and lepton number violating processes in the electroweak theory induced by gauge and Higgs fields passing the sphaleron solution at finite temperature. We show that for temperatures larger than 19 GeV the anomalous baryon and lepton number violating processes are suppressed by the Boltzmann factor exp (−βE sp), whereE sp is the sphaleron energy, rather than by the instanton tunneling factor exp (−8π2/g 2). We caculate the rate of baryon and lepton number violating processes at finite temperature and determine the freezing temperature of the anomalous processes in the early universe as a function of the Higgs mass. We compare the freezing temperature with the critical temperature of the electroweak phase transition infered from the one-loop finite-temperature effective potential. We obtain a critical Higgs mass of the order of 100 GeV, slightly depending on the top mass and the magnitude of the pre-exponential factor in the rate of theB non-conservation, above which the anomalous processes are certainly in equilibrium after the electroweak phase transition. Assuming that the temperature-dependence of the sphaleron energy is given by that found from the one-loop finitetemperature effective potential, this critical Higgs mass is lowered to a value of the order of 50 GeV.

Rate of anomalous electroweak baryon and lepton number violation at finite temperature

Using Langer's statistical theory of the decay of metastable states we calculate the rate of the anomalous electroweak baryon and lepton number violating processes in the case that the electroweak phase transition is of second order. Our formulas are valid in a temperature range between M w and the critical temperature T c. We get a dissipation of the baryon and lepton number of the order of exp(4.6×10 9).

Comments on the possibility of electroweak baryon number violation at high temperatures

Abstract Anomalous baryon number violation in the standard electroweak theory was first discussed by 't Hooft, who found it to be suppressed by a large factor exp(−8 π 2 / g 2 ) at zero temperature, due to a large energy barrier separating vacua with different baryon number. One might have expected that in the early Universe or in high-energy collisions, this process would become unsuppressed when the energies involved became comparable to the barrier height. We argue here that in both cases processes violating baryon number are likely to remain suppressed by a least the same zero-temperature factor.