In the absence of a tree-level scalar-field mass, renormalization-group methods permit the explicit summation of leading-logarithm contributions to all orders of the perturbative series within the effective potential for SU(2)xU(1) electroweak symmetry. This improvement of the effective potential function is seen to reduce residual dependence on the renormalization mass scale. The all-orders summation of leading-logarithm terms involving the dominant three couplings contributing to radiative corrections is suggestive of a potential characterized by a plausible Higgs boson mass of 216 GeV. However, the tree potential's local minimum at phi=0 is restored if QCD is sufficiently strong.

For any perturbative series that is known to k-subleading orders of perturbation theory, we utilize the process-appropriate renormalization-group ͑RG͒ equation in order to obtain all-orders summation of series terms proportional to ␣ n log nϪk ( 2 ) with kϭ͕0,1,2,3͖, corresponding to the summation to all orders of the leading and subsequent-three-subleading logarithmic contributions to the full perturbative series. These methods are applied to the perturbative series for semileptonic b decays in both MS and pole-mass schemes, and they result in RG-summed series for the decay rates which exhibit greatly reduced sensitivity to the renormalization scale . Such summation via RG methods of all logarithms accessible from known series terms is also applied to perturbative QCD series for vector-and scalar-current correlation functions, the perturbative static potential function, the ͑single-doublet standard-model͒ Higgs decay amplitude into two gluons, as well as the Higgs-mediated high-energy cross section for W ϩ W Ϫ →ZZ scattering. The resulting RG-summed expressions are also found to be much less sensitive to the renormalization scale than the original series for these processes.and the successive-order series coefficients within S͓x,L͔, as defined by Eq. ͑1.1͒, are ͓1͔T 0,0 ϭ1, T 1,0 ϭ4.25360, T 1,1 ϭ5, T 2,0 ϭ26.7848, T 2,1 ϭ36.9902, T 2,2 ϭ17.2917. ͑1.5͒The five active-flavor pole-mass expression for the same rate is obtained by replacing m b () with the renormalizationscale independent pole mass m b pole in Eqs. ͑1.4͒ and ͑1.2͒, as well as a concomitant alteration of the following series coefficients ͓1͔: PHYSICAL REVIEW D 66, 014010 ͑2002͒

The canonical structure of the Einstein-Hilbert Lagrange density L = √ −gR is examined in two spacetime dimensions, using the metric density h µν ≡ √ −gg µν and symmetric affine connection Γ λ σβ as dynamical variables. The Hamiltonian reduces to a linear combination of three first class constraints with a local SO(2, 1) algebra. The first class constraints are used to find a generator of gauge transformations that has a closed off-shell algebra and which leaves the Lagrangian and det(h µν ) invariant. These transformations are distinct from diffeomorphism invariance, and are gauge transformations characterized by a symmetric matrix ζ µν .

Using the Dirac constraint formalism, we examine the canonical structure of the Einstein-Hilbert action S d = 1 16πG d d x √ −gR, treating the metric g αβ and the symmetric affine connection Γ λ µν as independent variables. For d > 2 tertiary constraints naturally arise; if these are all first class, there are d(d − 3) independent variables in phase space, the same number that a symmetric tensor gauge field φ µν possesses. If d = 2, the Hamiltonian becomes a linear combination of first class constraints obeying an SO(2, 1) algebra. These constraints ensure that there are no independent degrees of freedom. The transformation associated with the first class constraints is not a diffeomorphism when d = 2; it is characterized by a symmetric matrix ξ µν . We also show that the canonical analysis is different if h αβ = √ −g g αβ is used in place of g αβ as a dynamical variable when d = 2, as in d dimensions, det h αβ = −( √ −g) d−2 . A comparison with the formalism used in the ADM analysis of the Einstein-Hilbert action in first order form is made by applying this approach in the two dimensional case with h αβ and Γ λ µν taken to be independent variables.