By starting from the Fermi theory and requirement of the tree-unitarity of this theory (it is similar to renormalizability, but only on a tree level) you may build theory of electroweak interactions (even with Higgs boson). I'm only show you how does it work.

Fermi theory predicts growth the matrix element of neutrino-lepton scattering as $E^{2}$ (or $\left(\frac{s}{4}\right)$). This tells us that Fermi theory is only the effective theory. But it is very similar to the low-energy limit of the second order of perturbative theory with lagrangian
$$
L = g \bar{l}_{L}\gamma_{\mu}\nu_{L}W^{\mu} + h.c.
$$
Indeed, if we discuss neutrino-lepton scattering, we will get
$$
M = -g^{2}\bar{l}_{L}(p_{4})\gamma^{\mu}\nu_{L} (p_{2})\bar{\nu}_{L}(p_{3})\gamma^{\nu}l_{L}(p_{1})\frac{\left( g_{\mu \nu} - \frac{q_{\mu}q_{\nu}}{m_{w}^{2}}\right)}{q^{2} - m_{W}^{2}}.
$$
At the limit $E^{2} <<m_{W}^{2}$ you will get
$$
M = \frac{g^{2}}{m_{W}^{2}}\bar{l}_{L}(p_{4})\gamma^{\mu}\nu_{L} (p_{2})\bar{\nu}_{L}(p_{3})\gamma_{\mu}l_{L}(p_{1}).
$$
So we have made the first step from effective theory to renormalizing theory.

The second one is to add EM interactions for $W$-boson. But the requirement of the unitarity in the process $W^{-}W^{+}$ tells us that we can't restrict ourselves to the minimal Lagrangian (the minimal lagrangian can be earned by the elongation of the derivative $\partial_{\mu} \to \partial_{\mu} - ieA_{\mu}$ in the free one for $W$-boson); we also must add the term $-iW^{\mu}(W^{\dagger})^{\nu}F_{\mu \nu}$.

The third step is to discuss the unitarity of some processes included leptons and W-bosons. For example, we can see that both of processes $W^{-}W^{-} \to W^{-}W^{-}$ and $l^{+}l^{-} \to W^{+}W^{-}$ violate the unitarity (first is proportional to $E^{4}$ while the second one is proportional to $E^{2}$). We can add the interaction with real spin-one massive particle $Z$, with real spin-zero massive particle $h$ and/or with charged spin-one-half massive particle. As it can be shown, the last case contradicts the experimental data (because it doesn't predict neutral current and the correct value of mass of particle), while the first two delete $E^{4}$- and $E^{2}$-terms respectively.

But how exactly to build this interaction? We need to include all of the terms in the lagrangian which don't have coupling constant with negative dimension. For example, $W,Z$ interaction may be written in a form
$$
g_{1}W^{2}Z^{2} + g_{2}(W^{+} \cdot Z )(W^{-} \cdot Z) + g_{WWZ}(EM-type),
$$
where "EM-type"-terms coincide with corresponding $W,A$-three-vertex terms;

$Z, W, A$-interaction we may written in a form
$$
g_{3}(W^{-} \cdot A )(W^{+} \cdot Z) + g_{4}W^{2}(A \cdot Z) + g_{5}(W^{-}\cdot Z)(W^{+} \cdot A),
$$
$W, h$ (similarly as $Z, h$) - in a form
$$
g_{6}W^{2}h^{2} + g_{7}W^{2}h.
$$
Also we need the self-interaction terms (for scalar boson and for $W$-boson):
$$
g_{8}(W^{+} \cdot W^{-})^{2} + g_{9}(W^{-})^{2}(W^{+})^{2} + g_{10}h^{4} + g_{11}h^{3}.
$$
The interaction of $Z, h$ with leptons may be written in a form
$$
g_{12}\bar{\nu}_{L}\gamma^{\alpha}\nu_{L}Z_{\alpha} + (g_{L}\bar{l}_{L}\gamma^{\alpha} l_{L} + g_{R}\bar{l}_{R}\gamma^{\alpha}l_{R})Z_{\alpha}
$$
I repeat once more: all these terms are invented not accidental. Each of them delete some violating-unitary terms in the amplitude of some process (by fixing the value of constants). For example, requirement of the unitarity of processes $e^{+}e^{-} \to W^{+}W^{-}, \bar{\nu}\nu \to W^{+}W^{-}, e\bar{\nu} \to W^{-}Z$ leads us to the system of equations:
$$
-\frac{1}{2}g^{2}+ g_{WWZ}g_{12} = 0, \quad -\frac{1}{2}g^{2} + e^{2} - g_{L}g_{WWZ} = 0,
$$
$$
e^{2} -g_{R}g_{WWZ} = 0 = 0, \quad -g_{L} + g_{12} - g_{WWZ} = 0,
$$
$$
g_{R} - g_{12}+ g_{WWZ}\left( 1 - \frac{m_{Z}^{2}}{2m_{W}^{2}}\right) = 0.
$$
From this set of equations we can determine constants $g_{L},g_{R}, g_{12},g_{WWZ}$ and get relation $m_{W} = m_{Z}\sqrt{1 - \frac{e^{2}}{g^{2}}}$.

This post imported from StackExchange Physics at 2014-08-07 15:35 (UCT), posted by SE-user Andrew McAddams