Strong and Weak Phases from Time-Dependent Measurements of B→ππB \to \pi \pi

Time-dependence in $B^0(t) \to \pi^+ \pi^-$ and \ob(t) \to \pi^+ \pi^- is utilized to obtain a maximal set of information on strong and weak phases. One can thereby check theoretical predictions of a small strong phase $\delta$ between penguin and tree amplitudes. A discrete ambiguity between $\delta \simeq 0$ and $\delta \simeq \pi$ may be resolved by comparing the observed charge-averaged branching ratio predicted for the tree amplitude alone, using measurements of $B \to \pi l \nu$ and factorization, or by direct comparison of parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix with those determined by other means. It is found that with 150 fb$^{-1}$ from BaBar and Belle, this ambiguity will be resolvable if no direct CP violation is found. In the presence of direct CP violation, the discrete ambiguity between $\delta$ and $\pi - \delta$ becomes less important, vanishing altogether as $|\delta| \to \pi/2$. The role of measurements involving the lifetime difference between neutral $B$ eigenstates is mentioned briefly.Comment: 14 pages, LaTeX, 5 figures, to be published in Phys. Rev. D. Updated version with one reference change

Adjoining a universal inner inverse to a ring element

Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\langle q\mid pqp= p\rangle.$ We obtain normal forms for elements of $R',$ and for elements of $R'$-modules arising by extension of scalars from $R$-modules. The details depend on where in the chain $pR\cap Rp \subseteq pR\cup Rp \subseteq pR + Rp \subseteq R$ the unit $1$ of $R$ first appears. This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant. We end with a normal form result for the algebra obtained by tying together a $k$-algebra $R$ given with a nonzero element $p$ satisfying $1\notin pR+Rp$ and a $k$-algebra $S$ given with a nonzero $q$ satisfying $1\notin qS+Sq,$ via the pair of relations $p=pqp,$ $q=qpq.$Comment: 28 pages. Results on mutual inner inverses added at end of earlier version, and much clarification of wording etc.. After publication, any updates, errata, related references etc. found will be recorded at http://math.berkeley.edu/~gbergman/paper

Lie polynomials in an algebra defined by a linearly twisted commutation relation

We present an elementary approach in characterizing Lie polynomials in the generators $A,B$ of an algebra with a defining relation that is in the form of a deformed or twisted commutation relation $AB=\sigma(BA)$ where the deformation or twisting map $\sigma$ is a linear polynomial with a slope parameter that is not a root of unity. The class of algebras defined as such encompasses $q$-deformed Heisenberg algebras, rotation algebras, and some types of $q$-oscillator algebras whose deformation parameters are not roots of unity, and so we have a general solution for the Lie polynomial characterization problem for these algebras

Quantum automorphism groups and SO(3)-deformations

We show that any compact quantum group having the same fusion rules as the ones of $SO(3)$ is the quantum automorphism group of a pair $(A, \varphi)$, where $A$ is a finite dimensional $C^*$-algebra endowed with a homogeneous faithful state. We also study the representation category of the quantum automorphism group of $(A, \varphi)$ when $\varphi$ is not necessarily positive, generalizing some known results, and we discuss the possibility of classifying the cosemisimple (not necessarily compact) Hopf algebras whose corepresentation semi-ring is isomorphic to that of $SO(3)$.Comment: Comments are welcom