On One-Loop Gap Equations for the Magnetic Mass in d=3 Gauge Theory

Recently several workers have attempted determinations of the so-called magnetic mass of d=3 non-Abelian gauge theories through a one-loop gap equation, using a free massive propagator as input. Self-consistency is attained only on-shell, because the usual Feynman-graph construction is gauge-dependent off-shell. We examine two previous studies of the pinch technique proper self-energy, which is gauge-invariant at all momenta, using a free propagator as input, and show that it leads to inconsistent and unphysical result. In one case the residue of the pole has the wrong sign (necessarily implying the presence of a tachyonic pole); in the second case the residue is positive, but two orders of magnitude larger than the input residue, which shows that the residue is on the verge of becoming ghostlike. This happens because of the infrared instability of d=3 gauge theory. A possible alternative one-loop determination via the effective action also fails. The lesson is that gap equations must be considered at least at two-loop level.Comment: 21 pages, LaTex, 2 .eps figure

Land and politics in the Transvaal in the 1880s

African Studies Seminar series. Paper presented October, 1972It is evident that the population of the South African Republic did not constitute a single homogenous group, devoid of variation in wealth, education and life-style. The diversity of Afrikaner society has frequently been overlooked by historians however, and the courses of social differentation completely ignored. The State Archives at Pretoria do contain some enormous collections of largely unused material however, which for all their superficial dullness embody a vast amount of detailed and valuable data relating to this very problem HAD one the time and equipment to analyse in detail the information which is to be found in the land-registers, estates, death notices and wills, there can be no question but that the result would represent a most notable contribution to the social and economic history of South Africa. If in addition it were possible to collect the petitions, ballot papers and voting list, many of them still extant then another dimension could be superimposed. These would however, be projects of vast size, and although they will, hopefully, be attempted in the future, for the present we shall have to be satisfied with a more modest harvest of information. There is no expectation that however complete, the bare bones of statistical data will answer all of our questions. They can only be used to suggest the patterns of life extending beyond the range of the livelier facts called from volksraad minutes, newspapers, petitions, private correspondence and reminiscences. The major part of this paper will be concerned with the social and economic diversity reflected in landownership and in particular with its causes, general and local. No claim is made to completeness in dealing with the origins of the phenomenon, and attention is paid largely to that material which illustrates aspects of the problem which have been previously neglected. Practical considerations obviously restrict the scope of all research, and for the purposes of this paper most of the evidence relates to the district of Wakkerstroom

On The Phase Transition in D=3 Yang-Mills Chern-Simons Gauge Theory

$SU(N)$ Yang-Mills theory in three dimensions, with a Chern-Simons term of level $k$ (an integer) added, has two dimensionful coupling constants, $g^2 k$ and $g^2 N$; its possible phases depend on the size of $k$ relative to $N$. For $k \gg N$, this theory approaches topological Chern-Simons theory with no Yang-Mills term, and expectation values of multiple Wilson loops yield Jones polynomials, as Witten has shown; it can be treated semiclassically. For $k=0$, the theory is badly infrared singular in perturbation theory, a non-perturbative mass and subsequent quantum solitons are generated, and Wilson loops show an area law. We argue that there is a phase transition between these two behaviors at a critical value of $k$, called $k_c$, with $k_c/N \approx 2 \pm .7$. Three lines of evidence are given: First, a gauge-invariant one-loop calculation shows that the perturbative theory has tachyonic problems if $k \leq 29N/12$.The theory becomes sensible only if there is an additional dynamic source of gauge-boson mass, just as in the $k=0$ case. Second, we study in a rough approximation the free energy and show that for $k \leq k_c$ there is a non-trivial vacuum condensate driven by soliton entropy and driving a gauge-boson dynamical mass $M$, while both the condensate and $M$ vanish for $k \geq k_c$. Third, we study possible quantum solitons stemming from an effective action having both a Chern-Simons mass $m$ and a (gauge-invariant) dynamical mass $M$. We show that if M \gsim 0.5 m, there are finite-action quantum sphalerons, while none survive in the classical limit $M=0$, as shown earlier by D'Hoker and Vinet. There are also quantum topological vortices smoothly vanishing as $M \rightarrow 0$.Comment: 36 pages, latex, two .eps and three .ps figures in a gzipped uuencoded fil

Center Vortices, Nexuses, and Fractional Topological Charge

It has been remarked in several previous works that the combination of center vortices and nexuses (a nexus is a monopole-like soliton whose world line mediates certain allowed changes of field strengths on vortex surfaces) carry topological charge quantized in units of 1/N for gauge group SU(N). These fractional charges arise from the interpretation of the standard topological charge integral as a sum of (integral) intersection numbers weighted by certain (fractional) traces. We show that without nexuses the sum of intersection numbers gives vanishing topological charge (since vortex surfaces are closed and compact). With nexuses living as world lines on vortices, the contributions to the total intersection number are weighted by different trace factors, and yield a picture of the total topological charge as a linking of a closed nexus world line with a vortex surface; this linking gives rise to a non-vanishing but integral topological charge. This reflects the standard 2\pi periodicity of the theta angle. We argue that the Witten-Veneziano relation, naively violating 2\pi periodicity, scales properly with N at large N without requiring 2\pi N periodicity. This reflects the underlying composition of localized fractional topological charge, which are in general widely separated. Some simple models are given of this behavior. Nexuses lead to non-standard vortex surfaces for all SU(N) and to surfaces which are not manifolds for N>2. We generalize previously-introduced nexuses to all SU(N) in terms of a set of fundamental nexuses, which can be distorted into a configuration resembling the 't Hooft-Polyakov monopole with no strings. The existence of localized but widely-separated fractional topological charges, adding to integers only on long distance scales, has implications for chiral symmetry breakdown.Comment: 15 pages, revtex, 6 .eps figure

