First Observation of Cabibbo-Suppressed Ξ

Definition \thedefinition .

Given a real constant $k$ , we let $\textbf{{s}}_{k}$ denote the solution to the ordinary differential equation

\left\{\begin{aligned} \displaystyle\phi^{\prime\prime}+k\phi&\displaystyle=0,% \\ \displaystyle\phi(0)=0,&\displaystyle\phi^{\prime}(0)=1\end{aligned}\right.

(1.1)

Setting $\textbf{{c}}_{k}(t)=\textbf{{s}}^{\prime}_{k}(t)$ , we clearly get $\textbf{{c}}^{\prime}_{k}(t)=-k\textbf{{s}}_{k}(t)$ and $\textbf{{c}}_{k}$ satisfies $\textbf{{c}}_{k}^{\prime\prime}+k\textbf{{c}}_{k}=0$ , with initial condition $\textbf{{c}}_{k}(0)=1,\textbf{{c}}^{\prime}_{k}(0)=0$ .

Definition 1 (PEL) .

PEL is the class of functions that can be described by

f(n)={}^{p(n)}2,

where $p$ can be any polynomial and the left superscript denotes tetration.

The complexity class PEL is the class of languages recognisable by a TM in time $f(n)$ , where $f(n)$ is in PEL and $n$ is the bit-length of the input. Equivalently, PEL is the class of languages recognisable by a TM working on a tape of size $f(n)$ , where $f(n)$ is in PEL.

PEL can therefore also be described as either PEL-TIME or PEL-SPACE.

Definition 2.3 .

Two factors of automorphy $\alpha$ and $\beta$ are called equivalent if there is a holomorphic function $h:\widetilde{U}\rightarrow\mathbb{C}^{*}$ satisfying

\beta(\lambda,z)=h(z+\lambda)\alpha(\lambda,z)h^{-1}(z).

Two summands of automorphy are called equivalent if the induced factors of automorphy are equivalent.

Definition 3.14

A map $f:{\mathbb{R}}^{3}\longrightarrow{\mathbb{R}}^{3}$ such that

f(p)=f(-p)

is said to have the antipodal property.

Definition 10

A reduction $\rho$ between two chain complexes $C_{\ast}=(C_{n},d_{n})_{n\in\mathbb{Z}}$ and $D_{\ast}=(D_{n},\hat{d}_{n})_{n\in\mathbb{Z}}$ , denoted by $\rho:C_{\ast}\mbox{\,$\Rightarrow\hskip-9.0pt\Rightarrow$\,}D_{\ast}$ , is a triple $\rho=(f,g,h)$ where $f:C_{\ast}\rightarrow D_{\ast}$ and $g:D_{\ast}\rightarrow C_{\ast}$ are chain complex morphisms, $h=\{h_{n}:C_{n}\rightarrow C_{n+1}\}_{n\in\mathbb{Z}}$ is a family of module morphism, and the following properties are satisfied:

1)

$f\circ g=id$ ;
2)

$g\circ f+d\circ h+h\circ d=id$ ;
3)

$f\circ h=0$ ; $h\circ g=0$ ; $h\circ h=0$ .

Definition 1 (Clifford geometric algebra) .

A Clifford geometric algebra $\mathcal{G}_{p,q}$ is defined by the associative geometric product of elements of a quadratic vector space $\mathbb{R}^{p,q}$ , their linear combination and closure. $\mathcal{G}_{p,q}$ includes the field of real numbers $\mathbb{R}$ and the vector space $\mathbb{R}^{p,q}$ as subspaces. The geometric product of two vectors is defined as

\mathrm{\mathbf{a}}\mathrm{\mathbf{b}}=\mathrm{\mathbf{a}}\cdot\mathrm{\mathbf% {b}}+\mathrm{\mathbf{a}}\wedge\mathrm{\mathbf{b}},

