Staggered Chiral Perturbation Theory

We discuss how to formulate a staggered chiral perturbation theory. This amounts to a generalization of the Lee-Sharpe Lagrangian to include more than one flavor (i.e. multiple staggered fields), which turns out to be nontrivial. One loop corrections to pion and kaon masses and decay constants are computed as examples in three cases: the quenched, partially quenched, and full (unquenched) case. The results for the one loop mass and decay constant corrections have already been presented in Ref. [1].Comment: talk presented by C. Aubin at Lattice2002(spectrum); 3 pages, 1 figur

Light Light by Julie Joosten

Mathieu Aubin\u27s review of Light Light by Julie Joosten

Generalised L\"uroth expansions and a family of Minkowski's Question-Mark functions

The Minkowski's Question-Mark function is a singular homeomorphism of the unit interval that maps the set of quadratic surds into the rationals. This function has deserved the attention of several authors since the beginning of the twentieth century. Using different representations of real numbers by infinite sequences of integers, called $\alpha$-L\"uroth expansions, we obtain different instances of the standard shift map on infinite symbols, all of them topologically conjugated to the Gauss Map. In this note we prove that each of these conjugations share properties with the Minkowski's Question-Mark function: all of them are singular homeomorphisms of the interval, and in the "rational" cases, they map the set of quadratic surds into the set of rational numbers. In this sense, this family is a natural generalisation of the Minkowski's Question-Mark function

Comment on "Chiral anomalies and rooted staggered fermions"

In hep-lat/0701018, Creutz claims that the rooting trick used in simulations of staggered fermions to reduce the number of tastes misses key physics whenever the desired theory has an odd number of continuum flavors, and uses this argument to call into question the rooting trick in general. Here we show that his argument fails as the continuum limit is approached, and therefore does not imply any problem for staggered simulations. We also show that the cancellations necessary to restore unitarity in physical correlators in the continuum limit are a straightforward consequence of the restoration of taste symmetry.Comment: 11 pages, version 3 (4/13/07): Revisions to correspond to Creutz's latest posting, including a change in the title. Version to appear in Physics Letters

"Audacity or Precision": The Paradoxes of Henri Villat's Fluid Mechanics in Interwar France

In Interwar France, Henri Villat became the true leader of theoretical researches on fluid mechanics. Most of his original work was done before the First World War; it was highly theoretical and its applicability was questioned. After having organized the first post-WWI International Congress of Mathematicians in 1920, Villat became the editor of the famous Journal de math\'ematiques pure et appliqu\'es and the director of the influential book series "M\'emorial des sciences math\'ematiques." From 1929 on, he held the fluid mechanics chair established by the Air Ministry at the Sorbonne in Paris and was heading the government's critical effort invested in fluid mechanics. However, while both his wake theory and his turbulence theory seemingly had little success outside France or in the aeronautical industry (except in the eyes of his students), applied mathematics was despised by the loud generation of Bourbaki mathematicians coming of age in the mid 1930s. How are we to understand the contrasted assessments one can make of Villat's place in the history of fluid mechanics

Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

Staggered Chiral Perturbation Theory at Next-to-Leading Order

We study taste and Euclidean rotational symmetry violation for staggered fermions at nonzero lattice spacing using staggered chiral perturbation theory. We extend the staggered chiral Lagrangian to O(a^2 p^2), O(a^4) and O(a^2 m), the orders necessary for a full next-to-leading order calculation of pseudo-Goldstone boson masses and decay constants including analytic terms. We then calculate a number of SO(4) taste-breaking quantities, which involve only a small subset of these NLO operators. We predict relationships between SO(4) taste-breaking splittings in masses, pseudoscalar decay constants, and dispersion relations. We also find predictions for a few quantities that are not SO(4) breaking. All these results hold also for theories in which the fourth-root of the fermionic determinant is taken to reduce the number of quark tastes; testing them will therefore provide evidence for or against the validity of this trick.Comment: 39 pages, 6 figures (v3: corrected technical error in enumeration of a subset of NLO operators; final conclusions unchanged