Topics on euclidean classical field equations with unique vacuua
We discuss the real classical field equation (-δ+μ2)φ+λF(φ) = f, where μ2>o, λ≥o, F ∈ c∞(R), and aF (a)≥0 for all a ∈ R, and where the source function f belongs to various function spaces contained in the Sobolev space H−1 (Rd). We review a number of results whose proofs are to appear elsewhere, on existence of a solution φ ∈ Hl(Rd), the correspondences between the function spaces of f and φ (regularity properties), contractivity properties and uniqueness of φ, and analytic dependence on λ and functional differentiability in f. We mention some of the ideas from the proofs, including a few alternate methods we have not discussed elsewhere. We prove a new result on positivity preservingness: if f is a nonnegative measure in H−1(Rd), then 0 ≤ φ ∈ H1(Rd), a result that is also valid for μ2 = 0. Finally, we give an intuitive interpretation of some of the results for the case of d = 1 dimension.

title: Euclidean nonlinear classical field equations with unique vacuum
abstract: AbstractWe study the real, Euclidean, classical field equation
$$( )\varphi + \lambda F(\varphi ) = f,\mu ^2 > 0$$
 where φ: ℝd→ℝ is suitably small at infinity. We study existence and regularity assuming that λ≧0,F∈C∞(ℝ), andaF(a)≧0∀a∈∝. These hypotheses allow strongly nonlinearF and nonunique solutions forf≠0. WhenF′≧0, we prove uniqueness, various contractivity properties, analytic dependence on the coupling constant λ, and differentiability in the external sourcef. For applications in the loop expansion in quantum field theory, it is useful to know that φ is in the Schwartz classL wheneverf is, and we provide a proof of this fact. The technical innovations of the problem lie in treating the noncompactness of Rd, the strong nonlinearity ofF, and the polynomial weights in the seminorms definingL.

title: Euclidean Fermi fields with a hermitean Feynman-Kac-Nelson formula. I
abstract: We construct free, Euclidean, spin one-half, quantum fields with the following properties: (i) CAR; (ii) Symanzik positivity; (iii) Osterwalder-Schrader positivity; (iv) no doubling of particle or spin states. They admit the recovery of the relativistic Dirac field by the Osterwalder-Schrader technique. We then formally parametrize interacting theories by a natural class of Hermitean, Euclidean actions, and obtain a simple, Hermitean, Feynman-Kac-Nelson formula. The interacting theory formally obeys all the properties (i)–(iv), and admits the reconstruction of a physical Hilbert space, including a Hermitean, contraction semigroup for the Wick rotated time evolution. We propose a system of axioms for the interacting theory.

title: Large width in the Lee model for the Higgs-Goldstone sector
abstract: We discuss a variant of the unstable particle Lee model motivated by the problem of large width in Higgs meson decay. The essential result is that, in the point-interaction model, the width for decay into Goldstone bosons (W's and Z's), defined from the second-sheet unstable particle pole, obeys Γ0 ⩽ Γ <32Γ0, where Γ0 is the naive perturbative width, and the ordering is from weak coupling at the lower end to strong coupling at the upper end. If the point interaction of the Lee model is modified to make the tree scattering amplitude for Goldstone particles relativistically invariant, a large-coupling upper bound for the width results. Γ < 10.7 mH, where by comparison, the naive perturbative width grows like mH3. 
 
We find that defining the renormalized coupling at the complex Higgs pole is reasonable in the Lee model, even when the pole moves far from the real axis. Furthermore, scattering effects remain strong, in the sense that the unitarity bound is roughly approached over a large range of energies, although the scattering amplitude never reaches more than about 80% of saturation in the point-interaction model, contrary to naive perturbation theory. We also point out, without discussing it in this paper, what appears to be a new fact about the Lee model, that the Higgs propagator has a bound-state pole in addition to the resonance pole.

title: On the lattice approach to the maximum Higgs boson mass
abstract: If the triviality upper bound on the Higgs boson mass mH occurs for strong self-coupling, inferring properties of the Higgs from the euclidean propagator is in principle theoretically difficult whether in coordinate or momentum space. In that case, common methods of identifying mH in lattice field theory simulations may produce a value for which is at best distantly related to the true upper limit. We discuss some shortcomings and ambiguities of recent results suggesting that the maximum occurs for weak coupling and emphasize potential complications due to finite-size and non-Lorentz-invariant effects of the lattice. The situation is illustrated by reference to the behavior in an analytically soluble approximation based on a 1/N expansion.

title: Instability of the existence of non-abelian vector potentials
abstract: We show the instability of the existence of vector potentials against variations of the field tensor in regions where it is “strongly non-abelian”.

title: Search for pair production of the scalar top quark in the electron+muon final state
abstract: We report the result of a search for the pair production of the lightest supersymmetric partner of the top quark ($\tilde{t}_1$) in $p\bar{p}$ collisions at a center-of-mass energy of 1.96 TeV at the Fermilab Tevatron collider corresponding to an integrated luminosity of 5.4 fb$^{-1}$. The scalar top quarks are assumed to decay into a $b$ quark, a charged lepton, and a scalar neutrino ($\tilde{\nu}$), and the search is performed in the electron plus muon final state. No significant excess of events above the standard model prediction is detected, and improved exclusion limits at the 95% C.L. are set in the the ($M_{\tilde{t}_1}$,$M_{\tilde{\nu}}$) mass plane.