Recursion Relations in pp-adic Mellin Space

In this work, we formulate a set of rules for writing down $p$-adic Mellin amplitudes at tree-level. The rules lead to closed-form expressions for Mellin amplitudes for arbitrary scalar bulk diagrams. The prescription is recursive in nature, with two different physical interpretations: one as a recursion on the number of internal lines in the diagram, and the other as reminiscent of on-shell BCFW recursion for flat-space amplitudes, especially when viewed in auxiliary momentum space. The prescriptions are proven in full generality, and their close connection with Feynman rules for real Mellin amplitudes is explained. We also show that the integrands in the Mellin-Barnes representation of both real and $p$-adic Mellin amplitudes, the so-called pre-amplitudes, can be constructed according to virtually identical rules, and that these pre-amplitudes themselves may be re-expressed as products of particular Mellin amplitudes with complexified conformal dimensions.Comment: 45 pages + appendices, several figure

Developing Competences Designed to Create Customer Value

This paper focuses on how small, specialized suppliers can gain competitive advantage by acting as a potential for their scale intensive producing customers in achieving competitive advantage. Of special interest is how a shared under-standing of ‘value’ for the customer is obtained, transferred and implemented in the specialized supplier’s production of process equipment. The study draws on theory on networks and specialized suppliers as well as interviews with key in-formants in three specialized supplier companies for the aluminum industry. An important finding is that the constellation of the specialized supplier’s network changes as the project moves from planning to production. With these changes, the role of specialized suppliers in the value creation process also changes. It seems to be an important competence for small, specialized supplier to be able to draw on and manage this network in their value creation process.

O(N) and O(N) and O(N)

Three related analyses of $\phi^4$ theory with $O(N)$ symmetry are presented. In the first, we review the $O(N)$ model over the $p$-adic numbers and the discrete renormalization group transformations which can be understood as spin blocking in an ultrametric context. We demonstrate the existence of a Wilson-Fisher fixed point using an $\epsilon$ expansion, and we show how to obtain leading order results for the anomalous dimensions of low dimension operators near the fixed point. Along the way, we note an important aspect of ultrametric field theories, which is a non-renormalization theorem for kinetic terms. In the second analysis, we employ large $N$ methods to establish formulas for anomalous dimensions which are valid equally for field theories over the $p$-adic numbers and field theories on $\mathbb{R}^n$. Results for anomalous dimensions agree between the first and second analyses when they can be meaningfully compared. In the third analysis, we consider higher derivative versions of the $O(N)$ model on $\mathbb{R}^n$, the simplest of which has been studied in connection with spatially modulated phases. Our general formula for anomalous dimensions can still be applied. Analogies with two-derivative theories hint at the existence of some interesting unconventional field theories in four real Euclidean dimensions.Comment: 44 pages, 8 figure