Renormalization Group Equations for the CKM matrix

We derive the one loop renormalization group equations for the Cabibbo-Kobayashi-Maskawa matrix for the Standard Model, its two Higgs extension and the minimal supersymmetric extension in a novel way. The derived equations depend only on a subset of the model parameters of the renormalization group equations for the quark Yukawa couplings so the CKM matrix evolution cannot fully test the renormalization group evolution of the quark Yukawa couplings. From the derived equations we obtain the invariant of the renormalization group evolution for three models which is the angle $\alpha$ of the unitarity triangle. For the special case of the Standard Model and its extensions with $v_{1}\approx v_{2}$ we demonstrate that also the shape of the unitarity triangle and the Buras-Wolfenstein parameters $\bar{\rho}=(1-{1/2}\lambda^{2})\rho$ and $\bar{\eta}=(1-{1/2}\lambda^{2})\eta$ are conserved. The invariance of the angles of the unitarity triangle means that it is not possible to find a model in which the CKM matrix might have a simple, special form at asymptotic energies.Comment: 9 page

The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions

We consider the problem of integrability of the Poisson equations describing spatial motion of a rigid body in the classical nonholonomic Suslov problem. We obtain necessary conditions for their solutions to be meromorphic and show that under some further restrictions these conditions are also sufficient. The latter lead to a family of explicit meromorphic solutions, which correspond to rather special motions of the body in space. We also give explicit extra polynomial integrals in this case. In the more general case (but under one restriction), the Poisson equations are transformed into a generalized third order hypergeometric equation. A study of its monodromy group allows us also to calculate the "scattering" angle: the angle between the axes of limit permanent rotations of the body in space

Criteria equivalent to the Riemann Hypothesis

We give a brief overview of a few criteria equivalent to the Riemann Hypothesis. Next we concentrate on the Riesz and B{\'a}ez-Duarte criteria. We proof that they are equivalent and we provide some computer data to support them. It is not compressed to six pages version of the talk delivered by M.W. during the XXVII Workshop on Geometrical Methods in Physics, 28 June -- 6 July, 2008, Bia{\l}owie{\.z}a, Poland.Comment: It is not compressed to six pages version of the talk delivered by M.W. during the XXVII Workshop on Geometrical Methods in Physics, 28 June -- 6 July, 2008, Bia{\l}owie{\.z}a, Poland. New Fig.1 is include

RG Invariance of the Pole Mass in the Minimal Subtraction Scheme

We prove the renormalization group(RG) invariance of the pole mass with respect to the RG functions of the minimal subtraction(MS) scheme and illustrate this in case of the the neutral scalar field theory both in the symmetric and in the broken symmetry phase

The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part II: The analytic continuation of the Lippmann-Schwinger bras and kets

The analytic continuation of the Lippmann-Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann-Schwinger equation is the rigged Hilbert space rather than just the Hilbert space. It is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution, resulting into a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.Comment: 32 pages, 3 figure

Testing the Standard Model and Schemes for Quark Mass Matrices with CP Asymmetries in B Decays

The values of $\sin (2 \alpha)$ and $\sin (2 \beta)$, where $\alpha$ and $\beta$ are angles of the unitarity triangle, will be readily measured in a B factory (and maybe also in hadron colliders). We study the standard model constraints in the $\sin (2 \alpha) - \sin (2 \beta)$ plane. We use the results from recent analyses of $f_B$ and $\tau_b|V_{cb}|^2$ which take into account heavy quark symmetry considerations. We find $\sin (2 \beta) \geq 0.15$ and most likely \sin (2 \beta) \roughly{>} 0.6, and emphasize the strong correlations between $\sin (2 \alpha)$ and $\sin (2 \beta)$. Various schemes for quark mass matrices allow much smaller areas in the $\sin (2 \alpha) - \sin (2 \beta)$ plane. We study the schemes of Fritzsch, of Dimopoulos, Hall and Raby, and of Giudice, as well as the ``symmetric CKM'' idea, and show how CP asymmetries in B decays will crucially test each of these schemes.Comment: 11 pages and 4 postscript figures available on request, LaTeX, WIS-92/52/Jun-PH, LBL-3256

Wolfenstein Parametrization Re-examined

The Wolfenstein parametrization of the $3\times 3$ Kobayashi-Maskawa (KM) matrix $V$ is modified by keeping its unitarity up to the accuracy of $O(\lambda^{6})$. This modification can self-consistently lead to the off-diagonal asymmetry of $V$: $|V_{ij}|^{2}-|V_{ji}|^{2}$ = $Z\displaystyle\sum_{k}\epsilon^{~}_{ijk}$ with $Z=\approx A^{2}\lambda^{6} (1-2\rho)$, which is comparable in magnitude with the Jarlskog parameter of $CP$ violation $J\approx A^{2}\lambda^{6}\eta$. We constrain the ranges of $J$ and $Z$ by using the current experimental data, and point out that the possibility of a symmetric KM matrix has almost been ruled out.Comment: 5 Latex pages including a figure; Two references are adde

On the inconsistency of the Bohm-Gadella theory with quantum mechanics

The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann-Schwinger equation nor they should appear. It is also shown that the Bohm-Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm-Gadella theory must therefore be abandoned.Comment: 16 page

The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part I

We exemplify the way the rigged Hilbert space deals with the Lippmann-Schwinger equation by way of the spherical shell potential. We explicitly construct the Lippmann-Schwinger bras and kets along with their energy representation, their time evolution and the rigged Hilbert spaces to which they belong. It will be concluded that the natural setting for the solutions of the Lippmann-Schwinger equation--and therefore for scattering theory--is the rigged Hilbert space rather than just the Hilbert space.Comment: 34 pages, 1 figur

Reply to ``Comment on `On the inconsistency of the Bohm-Gadella theory with quantum mechanics'''

In this reply, we show that when we apply standard distribution theory to the Lippmann-Schwinger equation, the resulting spaces of test functions would comply with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov, and of the theory of ultradistributions were wrong. As well, we point out several differences between the ``standard method'' of constructing rigged Hilbert spaces in quantum mechanics and the method used in Time Asymmetric Quantum Theory.Comment: 13 page