Chern-Simons Number Research Articles

Sphaleron of a 4 dimensional SO(4) Higgs model

We construct the finite energy path between topologically distinct vacua of a 4 dimensional SO(4) Higgs model which is known to support an instanton, and show that there is a sphaleron with Chern–Simons number N CS = 1 2 at the top of the energy barrier. This is carried out using the original geometric loop construction of Manton.

Sphalerons withCP-violating Higgs potentials

We investigate the effect on the sphaleron in the two Higgs doublet electroweak theory of including CP violation in the Higgs potential. To have better control over the relation between the sphaleron energy and the physical quantities in the theory, we show how to parametrize the Higgs potential in terms of physical masses and mixing angles, one of which causes CP violation. By altering this CP violating angle (and keeping the other physical quantities fixed) the sphaleron energy increases by up to 10%. We also calculate the static minimum energy path between adjacent vacua as a function of Chern-Simons number, using the method of gradient flow. The only effect CP violation has on the barrier is the change in height. As a by-product of our work on parametrization of the potential, we demonstrate that CP violation in the Higgs sector favours nearly degenerate light Higgs masses.

The nonperturbative broken phase sphaleron rate

I present a technique for measuring the broken phase sphaleron rate nonperturbatively. There are three parts to the calculation: determination of the probability distribution of Chern-Simons number N CS; measurement of 〈| dN CS/ dt|〉 N CS= 1 2 , the mean rate of change of N CS at the barrier; and measurement of the “dynamical prefactor,” the fraction of barrier crossings which result in a permanent integer change in N CS.

Nexus solitons in the center vortex picture of QCD

It is very plausible that confinement in QCD comes from linking of Wilson loops to finite-thickness vortices with magnetic fluxes corresponding to the center of the gauge group. The vortices are solitons of a gauge-invariant QCD action representing the generation of gluon mass. There are a number of other solitonic states of this action. We discuss here what we call nexus solitons, in which for gauge group SU(N), up to N vortices meet a a center, or nexus, provided that the total flux of the vortices adds to zero (mod N). There are fundamentally two kinds of nexuses: Quasi-Abelian, which can be described as composites of Abelian imbedded monopoles, whose Dirac strings are cancelled by the flux condition; and fully non-Abelian, resembling a deformed sphaleron. Analytic solutions are available for the quasi-Abelian case, and we discuss variational estimates of the action of the fully non-Abelian nexus solitons in SU(2). The non-Abelian nexuses carry Chern-Simons number (or topological charge in four dimensions). Their presence does not change the fundamentals of confinement in the center-vortex picture, but they may lead to a modified picture of the QCD vacuum.

The Chern-Simons number as an order parameter: classical sphaleron transitions for SU(2)-Higgs field theories for [formula omitted

The classical transitions between topologically distinct vacua in a SU(2)–Higgs model, using a Higgs field of mass approximately 120 GeV , is examined to probe the crossover region between the symmetric and broken phase. Assuming the Higgs mass is constant, we find the width of this crossover region is approximately 20% of the average temperature. We suggest that this observable is a better parameter to explore this region of phase space than equal-time correlation functions.

Chern-Simons number diffusion with hard thermal loops

We construct an extension of the standard Kogut-Susskind lattice model for classical 3+1 dimensional Yang-Mills theory, in which ``classical particle'' degrees of freedom are added. We argue that this will correctly reproduce the ``hard thermal loop'' effects of hard degrees of freedom, while giving a local implementation which is numerically tractable. We prove that the extended system is Hamiltonian and has the same thermodynamics as dimensionally reduced hot Yang-Mills theory put on a lattice. We present a numerical update algorithm and study the abelian theory to verify that the classical gauge theory self-energy is correctly modified. Then we use the extended system to study the diffusion constant for Chern-Simons number. We verify the Arnold-Son-Yaffe picture that the diffusion constant is inversely proportional to hard thermal loop strength. Our numbers correspond to a diffusion constant of Gamma = 29 +- 6 alpha_w^5 T^4 for m_D^2 = 11 g^2 T^2/6.

Lattice Chern-Simons number without ultraviolet problems

We develop a topological method of measuring Chern-Simons number change in the real time evolution of classical lattice SU(2) and SU(2) Higgs theory. We find that the Chern-Simons number diffusion rate per physical four-volume is very heavily suppressed in the broken phase, and that it decreases with lattice spacing in pure Yang-Mills theory, although not as quickly as predicted by Arnold, Son, and Yaffe.

