Banks-Zaks fixed point analysis in momentum subtraction schemes

We analyse the critical exponents relating to the quark mass anomalous dimension and beta-function at the Banks-Zaks fixed point in Quantum Chromodynamics (QCD) in a variety of representations for the quark in the momentum subtraction (MOM) schemes of Celmaster and Gonsalves. For a specific range of values of the number of quark flavours, estimates of the exponents appear to be scheme independent. Using the recent five loop modified minimal subtraction (MSbar) scheme quark mass anomalous dimension and estimates of the fixed point location we estimate the associated exponent as 0.263-0.268 for the SU(3) colour group and 12 flavours when the quarks are in the fundamental representation.Comment: 33 latex pages, 25 tables, anc directory contains txt file with electronic version of renormalization group function

Fixed-Point Analysis of the Low-Energy Constants in the Pion-Nucleon Chiral Lagrangian

In the framework of heavy-baryon chiral perturbation theory, we investigate the fixed point structure of renormalization group equations (RGE) for the ratios of the renormalized low energy constants (LECs) that feature in the pion-nucleon chiral Lagrangian. The ratios of the LECs deduced from our RGE analysis are found to be in semi-quantitative agreement with those obtained from direct fit to the experimental data. The naturalness of this agreement is discussed using a simple dimensional analysis combined with Wilsonian RGEs.Comment: 10 page

Multi-shocks in asymmetric simple exclusions processes: Insights from fixed-point analysis of the boundary-layers

The boundary-induced phase transitions in an asymmetric simple exclusion process with inter-particle repulsion and bulk non-conservation are analyzed through the fixed points of the boundary layers. This system is known to have phases in which particle density profiles have different kinds of shocks. We show how this boundary-layer fixed-point method allows us to gain physical insights on the nature of the phases and also to obtain several quantitative results on the density profiles especially on the nature of the boundary-layers and shocks.Comment: 12 pages, 8 figure

Linear Asymptotic Convergence of Anderson Acceleration: Fixed-Point Analysis

We study the asymptotic convergence of AA($m$), i.e., Anderson acceleration with window size $m$ for accelerating fixed-point methods $x_{k+1}=q(x_{k})$, $x_k \in R^n$. Convergence acceleration by AA($m$) has been widely observed but is not well understood. We consider the case where the fixed-point iteration function $q(x)$ is differentiable and the convergence of the fixed-point method itself is root-linear. We identify numerically several conspicuous properties of AA($m$) convergence: First, AA($m$) sequences $\{x_k\}$ converge root-linearly but the root-linear convergence factor depends strongly on the initial condition. Second, the AA($m$) acceleration coefficients $\beta^{(k)}$ do not converge but oscillate as $\{x_k\}$ converges to $x^*$. To shed light on these observations, we write the AA($m$) iteration as an augmented fixed-point iteration $z_{k+1} =\Psi(z_k)$, $z_k \in R^{n(m+1)}$ and analyze the continuity and differentiability properties of $\Psi(z)$ and $\beta(z)$. We find that the vector of acceleration coefficients $\beta(z)$ is not continuous at the fixed point $z^*$. However, we show that, despite the discontinuity of $\beta(z)$, the iteration function $\Psi(z)$ is Lipschitz continuous and directionally differentiable at $z^*$ for AA(1), and we generalize this to AA($m$) with $m>1$ for most cases. Furthermore, we find that $\Psi(z)$ is not differentiable at $z^*$. We then discuss how these theoretical findings relate to the observed convergence behaviour of AA($m$). The discontinuity of $\beta(z)$ at $z^*$ allows $\beta^{(k)}$ to oscillate as $\{x_k\}$ converges to $x^*$, and the non-differentiability of $\Psi(z)$ allows AA($m$) sequences to converge with root-linear convergence factors that strongly depend on the initial condition. Additional numerical results illustrate our findings