Tadpole renormalization and relativistic corrections in lattice NRQCD

We make a comparison of two tadpole renormalization schemes in the context of the quarkonium hyperfine splittings in lattice NRQCD. Improved gauge-field and NRQCD actions are analyzed using the mean-link $u_{0,L}$ in Landau gauge, and using the fourth root of the average plaquette $u_{0,P}$. Simulations are done for $c\bar c$, $b\bar c$, and $b\bar b$ systems. The hyperfine splittings are computed both at leading and at next-to-leading order in the relativistic expansion. Results are obtained at lattice spacings in the range of about 0.14~fm to 0.38~fm. A number of features emerge, all of which favor tadpole renormalization using $u_{0,L}$. This includes much better scaling behavior of the hyperfine splittings in the three quarkonium systems when $u_{0,L}$ is used. We also find that relativistic corrections to the spin splittings are smaller when $u_{0,L}$ is used, particularly for the $c\bar c$ and $b\bar c$ systems. We also see signs of a breakdown in the NRQCD expansion when the bare quark mass falls below about one in lattice units. Simulations with $u_{0,L}$ also appear to be better behaved in this context: the bare quark masses turn out to be larger when $u_{0,L}$ is used, compared to when $u_{0,P}$ is used on lattices with comparable spacings. These results also demonstrate the need to go beyond tree-level tadpole improvement for precision simulations.Comment: 14 pages, 7 figures (minor changes to some phraseology and references

Noise propagation in urban and industrial areas

Noise propagation in streets and the discrepancies between theoretical analyses and field measurements are discussed. A cell-model is used to estimate the general background level of noise due to vehicular sources distributed over the urban area

Mean link versus average plaquette tadpoles in lattice NRQCD

We compare mean-link and average plaquette tadpole renormalization schemes in the context of the quarkonium hyperfine splittings in lattice NRQCD. Simulations are done for the three quarkonium systems $c\bar c$, $b\bar c$, and $b\bar b$. The hyperfine splittings are computed both at leading and at next-to-leading order in the relativistic expansion. Results are obtained at a large number of lattice spacings. A number of features emerge, all of which favor tadpole renormalization using mean links. This includes much better scaling of the hyperfine splittings in the three quarkonium systems. We also find that relativistic corrections to the spin splittings are smaller with mean-link tadpoles, particularly for the $c\bar c$ and $b\bar c$ systems. We also see signs of a breakdown in the NRQCD expansion when the bare quark mass falls below about one in lattice units (with the bare quark masses turning out to be much larger with mean-link tadpoles).Comment: LATTICE(heavyqk) 3 pages, 2 figure

Additive Entropies of degree-q and the Tsallis Entropy

The Tsallis entropy is shown to be an additive entropy of degree-q that information scientists have been using for almost forty years. Neither is it a unique solution to the nonadditive functional equation from which random entropies are derived. Notions of additivity, extensivity and homogeneity are clarified. The relation between mean code lengths in coding theory and various expressions for average entropies is discussed.Comment: 13 page

Quarkonium spin structure in lattice NRQCD

Numerical simulations of the quarkonium spin splittings are done in the framework of lattice nonrelativistic quantum chromodynamics (NRQCD). At leading order in the velocity expansion the spin splittings are of $O(M_Q v^4)$, where $M_Q$ is the renormalized quark mass and $v^2$ is the mean squared quark velocity. A systematic analysis is done of all next-to-leading order corrections. This includes the addition of $O(M_Q v^6)$ relativistic interactions, and the removal of $O(a^2 M_Q v^4)$ discretization errors in the leading-order interactions. Simulations are done for both S- and P-wave mesons, with a variety of heavy quark actions and over a wide range of lattice spacings. Two prescriptions for the tadpole improvement of the action are also studied in detail: one using the measured value of the average plaquette, the other using the mean link measured in Landau gauge. Next-to-leading order interactions result in a very large reduction in the charmonium splittings, down by about 60% from their values at leading order. There are further indications that the velocity expansion may be poorly convergent for charmonium. Prelimary results show a small correction to the hyperfine splitting in the Upsilon system.Comment: 16 pages, REVTEX v3.1, 5 postscript figures include