Renormalization Group Treatment of Nonrenormalizable Interactions

The structure of the UV divergencies in higher dimensional nonrenormalizable theories is analysed. Based on renormalization operation and renormalization group theory it is shown that even in this case the leading divergencies (asymptotics) are governed by the one-loop diagrams the number of which, however, is infinite. Explicit expression for the one-loop counter term in an arbitrary D-dimensional quantum field theory without derivatives is suggested. This allows one to sum up the leading asymptotics which are independent of the arbitrariness in subtraction of higher order operators. Diagrammatic calculations in a number of scalar models in higher loops are performed to be in agreement with the above statements. These results do not support the idea of the na\"ive power-law running of couplings in nonrenormalizable theories and fail (with one exception) to reveal any simple closed formula for the leading terms.Comment: LaTex, 11 page

Renormalizable 1/N_f Expansion for Field Theories in Extra Dimensions

We demonstrate how one can construct renormalizable perturbative expansion in formally nonrenormalizable higher dimensional field theories. It is based on $1/N_f$-expansion and results in a logarithmically divergent perturbation theory in arbitrary high space-time dimension. First, we consider a simple example of $N$-component scalar filed theory and then extend this approach to Abelian and non-Abelian gauge theories with $N_f$ fermions. In the latter case, due to self-interaction of non-Abelian fields the proposed recipe requires some modification which, however, does not change the main results. The resulting effective coupling is dimensionless and is running in accordance with the usual RG equations. The corresponding beta function is calculated in the leading order and is nonpolynomial in effective coupling. It exhibits either UV asymptotically free or IR free behaviour depending on the dimension of space-time. The original dimensionful coupling plays a role of a mass and is also logarithmically renormalized. We analyze also the analytical properties of a resulting theory and demonstrate that in general it acquires several ghost states with negative and/or complex masses. In the former case, the ghost state can be removed by a proper choice of the coupling. As for the states with complex conjugated masses, their contribution to physical amplitudes cancels so that the theory appears to be unitary.Comment: 32 pages, 20 figure

Character Expansion Methods for Matrix Models of Dually Weighted Graphs

We consider generalized one-matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices. We rederive in a simple fashion a character expansion formula for these models originally due to Itzykson and Di Francesco, and then demonstrate how to take the large N limit of this expansion. The relationship to the usual matrix model resolvent is elucidated. Our methods give as a by-product an extremely simple derivation of the Migdal integral equation describing the large $N$ limit of the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a number of models solvable by traditional means. We then proceed to solve a new model: a sum over planar graphs possessing even coordination numbers on both the original and the dual lattice. We conclude by formulating equations for the case of arbitrary sets of even, self-dual coupling constants. This opens the way for studying the deep problem of phase transitions from random to flat lattices.Comment: 22 pages, harvmac.tex, pictex.tex. All diagrams written directly into the text in Pictex commands. (Two minor math typos corrected. Acknowledgements added.

Mass enhancement, correlations, and strong coupling superconductivity in the beta-pyrochlore KOs2O6

To assess electron correlation and electron-phonon coupling in the recently discovered beta-pyrochlores KOs2O6 and RbOs2O6, we have performed specific heat measurements in magnetic fields up to 14 T. We present data from high quality single crystalline KOs2O6, showing that KOs2O6 is a strong coupling superconductor with a coupling parameter lambda_ep \approx 1.0 to 1.6 (RbOs2O6: lambda_ep \approx 1). The estimated Sommerfeld coefficient of KOs2O6, gamma=76 to 110 mJ/(mol K^2), is twice that of RbOs2O6 [gamma=44 mJ/(mol K^2)]. Using strong-coupling corrections, we extract useful thermodynamic parameters of KOs2O6. Quantifying lambda_ep allows us to determine the mass enhancement over the calculated band electronic density of states. A significant contribution in addition to the electron-phonon term of lambda_c=1.7 to 4.3 is deduced. In an effort to understand the origin of the enhancement mechanism, we also investigate an unusual energetically low-lying phonon. There are three phonon modes per RbOs2O6, suggestive of the phonon source being the rattling motion of the alkali ion. This dynamic instability of the alkali ions causes large scattering of the charge carriers which shows up in an unusual temperature dependence of the electrical resistivity.Comment: Accepted for publication in PR