Speculations on primordial magnetic helicity

We speculate that above or just below the electroweak phase transition magnetic fields are generated which have a net helicity (otherwise said, a Chern-Simons term) of order of magnitude $N_B + N_L$, where $N_{B,L}$ is the baryon or lepton number today. (To be more precise requires much more knowledge of B,L-generating mechanisms than we currently have.) Electromagnetic helicity generation is associated (indirectly) with the generation of electroweak Chern-Simons number through B+L anomalies. This helicity, which in the early universe is some 30 orders of magnitude greater than what would be expected from fluctuations alone in the absence of B+L violation, should be reasonably well-conserved through the evolution of the universe to around the times of matter dominance and decoupling, because the early universe is an excellent conductor. Possible consequences include early structure formation; macroscopic manifestations of CP violation in the cosmic magnetic field (measurable at least in principle, if not in practice); and an inverse-cascade dynamo mechanism in which magnetic fields and helicity are unstable to transfer to larger and larger spatial scales. We give a quasi-linear treatment of the general-relativistic MHD inverse cascade instability, finding substantial growth for helicity of the assumed magnitude out to scales $\sim l_M\epsilon^{-1}$, where $\epsilon$ is roughly the B+L to photon ratio and $l_M$ is the magnetic correlation length. We also elaborate further on an earlier proposal of the author for generation of magnetic fields above the EW phase transition.

Computing the strong sphaleron rate

We measure the diffusion constant for Chern-Simons number for classical, lattice SU(3) Yang-Mills theory, using a generalization of the topological definition of Chern-Simons number developed recently by Moore and Turok. The diffusion constant is much larger than that for SU(2), even before the ratio of coupling constants has been accounted for, which implies that chiral quark number is efficiently destroyed by strong processes during the electroweak phase transition. For the physical value of \alpha_s we estimate the decay time for chiral quark number to be about 80/T, although various systematics make this number uncertain by about a factor of 2.

Classical field dynamics of the electroweak phase transition

We investigate the thermodynamics and dynamics of the electroweak phase transition by modeling the infrared physics with classical Yang-Mills Higgs theory. We discuss the accuracy of this approach and conclude that, for quantities whose determination is dominated by the infrared, the classical method should be correct up to parametrically suppressed [i.e., $O(\ensuremath{\alpha})]$ corrections. For a Higgs self-coupling which at the tree level corresponds to ${m}_{H}\ensuremath{\simeq}50$ GeV, we determine the jump in the order parameter to be $\ensuremath{\delta}\ensuremath{\varphi}=1.5\mathrm{gT}$, the surface tension to be $\ensuremath{\sigma}{=0.07g}^{4}{T}^{3}$, and the friction coefficient on the moving bubble wall due to infrared bosons to be $\ensuremath{\eta}\ensuremath{\equiv}{P/v}_{w}=0.03\ifmmode\pm\else\textpm\fi{}{0.004g}^{6}{T}^{4}$. We also investigate the response of the Chern-Simons number to a spatially uniform chemical potential and find that it falls off a short distance inside the bubble wall, both in equilibrium and below the equilibrium temperature.

Zero Modes of the Dirac Operator for Regular Einstein-Yang-Mills Background Fields

The existence of normalizable zero modes of the twisted Dirac operator is proven for a class of static Einstein-Yang-Mills background fields with a half-integer Chern-Simons number. The proof holds for any gauge group and applies to Dirac spinors in an arbitrary representation of the gauge group. The class of background fields contains all regular, asymptotically flat, CP-symmetric configurations with a connection that is globally described by a time-independent spatial one-form which vanishes sufficiently fast at infinity. A subset is provided by all neutral, spherically symmetric configurations which satisfy a certain genericity condition, and for which the gauge potential is purely magnetic with real magnetic amplitudes.

Topological transitions at T > 0 in the euclidean 2d U(1)-Higgs model

The two-dimensional U(1)-gauged Higgs model is studied on an euclidean lattice of size L 1 × L 2, where the temperature T = L 2 −1 is of the order of the sphaleron mass. The simulation parameters are taken from zero temperature results [1]. By comparison with classical and semiclassical results I discuss, whether the sphaleron transition rate can be extracted from the behavior of the Chern-Simons number and from the formation of vortices in an euclidean simulation at high temperatures.

Improved Hamiltonian for Minkowski Yang-Mills theory

I develop an improved Hamiltonian for classical, Minkowski Yang-Mills theory, which evolves infrared fields with tree level corrections from lattice spacing a beginning at O( a 4). I use it to investigate the response of Chern-Simons number to a chemical potential, and to compute the maximal Lyapunov exponent. The Lyapunov exponent has a small a limit, and the Chern-Simons number response appears to be approaching one at the finest lattices considered. In both cases the limit is within 10% of the limit found using the unimproved (Kogut-Susskind) Hamiltonian. For the maximal Lyapunov exponent the limits differ between Hamiltonians by about 5%, significant at about 5σ, indicating that while a small a limit exists, its value depends on the specifics of the lattice cutoff. For Chern-Simons number the difference between Hamiltonians is within statistical errors of about 10%, which constitutes an upper bound on the lattice regulation dependence.

