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The Hadronic Current

The hadronic current can be divided in one part corresponding to conservation of strageness $(\Delta S = 0)$ and in another which changes strangeness $(\Delta s \neq 0)$:

h^{\sigma}(x) = h_{(0)}^{\sigma}(x) \cos \theta_{C}
+ h_{(1)}^{\sigma}(x) \sin \theta_{C}
\end{displaymath} (83)

where the mixing angle $\theta_{C}$ is constant and is called Cabbibo angle. This angle was introduced in order to take into account the fact that the reactions with $\Delta S \neq 0$ have much less amplitude than the ones preserving strtangeness. Universality in the hadronic world is then written:
$\displaystyle G^{2}$ $\textstyle =$ $\displaystyle G_{(0)}^{2}  \cos^{2} \theta_{C} + G_{(1)}^{2} \sin^{2} \theta_{C}$ (84)
$\displaystyle (leptonic)$   $\displaystyle (hadronic)$ (85)

The phenomenology indicates that each one of the parts of (85) has a vectorial and an axial component, namely

$\displaystyle h_{(0)}^{\sigma}(x)$ $\textstyle =$ $\displaystyle V_{(0)}^{\sigma}(x) - A_{(0)}^{\sigma}(x)$ (86)
$\displaystyle h_{(1)}^{\sigma}(x)$ $\textstyle =$ $\displaystyle V_{(1)}^{\sigma}(x) - A_{(1)}^{\sigma}(x)$ (87)

that is, has the structure (V-A) as in the leptonic case.

As a complete theory of the strong interactions of hadrons is not avaliable, it is not possible to give a simple expression of the hadronic current in terms of operators for hadrons. Nevertheless we can go deeper in the level of elementarity and take the quark model (then it will be the basis of the quark theory : QCD). The model of quarks comes from the hypothesis that the hadrons are bound states of quarks $(u,d,s,c,b,t)$.

In order to present the usual phenomenology we can restrict ourselves to a model with only light quarks $(u,d,s)$. We have then a unitary internal symmetry of flavor, $SU(3)_{f}$. Under this symmetry, the quark triplet

q(x) = \left( \begin{array}{c}
u(x)  d(x)  s(x)
\end{array} \right)
\end{displaymath} (88)

is transformed as
q'(x) = (1+ i \frac{\lambda_{k}}{2} \epsilon_{k}) q(x)
\end{displaymath} (89)

where $\lambda_{k}; (k=1,...,8)$ are the $3\times3$ matrices of $SU(3)$. If the masses of quarks are equal, the equation of motion of the free quarks is:
(i \gamma^{\sigma} \partial_{\sigma} - m) q(x) = 0
\end{displaymath} (90)

It is then found a conserved vector current

V_{k}^{\sigma} = \bar{q}(x) \gamma^{\sigma} \frac{\lambda_{k}}{2} q(x)
\end{displaymath} (91)

and a non conserved axial current
A_{k}^{\sigma} = \bar{q}(x) \gamma^{\sigma} \gamma^{5} \frac{\lambda_{k}}{2} q(x)
\end{displaymath} (92)

\partial_{\sigma} A_{k}^{\sigma}(x) = i m \bar{q}(x) \gamma^{5} \lambda_{k} q(x)
\end{displaymath} (93)

In terms of the currents so defined, the electromagnetic current is

J_{em}^{\sigma}(x) = V_{3}^{\sigma}(x) + \frac{1}{\sqrt{3}} V_{8}^{\sigma}(x)
\end{displaymath} (94)

meanwhile the weak components with $\Delta S = 0$ and $\Delta S = 1$ are:
$\displaystyle V_{(0)}^{\sigma(x)}$ $\textstyle =$ $\displaystyle V_{1}^{\sigma}(x) + i  V_{2}^{\sigma}(x)$ (95)
$\displaystyle A_{(0)}^{\sigma(x)}$ $\textstyle =$ $\displaystyle A_{1}^{\sigma}(x) + i  A_{2}^{\sigma}(x)$ (96)
$\displaystyle V_{(1)}^{\sigma(x)}$ $\textstyle =$ $\displaystyle V_{4}^{\sigma}(x) + i  V_{5}^{\sigma}(x)$ (97)
$\displaystyle A_{(1)}^{\sigma(x)}$ $\textstyle =$ $\displaystyle A_{4}^{\sigma}(x) + i  A_{5}^{\sigma}(x)$ (98)

The Cabbibo hypothesis for the hadronic current is written

V^{\sigma}(x) = a V_{(0)}^{\sigma}(x) + b V_{(1)}^{\sigma}(x)
\end{displaymath} (99)

with $a=\cos \theta_{C}$ and $b=\sin \theta_{C}$. Due to the fact that the only invariant under SU(3) is $(a^{2} + b^{2} )^{1/2}$, it results that the condition $a^{2}+b^{2}=1$ (Cabbibo universality) is also invariant, but as $\theta_{C}$ is experimentally fixed in
\sin \theta_{C} \simeq 0.21 \pm 0.03
\end{displaymath} (100)

one inmediatelly concludes that there is a breaking of $SU(3)_{f}$.

next up previous
Next: Unitarity Limit Up: Examples Previous: Examples
root 2001-01-22