Effective Beta-Functions for Effective Field Theory

We consider the problem of determining the beta-functions for any reduced effective field theory. Even though not all the Green's functions of a reduced effective field theory are renormalizable, unlike the full effective field theory, certain effective beta- functions for the reduced set of couplings may be calculated without having to introduce vertices in the Feynman rules for redundant operators. These effective beta-functions suffice to apply the renormalization group equation to any transition amplitude (i.e., S- matrix element), thereby rendering reduced effective field theories no more cumbersome than traditionally renormalizable field theories. These effective beta-functions may equally be regarded as the running of couplings for a particular redefinition of the fields.Comment: 13 pages, LaTeX (requires JHEP class). Version 3: additional references and a slight expansion of Sections 3 and 5. No substantive change

Instantons and SL(2,R) Symmetry in Type IIB Supergravity

We discuss the relation between the dual formulations of Type IIB supergravity emphasizing the differences between Lorentz and Euclidean signature. We demonstrate how the SL(2,R) symmetry of the usual action is manifested in the solution of the equations of motion with Euclidean signature for the dual theory.Comment: 4 pages, no figure

Induced Gravity II: Grand Unification

As an illustration of a renormalizable, asymptotically-free model of induced gravity, we consider an $SO(10)$ gauge theory interacting with a real scalar multiplet in the adjoint representation. We show that dimensional transmutation can occur, spontaneously breaking $SO(10)$ to $SU(5){\otimes}U(1),$ while inducing the Planck mass and a positive cosmological constant, all proportional to the same scale $v$. All mass ratios are functions of the values of coupling constants at that scale. Below this scale (at which the Big Bang may occur), the model takes the usual form of Einstein-Hilbert gravity in de Sitter space plus calculable corrections. We show that there exist regions of parameter space in which the breaking results in a local minimum of the effective action, and a {\bf positive} dilaton $(\hbox{mass})^2$ from two-loop corrections associated with the conformal anomaly. Furthermore, unlike the singlet case we considered previously, some minima lie within the basin of attraction of the ultraviolet fixed point. Moreover, the asymptotic behavior of the coupling constants also lie within the range of convergence of the Euclidean path integral, so there is hope that there will be candidates for sensible vacua. Although open questions remain concerning unitarity of all such renormalizable models of gravity, it is not obvious that, in curved backgrounds such as those considered here, unitarity is violated. In any case, any violation that may remain will be suppressed by inverse powers of the reduced Planck mass.Comment: 44 pages, 5 figures, 2 tables. v2 has new discussion concerning stability of SSB plus related appendix. Additional references added. v3 is version to be published; contains minor revision