(1)

where $\mathrm{\mathbf{a}}\cdot\mathrm{\mathbf{b}}$ indicates the standard inner product and the bivector $\mathrm{\mathbf{a}}\wedge\mathrm{\mathbf{b}}$ indicates Grassmann’s antisymmetric outer product. $\mathrm{\mathbf{a}}\wedge\mathrm{\mathbf{b}}$ can be geometrically interpreted as the oriented parallelogram area spanned by the vectors $\mathrm{\mathbf{a}}$ and $\mathrm{\mathbf{b}}$ . Geometric algebras are graded, with grades (subspace dimensions) ranging from zero (scalars) to $n=p+q$ (pseudoscalars, $n$ -volumes).

Definition 1.21 .

Given $\mathbf{v}\in{\mathcal{U}}^{n}$ , $\mathcal{A}(\mathbf{v})$ is the set of all $\mathbf{u}\in{\mathcal{U}}^{n}\setminus\mathbf{0}$ s.t.

	$\displaystyle\mathbf{u}[i]=0,\mbox{ if }\mathbf{v}[i]=0,$
	$\displaystyle\mathbf{u}[i]={\Delta\!}^{\!+},\mbox{ if }\mathbf{v}[i]={\Delta\!% }^{\!+},$
	$\displaystyle\mathbf{u}[i]={\Delta\!}^{\!+}\mbox{ or }\mathbf{u}[i]=0,\mbox{ % if }\mathbf{v}[i]=\Delta.$

Definition 6 (Semiring of series)

The set of formal power series with coefficients in a semiring $\mathcal{S}$ endowed with the following sum and Cauchy product:

	$\displaystyle s\oplus s^{\prime}:(s\oplus s^{\prime})(k)=s(k)\oplus s^{\prime}% (k),$
	$\displaystyle s\otimes s^{\prime}:(s\otimes s^{\prime})(k)=\bigoplus_{i+j=k}s(% i)\otimes s^{\prime}(j),$

is a semiring denoted $\mathcal{S}[\![z_{1},...,z_{p}]\!]$ . If $\mathcal{S}$ is complete, $\mathcal{S}[\![z_{1},...,z_{p}]\!]$ is complete. A series with a finite support is called a polynomial, and a monomial if there is only one element in the series. The greatest lower bound of series is given by :

s\wedge s^{\prime}:(s\wedge s^{\prime})(k)=s(k)\wedge s^{\prime}(k).

Theorem 3.2 .

Under conditions $({\cal A}_{1}),({\cal A}_{2})$ , $({\cal A}_{4})$ and $({\cal A}_{7})$ there exists a Radon measure $\varrho\in\mathcal{M}(X)$ and a measurable function $c:X\to\mathbb{R}$ such that

\pi(\mu,\,\cdot\,)=\varrho+c\mu.

(4)

In particular, $\varrho=\pi(0,\,\cdot\,)$ and $c$ is the (signed) density $c(x)=\pi(\delta_{x},\{x\})-\varrho(\{x\})$ .

Definition 3.1 .

Let $\mathfrak{g}$ be a Lie algebra whose centre is $\mathfrak{z}$ and let $\mathfrak{v}:=\mathfrak{z}^{\perp}$ .We say that $\mathfrak{g}$ is of Heisenberg-type (or simply $H$ -type) if

[\mathfrak{v},\mathfrak{v}]=\mathfrak{z}

and there exists an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak{g}$ with $\left<\mathfrak{z},\mathfrak{v}\right>=0$ such that for any $Z\in\mathfrak{z}$ , the map $J_{Z}:\mathfrak{v}\mapsto\mathfrak{v}$ given by

\langle J_{Z}X,Y\rangle=\langle[X,Y],Z\rangle,

for $X,Y\in\mathfrak{v}$ , is an orthogonal map whenever $\left<Z,Z\right>=1$ . An $H$ -type group is a connected and simply connected Lie group $\mathbb{G}$ whose Lie algebra is of $H$ -type.