Motion of Chern-Simons number at high temperatures under a chemical potential

I investigate the evolution of finite temperature, classical Yang-Mills field equations under the influence of a chemical potential for Chern-Simons number N cs. The rate of N cs diffusion,, Γ d , and the linear response of N cs to a chemical potential, Γ μ , are both computed; the relation Γ d = 2 Γ μ is satisfied numerically and the results agree with the recent measurement of Γ d by Ambjørn and Krasnitz. The response of N cs under chemical potential remains linear at least to μ = 6 T, which is impossible if there is a free energy barrier to the motion of N cs. The possibility that the result depends on lattice artefacts via hard thermal loops is investigated by changing the lattice action and by examining elongated rectangular lattices; provided that the lattice is fine enough, the result is weakly if at all dependent on the specifics of the cutoff. I also compare SU(2) with SU(3) and find Γ SU(3)∼ 7( α s/ α w ) 4 Γ SU(2) .

Chern-Simons diffusion rate near the electroweak phase transition for mH ≈ mW

The rate of B-violation in the standard model at finite temperature is closely related to the diffusion rate Γ of Chern-Simons number. We compute this rate for m H ≈ m W in the classical approximation in an effective SU(2)-Higgs model, using Krasnitz's algorithm. The parameters in the effective hamiltonian are determined by comparison with dimensional reduction. In the high temperature phase we find Γ/ V( α W T) 4 ≈ 1, neglecting a finite renormalization. In the low temperature phase near the transition we find the rate to be much larger than might be expected from previous analytic calculations based on the sphaleron.

ℤ 2 as the topological origin of B + L violation in the hot electroweak theory

The space of static finite-energy configurations in the electroweak theory admits a ℤ 2 2 topological structure. More precisely, we show that this space contains two disconnected sectors of unstable gauge-Higgs fields odd under a properly defined generalized parity. This classification extends the description of baryon- and lepton-number violating electroweak processes to the symmetric phase of the theory. Configurations with odd pure-gauge behaviour at spatial infinity, such as the sphaleron, multisphalerons, as well as an infinite surface of their equivalents, have half-integer Chern—Simons number and mediate B + L violating processes in the early universe. Finite-energy configurations with even pure-gauge behaviour, such as the S* new sphaleron, are topologically equivalent to the vacuum and are irrelevant for B + L violation. Electroweak string configurations comprise continuous families of sphaleron equivalents as well as configurations equivalent to the vacuum. We discuss the possible formation of B + L violating quark-lepton condensates in the symmetric high-temperature phase of the electroweak theory.

The classical sphaleron transition rate exists and is equal to 1.1( αwT) 4

Results of a large scale numerical simulation show that the high temperature Chern-Simons number diffusion rate in the electroweak theory has a classical limit Γ = κ( α w T) 4,where κ = 1.09 ± 0.04 and α w is the weak fine structure constant.

Cosmic strings at the electroweak phase transition-an application

We consider the phase transition in the electroweak theory with two Higgs doublets in the presence of preexisting cosmic strings. The correction to the effective action coming from the triangle diagram consisting of one vector and two scalar legs is calculated. This correction biases the Chern-Simons number which in turn generates a baryon asymmetry which we estimate to be ∼ 10 −10 −10 −12 .

Chern-Simons number diffusion in (1+1)-dimensional Higgs theory

We study the Chern-Simons number diffusion rate in the (1+1)-dimensional latticeAbelian Higgs model at temperatures much higher than, as well as comparable to, the sphaleron energy. It is found that in the high-temperature limit the rate is likely to grow as power 2/3 of the temperature. In the intermediate-temperature regime, our numerical simulations show that very weak temperature dependence of the rate, found in previous work, persists at smaller lattice spacings. We discuss possibilities of relating the observed behavior of the rate to static finite-temperature properties of the model.

Thermal fluctuations of Chern-Simons numbers in the lattice SU(2) Higgs model.

We study the temperature dependence of the Chern-Simons number fluctuations in the SU(2) Higgs model on Euclidean lattices with spatial sizes up to ${20}^{3}$. Temperatures well above the Higgs phase transition ${\mathit{T}}_{\mathit{H}}$ are achieved on anisotropic lattices. Numerical results are compared to perturbative results on finite lattices as well as in continuum perturbation theory. We find qualitative agreement with perturbative estimates and see at high temperatures a tendency towards static configurations. Up to temperatures T\ensuremath{\simeq}2${\mathit{T}}_{\mathit{H}}$ we find an indication that tunneling between vacuums with different Chern-Simons numbers is still exponentially suppressed.