Fourth Painleve Equation and $PT$-Symmetric Hamiltonians

This paper is an addendum to earlier papers cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painleve I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the fourth Painleve transcendent are studied numerically and analytically. For a fixed initial value, say $y(0)=1$, a discrete set of initial slopes $y(0)=b_n$ give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y(0)=0$, a discrete set of initial values $y(0)=c_n$ give rise to separatrix solutions. For Painleve IV the large-$n$ asymptotic behavior of $b_n$ is $b_nsim B_{rm IV}n^{3/4}$ and that of $c_n$ is $c_nsim C_{rm IV} n^{1/2}$. The constants $B_{rm IV}$ and $C_{rm IV}$ are determined both numerically and analytically. The analytical values of these constants are found by reducing the nonlinear Painleve IV equation to the linear eigenvalue equation for the sextic $PT$-symmetric Hamiltonian $H=frac{1}{2} p^2+frac{1}{8} x^6$.

Relations between Heavy-light Meson and Quark Masses

The study of heavy-light meson masses should provide a way to determine renormalized quark masses and other properties of heavy-light mesons. In the context of lattice QCD, for example, it is possible to calculate hadronic quantities for arbitrary values of the quark masses. In this paper, we address two aspects relating heavy-light meson masses to the quark masses. First, we introduce a definition of the renormalized quark mass that is free of both scale dependence and renormalon ambiguities, and discuss its relation to more familiar definitions of the quark mass. We then show how this definition enters a merger of the descriptions of heavy-light masses in heavy-quark effective theory and in chiral perturbation theory ($chi$PT). For practical implementations of this merger, we extend the one-loop $chi$PT corrections to lattice gauge theory with heavy-light mesons composed of staggered fermions for both quarks. Putting everything together, we obtain a practical formula to describe all-staggered heavy-light meson masses in terms of quark masses as well as some lattice artifacts related to staggered fermions. In a companion paper, we use this function to analyze lattice-QCD data and extract quark masses and some matrix elements defined in heavy-quark effective theory.

Decay constants $f_B$ and $f_{B_s}$ and quark masses $m_b$ and $m_c$ from HISQ simulations

We present a progress report on our calculation of the decay constants $f_B$ and $f_{B_s}$ from lattice-QCD simulations with highly-improved staggered quarks. Simulations are carried out with several heavy valence-quark masses on $(2+1+1)$-flavor ensembles that include charm sea quarks. We include data at six lattice spacings and several light sea-quark masses, including an approximately physical-mass ensemble at all but the smallest lattice spacing, 0.03 fm. This range of parameters provides excellent control of the continuum extrapolation to zero lattice spacing and of heavy-quark discretization errors. Finally, using the heavy-quark effective theory expansion we present a method of extracting from the same correlation functions the charm- and bottom-quark masses as well as some low-energy constants appearing in the heavy-quark expansion.