Walking near a conformal fixed point

We investigate the 2-d O(3) model with a q-term as a toy model for slowly walking 4-d non-Abelian gauge theories. Using the very efficient meron-cluster algorithm, an accurate investigation of the scale dependence of the renormalized coupling is carried out for different values of the vacuum angle q. Approaching q = p, the infrared dynamics of the 2-d O(3) model is determined by a non-trivial conformal fixed point. We provide evidence for a slowly walking behavior near the fixed point and we perform a finite-size scaling analysis of the mass gap.

Asymptotic freedom, dimensional transmutation, and an infrared conformal fixed point for the δ-function potential in one-dimensional relativistic quantum mechanics

We consider the Schrödinger equation for a relativistic point particle in an external one-dimensional δ-function potential. Using dimensional regularization, we investigate both bound and scattering states, and we obtain results that are consistent with the abstract mathematical theory of self-adjoint extensions of the pseudodifferential operator H=p2+m2−−−−−−−√. Interestingly, this relatively simple system is asymptotically free. In the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point. Thus it can be used to illustrate nontrivial concepts of quantum field theory in the simpler framework of relativistic quantum mechanics.

Chiral-Scale Perturbation Theory About an Infrared Fixed Point

We review the failure of lowest order chiral SU(3)L ×SU(3)R perturbation theory χPT3 to account for amplitudes involving the f0(500) resonance and O(mK) extrapolations in momenta. We summarize our proposal to replace χPT3 with a new effective theory χPTσ based on a low-energy expansion about an infrared fixed point in 3-flavour QCD. At the fixed point, the quark condensate ⟨q̅q⟩vac ≠ 0 induces nine Nambu-Goldstone bosons: π,K,η and a QCD dilaton σ which we identify with the f0(500) resonance. We discuss the construction of the χPTσ Lagrangian and its implications for meson phenomenology at low-energies. Our main results include a simple explanation for the ΔI = 1/2 rule in K-decays and an estimate for the Drell-Yan ratio in the infrared limit.

Δ I = 1 / 2 rule for kaon decays derived from QCD infrared fixed point

This article gives details of our proposal to replace ordinary chiral SU(3)L×SU(3)R perturbation theory χPT3 by three-flavor chiral-scale perturbation theory χPTσ. In χPTσ, amplitudes are expanded at low energies and small u,d,s quark masses about an infrared fixed point αIR of three-flavor QCD. At αIR, the quark condensate ⟨q¯q⟩vac≠0 induces nine Nambu-Goldstone bosons: π,K,η, and a 0++ QCD dilaton σ. Physically, σ appears as the f0(500) resonance, a pole at a complex mass with real part ≲ mK. The ΔI=1/2 rule for nonleptonic K decays is then a consequence of χPTσ, with a KSσ coupling fixed by data for γγ→ππ and KS→γγ. We estimate RIR≈5 for the nonperturbative Drell-Yan ratio R=σ(e+e−→hadrons)/σ(e+e−→μ+μ−) at αIR and show that, in the many-color limit, σ/f0 becomes a narrow qq¯ state with planar-gluon corrections. Rules for the order of terms in χPTσ loop expansions are derived in Appendix A and extended in Appendix B to include inverse-power Li-Pagels singularities due to external operators. This relates to an observation that, for γγ channels, partial conservation of the dilatation current is not equivalent to σ-pole dominance.

Remarks on the round-off errors in iterative processes for fixed-point computers /

"May 10, 1962."

The recursive nature of descriptions : a fixed point /

"UIUC-ENG-R-75-2539."

Positive solutions for nonlinear m-point Eigenvalue problems

In this paper, we are concerned with determining values of lambda, for which there exist positive solutions of the nonlinear eigenvalue problem [GRAPHICS] where a, b, c, d is an element of [0, infinity), xi(i) is an element of (0, 1), alpha(i), beta(i) is an element of [0 infinity) (for i is an element of {1, ..., m - 2}) are given constants, p, q is an element of C ([0, 1], (0, infinity)), h is an element of C ([0, 1], [0, infinity)), and f is an element of C ([0, infinity), [0, infinity)) satisfying some suitable conditions. Our proofs are based on Guo-Krasnoselskii fixed point theorem. (C) 2004 Elsevier Inc. All rights reserved.

Quantum entanglement and fixed-point bifurcations

How does the classical phase-space structure for a composite system relate to the entanglement characteristics of the corresponding quantum system? We demonstrate how the entanglement in nonlinear bipartite systems can be associated with a fixed-point bifurcation in the classical dynamics. Using the example of coupled giant spins we show that when a fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum state-the ground state-achieves its maximum amount of entanglement near the critical point. We conjecture that this will be a generic feature of systems whose classical limit exhibits such a bifurcation.