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.

Two ways to common knowledge

It is not clear what a system for evidence-based common knowledge should look like if common knowledge is treated as a greatest fixed point. This paper is a preliminary step towards such a system. We argue that the standard induction rule is not well suited to axiomatize evidence-based common knowledge. As an alternative, we study two different deductive systems for the logic of common knowledge. The first system makes use of an induction axiom whereas the second one is based on co-inductive proof theory. We show the soundness and completeness for both systems.

The bulk transition of many-flavour QCD and the search for a UVFP at strong coupling

We explore the nature of the bulk transition observed at strong coupling in the SU(3) gauge theory with Nf=12 fermions in the fundamental representation. The transition separates a weak coupling chirally symmetric phase from a strong coupling chirally broken phase and is compatible with the scenario where conformality is restored by increasing the flavour content of a non abelian gauge theory. We explore the intriguing possibility that the observed bulk transition is associated with the occurrence of an ultraviolet fixed point (UVFP) at strong coupling, where a new theory emerges in the continuum.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).

Justifying induction on modal μ-formulae

We define a rank function for formulae of the propositional modal μ-calculus such that the rank of a fixed point is strictly bigger than the rank of any of its finite approximations. A rank function of this kind is needed, for instance, to establish the collapse of the modal μ-hierarchy over transitive transition systems. We show that the range of the rank function is ωω. Further we establish that the rank is computable by primitive recursion, which gives us a uniform method to generate formulae of arbitrary rank below ωω.

Full and hat inductive definitions are equivalent in NBG

A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA 0, those in second order set theory extending NBG (as well as those in n + 3-th order number theory extending the so-called Bernays−Gödel expansion of full n + 2-order number theory etc.) are. In this article, we establish the equivalence between Δ10\bf-LFP and Δ10\bf-FP, which assert the existence of a least and of a (not necessarily least) fixed point, respectively, for positive elementary operators (or between Δn+20\bf-LFP and Δn+20\bf-FP). Our proof also shows the equivalence between ID 1 and ^ID1, both of which are defined in the standard way but with the starting theory PA replaced by ZFC (or full n + 2-th order number theory with global well-ordering).