= Higgs conjugate doublet
{c}
{title2=$\widetilde\phi=i\tau_2\phi^*$}
Set $\varepsilon=i\tau_2$, whose <Pauli matrix> identity is $\varepsilon\tau_i^*=-\tau_i\varepsilon$. Exponentiation gives $\varepsilon U^*=U\varepsilon$ for $U\in SU(2)$. Hence complex conjugating a <Higgs doublet> and multiplying by $\varepsilon$ restores the <fundamental representation> of $SU(2)$, while reversing <hypercharge>. A Higgs field of hypercharge $1/2$ yields a conjugate doublet of hypercharge $-1/2$. Its product $\widetilde\phi^\dagger L$ with a lepton doublet $L$ of hypercharge $-1/2$ transforms in the <trivial representation> of the whole electroweak <gauge group>. At vacuum $(0,v)^T/\sqrt2$, this composite is $v\nu_L/\sqrt2$.
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