An Objective: the Weak Interactions

To establish an (or the) objective of the formalism of the gauge theories which we are going to develop it seems interesting to present a phenomenological introduction of the weak interactions preparatory to the unified electroweak model $2! = SU(2)_{L} \times U(1)_{Y}$.

The weak interactions are the responsible of: ex2html_comment_mark>54 0)bean

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The continuous spectra of electrons emitted in the $\beta$ decay of radioactive nuclei.

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Hadron decays.

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All the reactions induced by neutrinos.

These interactions are characterized by the Fermi coupling constant

$$G_{F}\simeq \frac{10^{-5}}{m_{p}^{2}} \simeq 1.4\;10^{-47} erg cm^{3}$$ (59)

and by violating the parity symmetry.

To maintain the validity of the energy-momentum conservation law in the weak processes, induced Pauli to postulate the existence of the neutrino: neutral particle, of spin $1/2$, massless (or almost) which only interacts weakly.

The ``basic" reaction, at the hadronic level, is the neutron decay

$$n\rightarrow p + e + \bar{\nu}_{e}$$ (60)

and for its phenomenological description, Fermi proposed the interaction Lagrangian

$${\cal L}_{F} = G_{F} {\cal J}_{\mu}^{(n p)}(x)\cdot {\cal J}^{\mu (e \bar{\nu})}$$ (61)

$$= G_{F}[\bar{\psi}_{p}(x) \gamma_{\mu} \psi_{n}(x)] [\bar{\psi}_{e}(x) \gamma^{\mu} \psi_{\nu_{e}}(x)]$$ (62)

where the first current is associated to the transition of one neutron in one proton and the second corresponds to the pair $(e, \bar{\nu}_{e})$. All the fields present in the previous currents are written in the same point corresponding to a point interaction. The Fermi proposal was certainly inspired by the Lagrangian of electromagnetic interaction.

$${\cal L}_{em}= e j_{em}^{\mu}(x) A_{\mu}(x) = e [\bar{\psi}(x) \gamma^{\mu} \psi(x)] [A_{\mu}(x)]$$ (63)

Later, experimental evidence of other weak reactions different from neutron decay (62) was found. Besides Dirac's theory indicates that other types of coupling are also possible. The generalization of the Fermi Lagrangian (64) is

$${\cal L} = G_{F} \sum_{i} \alpha_{i} [\bar{\psi}_{p}(x) \... ...si_{n}(x)] [\bar{\psi}_{e}(x) \Gamma_{i} \psi_{\nu_{e}}(x)]$$ (64)

with the 16 independent matrices $\Gamma_{i}$ constructed with Dirac's matrices $\gamma$ and the identity. The election of a given $\Gamma$ gives place to a precise transformation law of the corresponding current. For example:

$$\Gamma_{i} \bar{\psi} \Gamma_{i} \psi$$

$$1 S : \bar{\psi} \psi$$

$$\gamma_{\mu} V : \bar{\psi} \gamma_{\mu} \psi$$

$$\gamma_{5} P : \bar{\psi} \gamma_{5} \psi$$

$$\sigma_{\mu} \gamma_{5} A : \bar{\psi} \gamma_{\mu} \gamma_{5} \psi$$