Seeking an Even-Parity Mass Term for 3-D Gauge Theory

Mass-gap calculations in three-dimensional gauge theories are discussed. Also we present a Chern--Simons-like mass-generating mechanism which preserves parity and is realized non-perturbatively.Comment: 11 pages, revte

A conjecture on the infrared structure of the vacuum Schrodinger wave functional of QCD

The Schrodinger wave functional for the d=3+1 SU(N) vacuum is a partition function constructed in d=4; the exponent 2S in the square of the wave functional plays the role of a d=3 Euclidean action. We start from a gauge-invariant conjecture for the infrared-dominant part of S, based on dynamical generation of a gluon mass M in d=4. We argue that the exact leading term, of O(M), in an expansion of S in inverse powers of M is a d=3 gauge-invariant mass term (gauged non-linear sigma model); the next leading term, of O(1/M), is a conventional Yang-Mills action. The d=3 action that is the sum of these two terms has center vortices as classical solutions. The d=3 gluon mass, which we constrain to be the same as M, and d=3 coupling are related through the conjecture to the d=4 coupling strength, but at the same time the dimensionless ratio in d=3 of mass to coupling squared can be estimated from d=3 dynamics. This allows us to estimate the QCD coupling $\alpha_s(M^2)$ in terms of this strictly d=3 ratio; we find a value of about 0.4, in good agreement with an earlier theoretical value but a little low compared to QCD phenomenology. The wave functional for d=2+1 QCD has an exponent that is a d=2 infrared-effective action having both the gauge-invariant mass term and the field strength squared term, and so differs from the conventional QCD action in two dimensions, which has no mass term. This conventional d=2 QCD would lead in d=3 to confinement of all color-group representations. But with the mass term (again leading to center vortices), N-ality = 0 mod N representations are not confined.Comment: 15 pages, no figures, revtex

Fixed points and vacuum energy of dynamically broken gauge theories

We show that if a gauge theory with dynamical symmetry breaking has non-trivial fixed points, they will correspond to extrema of the vacuum energy. This relationship provides a different method to determine fixed points.Comment: 17 pages, uuencoded latex file, 3 figures, uses epsf and epsfig. Submitted to Mod. Phys. Lett.

On topological charge carried by nexuses and center vortices

In this paper we further explore the question of topological charge in the center vortex-nexus picture of gauge theories. Generally, this charge is locally fractionalized in units of 1/N for gauge group SU(N), but globally quantized in integral units. We show explicitly that in d=4 global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy $\Pi_3(S^3)\simeq Z$; this homotopy insures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy $\Pi_2(S^2)\simeq Z$, a homotopy usually associated with 't Hooft-Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We find a new type of nexus which can carry topological charge by linking to vortices or carry d=3 Chern-Simons number without center vortices present; the Chern-Simons number is connected with twisting and writhing of field lines, as the author had suggested earlier. In general, no topological charge in d=4 arises from these specific static configurations, since the charge is the difference of two (equal) Chern-Simons number, but it can arise through dynamic reconnection processes. We complete earlier vortex-nexus work to show explicitly how to express globally-integral topological charge as composed of essentially independent units of charge 1/N.Comment: Revtex4; 3 .eps figures; 18 page

Baryon number non-conservation and phase transitions at preheating

Certain inflation models undergo pre-heating, in which inflaton oscillations can drive parametric resonance instabilities. We discuss several phenomena stemming from such instabilities, especially in weak-scale models; generically, these involve energizing a resonant system so that it can evade tunneling by crossing barriers classically. One possibility is a spontaneous change of phase from a lower-energy vacuum state to one of higher energy, as exemplified by an asymmetric double-well potential with different masses in each well. If the lower well is in resonance with oscillations of the potential, a system can be driven resonantly to the upper well and stay there (except for tunneling) if the upper well is not resonant. Another example occurs in hybrid inflation models where the Higgs field is resonant; the Higgs oscillations can be transferred to electroweak (EW) gauge potentials, leading to rapid transitions over sphaleron barriers and consequent B+L violation. Given an appropriate CP-violating seed, we find that preheating can drive a time-varying condensate of Chern-Simons number over large spatial scales; this condensate evolves by oscillation as well as decay into modes with shorter spatial gradients, eventually ending up as a condensate of sphalerons. We study these examples numerically and to some extent analytically. The emphasis in the present paper is on the generic mechanisms, and not on specific preheating models; these will be discussed in a later paper.Comment: 10 pages, 7 figures included, revtex, epsf, references adde