Double scaling limit for the O(N)3-invariant tensor model

We study the double scaling limit of the O(N)3-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the 1/N expansion as a finite sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the 1/N expansion. This leads to a double scaling limit which picks up contributions from all orders in the 1/N expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of the Feynman graphs, as well as an analytic analysis of the singularities of the relevant generating series.

Dissecting the $$\Delta I=1/2$$ rule at large $$N_c$$

We study the scaling of kaon decay amplitudes with the number of colours, $$N_c$$Nc, in a theory with four degenerate flavours, $$N_f=4$$Nf=4. In this scenario, two current-current operators, $$Q^\pm $$Q±, mediate $$\Delta S=1$$ΔS=1 transitions, such as the two isospin amplitudes of non-leptonic kaon decays for $$K\rightarrow (\pi \pi )_{I=0,2}$$K→(ππ)I=0,2, $$A_0$$A0 and $$A_2$$A2. In particular, we concentrate on the simpler $$K\rightarrow \pi $$K→π amplitudes, $$A^\pm $$A±, mediated by these two operators. A diagrammatic analysis of the large-$$N_c$$Nc scaling of these observables is presented, which demonstrates the anticorrelation of the leading $${{\mathcal {O}}}(1/N_c)$$O(1/Nc) and $${{\mathcal {O}}}(N_f/N_c^2)$$O(Nf/Nc2) corrections in both amplitudes. Using our new $$N_f=4$$Nf=4 and previous quenched data, we confirm this expectation and show that these corrections are naturally large and may be at the origin of the $$\Delta I=1/2$$ΔI=1/2 rule. The evidence for the latter is indirect, based on the matching of the amplitudes to their prediction in Chiral Perturbation Theory, from which the LO low-energy couplings of the chiral weak Hamiltonian, $$g^\pm $$g±, can be determined. A NLO estimate of the $$K \rightarrow (\pi \pi )_{I=0,2}$$K→(ππ)I=0,2 isospin amplitudes can then be derived, which is in good agreement with the experimental value.

A diagrammatic analysis of the variational perturbation method for classical fluids

The statistical mechanics of classical fluids can be approached from the particle perspective, where the focus is on the positions of the particles, or from the field perspective, where the focus is on the form of the interaction fields generated by the particles. These two perspectives can be combined through the variational perturbation method.

Evaluator, Choreographer, Ideologue, Catalyst: The Disparate Reception Histories of Alexander Klein's Graphical Method

The graphical method propounded by Russian German Israeli architect Alexander Klein during the late 1920s evaluates the qualities of architectural plans through a process of diagrammatic analysis following purportedly objective criteria. In Evaluator, Choreographer, Ideologue, Catalyst: The Disparate Reception Histories of Alexander Klein's Graphical Method, Christoph Lueder examines the reception and adaptation of Klein's method. Ernst Löwitsch reinterpreted Klein's analytical notation as choreography of domestic life. Following Klein's forced emigration from Nazi Germany, Frank Gloor rediscovered Klein's graphical method and transformed and adapted it into a scientific method classifying degrees of flexibility. Catherine Bauer disseminated the method to the English-speaking world under a new title, “Functional Housing for Frictionless Living,” which led to Robin Evans's enduring indictment of Klein's diagrams as emblematic of reductive functionalism. Throughout its reception, the graphical method has been viewed at various times as a methodology of scientific evaluation, a choreography of everyday life, an indictment of functionalist ideology, and a catalyst for new working methodologies.