253 research outputs found
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
Induced Gravity II: Grand Unification
As an illustration of a renormalizable, asymptotically-free model of induced
gravity, we consider an gauge theory interacting with a real scalar
multiplet in the adjoint representation. We show that dimensional transmutation
can occur, spontaneously breaking to while
inducing the Planck mass and a positive cosmological constant, all proportional
to the same scale . 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 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
Zero modes in de Sitter background
There are five well-known zero modes among the fluctuations of the metric of
de~Sitter (dS) spacetime. For Euclidean signature, they can be associated with
certain spherical harmonics on the sphere, viz., the vector
representation of the global isometry. They appear, for example,
in the perturbative calculation of the on-shell effective action of dS space,
as well as in models containing matter fields. These modes are shown to be
associated with collective modes of corresponding to certain coherent
fluctuations. When dS space is embedded in flat five dimensions they may
be seen as a legacy of translation of the center of the sphere. Rigid
translations of the -sphere on leave the classical action invariant
but are unobservable displacements from the point of view of gravitational
dynamics on Thus, unlike similar moduli, the center of the sphere is not
promoted to a dynamical degree of freedom. As a result, these zero modes do not
signify the possibility of physically realizable fluctuations or flat
directions for the metric of dS space. They are not associated with Killing
vectors on but can be with certain non-isometric, conformal Killing forms
that locally correspond to a rescaling of the volume element
For convenience, we frame our discussion in the context of renormalizable
gravity, but the conclusions apply equally to the corresponding zero modes in
Einstein gravity. We expect that these zero modes will be present to all orders
in perturbation theory. They will occur for Lorentzian signature as well, so
long as the hyperboloid is locally stable, but there remain certain
infrared issues that need to be clarified. We conjecture that they will appear
in any gravitational theory having dS background as a locally stable solution
of the effective action, regardless of whether additional matter is included.Comment: v4, 28pages, no figures; final journal form, minor changes in text
and refs from v
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