Unlocking Undergraduate Problem Solving

It is difficult to find good problems for undergraduates. In this article, we explore an interesting problem that can be used in virtually any mathematics course. We then offer natural generalizations, state and prove some related results, and ultimately end with several open problems suitable for undergraduate research. Finally, we attempt to shed some light on what makes a problem interesting

Supersymmetric Electroweak Baryogenesis in the WKB approximation

We calculate the baryon asymmetry generated at the electroweak phase transition in the minimal supersymmetric standard model, treating the particles in a WKB approximation in the bubble wall background. A set of diffusion equations for the particle species relevant to baryon generation, including source terms arising from the CP violation associated with the complex phase $\delta$ of the $\mu$ parameter, are derived from Boltzmann equations, and solved. The conclusion is that $\delta$ must be \gsim 0.1 to generate a baryon asymmetry consistent with nucleosynthesis. We compare our results to several other recent computations of the effect, arguing that some are overestimates.Comment: 12 pages, 1 figure, corrected some criticisms of hep-ph/9702409; to appear in Phys. Lett.

No new limit on the size distribution of gamma-ray bursts

The results of a study (Carter et. al.) of gamma ray bursts using long duration balloon exposure are analyzed. Arguments are presented against the conclusion that the size spectrum extrapolates to a power law with index from -1.0 to -0.5, and that therefore the gamma ray bursts are of galactic origin. It is claimed that the data are consistent with an upper limit over 100 times that proposed, and that therefore no conclusion can be drawn from the measurements regarding the nature or origin of gamma ray bursts. The resulting upper limit to the rate of occurrence of small bursts lies above the -1.5 index power law extrapolation of the size spectrum of known events, i.e., greater than the rate expected from an infinitely extended source region

Testing for Features in the Primordial Power Spectrum

Well-known causality arguments show that events occurring during or at the end of inflation, associated with reheating or preheating, could contribute a blue component to the spectrum of primordial curvature perturbations, with the dependence k^3. We explore the possibility that they could be observably large in CMB, LSS, and Lyman-alpha data. We find that a k^3 component with a cutoff at some maximum k can modestly improve the fits (Delta chi^2=2.0, 5.4) of the low multipoles (l ~ 10 - 50) or the second peak (l ~ 540) of the CMB angular spectrum when the three-year WMAP data are used. Moreover, the results from WMAP are consistent with the CBI, ACBAR, 2dFGRS, and SDSS data when they are included in the analysis. Including the SDSS galaxy clustering power spectrum, we find weak positive evidence for the k^3 component at the level of Delta chi' = 2.4, with the caveat that the nonlinear evolution of the power spectrum may not be properly treated in the presence of the k^3 distortion. To investigate the high-k regime, we use the Lyman-alpha forest data (LUQAS, Croft et al., and SDSS Lyman-alpha); here we find evidence at the level Delta chi^2' = 3.8. Considering that there are two additional free parameters in the model, the above results do not give a strong evidence for features; however, they show that surprisingly large bumps are not ruled out. We give constraints on the ratio between the k^3 component and the nearly scale-invariant component, r_3 < 1.5, over the range of wave numbers 0.0023/Mpc < k < 8.2/Mpc. We also discuss theoretical models which could lead to the k^3 effect, including ordinary hybrid inflation and double D-term inflation models. We show that the well-motivated k^3 component is also a good representative of the generic spikelike feature in the primordial perturbation power spectrum.Comment: 23 pages, 6 figures; added new section on theoretical motivation for k^3 term, and discussion of double D-term hybrid inflation models; title changed, added a new section discussing the generic spikelike features, published in IJMP

Semiclassical force for electroweak baryogenesis: three-dimensional derivation

We derive a semiclassical transport equation for fermions propagating in the presence of a CP-violating planar bubble wall at a first order electroweak phase transition. Starting from the Kadanoff-Baym (KB) equation for the two-point (Wightman) function we perform an expansion in gradients, or equivalently in the Planck constant h-bar. We show that to first order in h-bar the KB equations have a spectral solution, which allows for an on-shell description of the plasma excitations. The CP-violating force acting on these excitations is found to be enhanced by a boost factor in comparison with the 1+1-dimensional case studied in a former paper. We find that an identical semiclassical force can be obtained by the WKB method. Applications to the MSSM are also mentioned.Comment: 19 page

Spontaneous Symmetry Breaking in General Relativity. Vector Order Parameter

Gravitational properties of a hedge-hog type topological defect in two extra dimensions are considered in General Relativity employing a vector as the order parameter. All previous considerations were done using the order parameter in the form of a multiplet in a target space of scalar fields. The difference of these two approaches is analyzed and demonstrated in detail. Regular solutions of the Einstein equations are studied analytically and numerically. It is shown that the existence of a negative cosmological constant is sufficient for the spontaneous symmetry breaking of the initially plain bulk. Regular configurations have a growing gravitational potential and are able to trap the matter on the brane. If the energy of spontaneous symmetry breaking is high, the gravitational potential has several points of minimum. Identical in the uniform bulk spin-less particles, being trapped within separate minima, acquire different masses and appear to the observer on brane as different particles with integer spins.Comment: 23 pages, 6 figure

The protein import apparatus of chloroplasts

Routing of cytosolically synthesized precursor proteins into chloroplasts is a specific process which involves a multitude of soluble and membrane components. In this review we wil1 focus on early events of the translocation pathway of nuclear coded plastidic precursor proteins and compare import routes for polypeptide of the outer chloroplast envelope to that of internal chloroplast compartments. A number of proteins housed in the chloroplast envelopes have been implied to be involved in the translocation process, but so far a certain function has not been assigned to any of these proteins. The only exception could be an envelope localized hsc 70 homologue which could retain the import competence of a precursor protein in transit into the organelle