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

Vertices and the CJT Effective Potential

The Cornwall-Jackiw-Tomboulis effective potential is modified to include a functional dependence on the fermion-gauge particle vertex, and applied to a quark confining model of chiral symmetry breaking.Comment: 10 pages (latex), PURD-TH-93-1

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

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.Comment: LateX, 24 pages, 2 .eps figure

Center Vortices, Nexuses, and the Georgi-Glashow Model

In a gauge theory with no Higgs fields the mechanism for confinement is by center vortices, but in theories with adjoint Higgs fields and generic symmetry breaking, such as the Georgi-Glashow model, Polyakov showed that in d=3 confinement arises via a condensate of 't Hooft-Polyakov monopoles. We study the connection in d=3 between pure-gauge theory and the theory with adjoint Higgs by varying the Higgs VEV v. As one lowers v from the Polyakov semi- classical regime v>>g (g is the gauge coupling) toward zero, where the unbroken theory lies, one encounters effects associated with the unbroken theory at a finite value v\sim g, where dynamical mass generation of a gauge-symmetric gauge- boson mass m\sim g^2 takes place, in addition to the Higgs-generated non-symmetric mass M\sim vg. This dynamical mass generation is forced by the infrared instability (in both 3 and 4 dimensions) of the pure-gauge theory. We construct solitonic configurations of the theory with both m,M non-zero which are generically closed loops consisting of nexuses (a class of soliton recently studied for the pure-gauge theory), each paired with an antinexus, sitting like beads on a string of center vortices with vortex fields always pointing into (out of) a nexus (antinexus); the vortex magnetic fields extend a transverse distance 1/m. An isolated nexus with vortices is continuously deformable from the 't Hooft-Polyakov (m=0) monopole to the pure-gauge nexus-vortex complex (M=0). In the pure-gauge M=0 limit the homotopy $\Pi_2(SU(2)/U(1))=Z_2$ (or its analog for SU(N)) of the 't Hooft monopoles is no longer applicable, and is replaced by the center-vortex homotopy $\Pi_1(SU)N)/Z_N)=Z_N$.Comment: 27 pages, LaTeX, 3 .eps figure

Relativistic center-vortex dynamics of a confining area law

We offer a physicists' proof that center-vortex theory requires the area in the Wilson-loop area law to involve an extremal area. Area-law dynamics is determined by integrating over Wilson loops only, not over surface fluctuations for a fixed loop. Fluctuations leading to to perimeter-law corrections come from loop fluctuations as well as integration over finite -thickness center-vortex collective coordinates. In d=3 (or d=2+1) we exploit a contour form of the extremal area in isothermal which is similar to d=2 (or d=1+1) QCD in many respects, except that there are both quartic and quadratic terms in the action. One major result is that at large angular momentum \ell in d=3+1 the center-vortex extremal-area picture yields a linear Regge trajectory with Regge slope--string tension product \alpha'(0)K_F of 1/(2\pi), which is the canonical Veneziano/string value. In a curious effect traceable to retardation, the quark kinetic terms in the action vanish relative to area-law terms in the large-\ell limit, in which light-quark masses \sim K_F^{1/2} are negligible. This corresponds to string-theoretic expectations, even though we emphasize that the extremal-area law is not a string theory quantum-mechanically. We show how some quantum trajectory fluctuations as well as non-leading classical terms for finite mass yield corrections scaling with \ell^{-1/2}. We compare to old semiclassical calculations of relativistic q\bar{q} bound states at large \ell, which also yield asymptotically-linear Regge trajectories, finding agreement with a naive string picture (classically, not quantum-mechanically) and disagreement with an effective-propagator model. We show that contour forms of the area law can be expressed in terms of Abelian gauge potentials, and relate this to old work of Comtet.Comment: 20 pages RevTeX4 with 3 .eps figure

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.Comment: Latex, 23 page

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

Finite VEVs from a Large Distance Vacuum Wave Functional

We show how to compute vacuum expectation values from derivative expansions of the vacuum wave functional. Such expansions appear to be valid only for slowly varying fields, but by exploiting analyticity in a complex scale parameter we can reconstruct the contribution from rapidly varying fields.Comment: 39 pages, 16 figures, LaTeX2e using package graphic