Electroweak theory has two coupling constants before and after Spontaneous Symmetry Breaking (SSB) each one for $SU(2)_L$ and $U(1)_Y$, though they are connected by Weinberg angle after SSB. My question is, how is unification complete with two independent couplings before SSB. The motive for unification is a single unified force with certain range and (coupling)strength.
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6$\begingroup$ You are indeed correct that the $SU(2)_L\times U(1)$ is not a complete unification. This is however achieved when one embed $SU(2)_L\times U(1)$ in a simple Lie Group, for example $SU(5)$. The latter is called the Georgi-Glashow model:en.wikipedia.org/wiki/Georgi%E2%80%93Glashow_model: In this model the standard model gauge groups $SU(3) × SU(2) × U(1)$ are combined into a single simple gauge group $SU(5)$. $\endgroup$– Anne O'NymeCommented Jun 1, 2014 at 8:24
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2 Answers
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The two parts of the electroweak gauge group do not separately describe the weak and the electromagnetic force.
The unified group here is not $$ SU(2)_\mathrm{weak} \times U(1)_\mathrm{em}$$ but rather $$ SU(2)_L \times U(1)_Y $$ and the electric charge arises as a linear combination of hypercharge and weak isospin.
Therefore, although the group is not simple, the weak and electromagnetic forces have been unified, giving rise to two other forces.
Edit: To adress the matter of coupling constants:
Indeed, before SSB there are two independent coupling constants $g'$ (for the $U(1)$) and $g$ (for $SU(2)$). One way to relate them to parameters after SSB is to think of the couplings constant $g$ vanishing, but a new parameter arising: the Weinberg angle. The Weinberg angle $\theta_W$ determines, what linear combination of of the neutral vector bosons $W_3$ from $SU(2)$ and $B$ from $U(1)$ turn into the massive $Z$ boson and what combination turns into the massless $\gamma$.
The Weinberg angle is determined through the gauge couplings as $$ \cos\theta_w = \frac{g}{\sqrt{g^2 + g'^2}}.$$ In other words, in the breaking $SU(2)_L \times U(1)_Y$ both groups get broken, but there exists a linear combination of generators that remains unbroken. The $U(1)$ spanned by this generator does not relate 1:1 to either gauge group before SSB, though!
The coupling constant for the photon now relates to the couplings before SSB through the Weinberg angle $$ e = g \sin\theta_w = g' \cos\theta_w.$$
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$\begingroup$ also, before SSB, we have a U(1) coupling constant g' and hypercharge Y in the covariant term. How are they connected to each other? In QED covariant derivative (e.g.), we have electric charge as the only coupling constant. $\endgroup$ Commented Jul 21, 2014 at 9:20
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Unification in the EW theory means that weak interactions that are short range are not very difference at short distances than electromagnetism, both are described by spin-1 gauge boson exchange with couplings that are similar in size. They differ mainly at large distances where the mass of the mediators becomes important, killing the interactions at distances $r>1/m_{W}$. In this sense, EW unification is just meant for the conceptual understanding of these two seemingly different forces that are instead described by very similar equations and coupling constants.
