We study idealizations of the full nonlinear Schwinger-Dyson equations for the asymptotically free theory of φ 3 in six dimensions in its meta-stable vacuum. We begin with the cubic nonlinearity and go on to all-order nonlinearities which contain instanton effects. In an asymptotically free theory the relevant Schwinger-Dyson equations are homogeneous and ultraviolet finite and perturbative methods fail from the outset. We show how our toy models of the cubic Schwinger-Dyson equations contain the usual diseases of perturbation theory in the massless limit (e.g., factorially-divergent β-functions, singular Borel-transform kernels associated with infrared renormalons) and show how these models yield specific mechanisms for removing such singularities when there is a mass gap. The solutions to these homogeneous equations, in spite of being ultraviolet finite, still depend on an undetermined parameter equivalent to the perturbative renormalization scale µ. In the all-order nonlinear equation we show how to recover the usual renormalization-groupimproved instanton effects and associated factorial divergences.
UCLA/95/TEP/20June 1995 * Electronic address: cornwall@physics.ucla.edu † Electronic address: dmorris@physics.ucla.edu
I. INTRODUCTIONAsymptotically free theories appear to have perturbatively calculable properties in the ultraviolet (UV) where the running coupling gets small. The well-known price paid for this convenience of perturbation theory in the UV is its complete breakdown at some low momentum scale, usually thought of as being of O(Λ RG ) where Λ RG is the renormalization group (RG) mass. However, as one goes to higher orders N in perturbation theory, the critical momentum scale grows exponentially in N essentially because the number of graphs grows as N!. That is to say, the contribution of the N th order term in perturbation theory behaves like N!(aḡ 2 ) N whereḡ ∼ (ln k 2 ) −1/2 is the running charge; to keep this term small as N grows requires an exponential increase in k 2 . There are also factorial divergence associated with renormalons [1]. Consequently, perturbation theory to all orders cannot be used for any momentum scale, however large, in an asymptotically free theory without understanding how to deal with non-Borel-summable factorial divergences. This is currently not a practical limit on perturbation theory in QCD, where the critical momentum does not creep into the UV until N is rather large, say N ≥ 10.A more practical issue is understanding QCD processes at infrared (IR) scales k ∼ Λ RG and the attendant phenomena of confinement, condensates, renormalons, large instantons, etc.[2] The purpose of the present paper is to discuss the factorial divergences mentioned above, as well as IR issues (except for confinement) in a toy model of asymptotic freedom [3]: φ 3 in six dimensions (φ 3 6 ). This theory is terminally ill because the Hamiltonian is unbounded below but we can go quite far before encountering pathologies. The idea is to study the theory at all momentum scales from the I...
We examine contributions to the anomalous magnetic moment of the muon from weak-isosinglet squarks found in E, superstring models. We find that such contributions are up to 2 orders of magnitude larger than those previously calculated and correspondingly require smaller Yukawa couplings in order to maintain agreement with the measured muon anomalous magnetic moment.
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