Hypercharge of the complex Higgs doublet

I was wondering how one can see that the hypercharge of the complex Higgs doublet is $Y=\frac{1}{2}$. Complex Higgs doublet. $$\Phi(x) = \begin{pmatrix}\ \Phi^{+}(x)\\ \Phi^{0}(x) \end{pmatrix}$$ Lagrangian density: $$L = \frac{1}{2}\partial_{\mu}\Phi^{\dagger}\partial_{\mu}\Phi + \frac{m^2}{2}\vert{\Phi(x)}\vert^2 + \frac{\lambda}{4!}\vert{\Phi(x)}\vert^4$$ There is now a local $U(1)_Y$ symmetry called hypercharge, given by $\Phi'(x) = e^{-i\tfrac{1}{2}\varphi}\Phi(x)$. But how is the hypercharged deduced? I would really ask more specific questions, but I dont really get it and I am learning this stuff just for me, because I am interested how it works.

• The $SU(2)_L \times U(1)_Y$ is a (local) gauge symmetry, not a global symmetry. Use the formula $Q= T_3 + \frac{Y}{2}$ to obtain the hypercharge. Here $Q$ is the electric charge and $T_3$ is the eigenvalue of the $SU(2)$ $T_3$ generator ($\frac{1}{2}$ for the first member of the doublet and $-\frac{1}{2}$ for the second member of the doublet ). Your doublet is not "correct" because you have a different value of $Y$ for the two members of the doublet. You have to choose, for instance $(\Phi_+,\Phi_0)$, corresponding to $Y = 1$ or $(\Phi_0,\Phi_-)$, corresponding to $Y = -1$ – Trimok Jul 17 '14 at 9:53
• thanks for the remarks. I see that with $T_3 = \tfrac{1}{2}\sigma^3$ the eigenvalues are $\tfrac{1}{2}, -\tfrac{1}{2}$. Using this one gets Q=1 for $\Phi^{+}$ and Q = 0 for $\Phi^{0}$ with Y = 1. But why is there written $\tfrac{1}{2}$ and $-\tfrac{1}{2}$ ? – nerdizzle Jul 17 '14 at 10:59
• Sorry, I don't understand clearly your question... – Trimok Jul 17 '14 at 11:19
• is there a convention where one uses $Y_{\Phi}=\tfrac{1}{2}$ and $Y_{\tilde{\Phi}}=-\tfrac{1}{2}$? becuase that is what is written in my notes... – nerdizzle Jul 17 '14 at 11:31
• OK. I think some authors describe the hypercharge by $Q= T_3+Y$ instead of $Q= T_3 + \frac{Y}{2}$. So you have to check the definition in your notes... – Trimok Jul 17 '14 at 11:41

So, I reformulate your question: Why do we choose the hypercharge of the Higgs doublet to be $Y=1$? The idea is that we want to respect the Gell-Mann–Nishijima formula $Y=2(Q-T_3)$. Since we know the charges and the values of $T_3=\pm 1/2$ for the doublet, it is straightforward to find that $Y=1$.
There are several ways to see why the formula $Y=2(Q-T_3)$ must be respected:
• One possibility is to compute the commutator of the generators $Q$ and $T_i$. You will notice that $[Q,T_i]=[T_3,T_i]\neq0$, but $[Q-T_3,T_i]=0$. This means that the global symmetries $SU(2)$ and $U(1)_{\rm em}$ cannot be simultaneously satisfied, but we can define the hypercharge $Y$ as new Abelian symmetry.
• Other possibility is to try to assign Abelian charges for the fermionic multiplets of the Standard Model (knowing that the charge must be the same for all members of a multiplet). You will conclude that $Y=2(Q-T_3)$ is the only possible choice up to a multiplicative constant.