In the convention $(SU(3)_c,SU(2)_L)_Y$, one <Standard Model fermion> generation including a <right-handed neutrino> is
$$
Q_L:(\mathbf3,\mathbf2)_{1/6},\quad
u_R:(\mathbf3,\mathbf1)_{2/3},\quad
d_R:(\mathbf3,\mathbf1)_{-1/3},
$$
$$
L_L:(\mathbf1,\mathbf2)_{-1/2},\quad
e_R:(\mathbf1,\mathbf1)_{-1},\quad
\nu_R:(\mathbf1,\mathbf1)_0.
$$
A bare <Dirac mass term> pairs left- and right-handed fields in the same gauge representation, but every charged left-handed Standard Model fermion is an $SU(2)_L$ doublet while its right-handed partner is a singlet. The <Higgs field> $H:(\mathbf1,\mathbf2)_{1/2}$ and $\widetilde H=i\sigma^2H^*:(\mathbf1,\mathbf2)_{-1/2}$ permit the gauge-invariant <Yukawa interactions>
$$
y_d\bar Q_LHd_R+y_u\bar Q_L\widetilde Hu_R
+y_e\bar L_LHe_R+y_\nu\bar L_L\widetilde H\nu_R+\text{h.c.}
$$
The neutral Higgs vacuum expectation value turns them into masses after <electroweak symmetry breaking>.

Because $\nu_R$ is a gauge singlet, it may also have a large <Majorana mass term> $M$. Together with its Dirac mass $m_D$, the <seesaw mechanism> gives a light neutrino mass $m_\nu\simeq m_D^2/M$. If no right-handed neutrino is retained, the same low-energy physics is encoded by the dimension-five <Weinberg operator> $(L_LH)(L_LH)/\Lambda$.

Now let $\phi$ be an <electroweak scalar triplet> with $Y=1$. Its $T^3$ weights are $1,0,-1$, and electric charge is $Q=T^3+Y$, so its components have charges
$$
\boxed{\phi^{++}:Q=2,\qquad\phi^+:Q=1,\qquad\phi^0:Q=0}.
$$
In the given matrix convention,
$$
\phi^{++}=\frac{\phi_1-i\phi_2}{2},
\qquad
\phi^+=\frac{\phi_3}{2},
\qquad
\boxed{\phi^0=\frac{\phi_1+i\phi_2}{2}}.
$$
Thus the electromagnetic-neutral direction satisfies $\phi_3=0$ and $\phi_1-i\phi_2=0$.

The gauge-invariant <type-II seesaw mechanism> Yukawa interaction is
$$
\boxed{\mathcal L_Y=-\frac12y_\phi
L_L^TC\,i\sigma^2\phi L_L+\text{h.c.}}.
$$
The two lepton doublets have total hypercharge $-1$, which is cancelled by $Y(\phi)=1$, and the displayed $SU(2)$ contraction is a singlet. Expanding it contains
$$
-\frac{y_\phi}{4}(\phi_1+i\phi_2)\nu_L^TC\nu_L+\cdots.
$$
Therefore a neutral condensate $\langle\phi^0\rangle\ne0$ preserves electromagnetism and generates a left-handed-neutrino <Majorana mass term> proportional to $y_\phi\langle\phi^0\rangle$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 4 / Solution

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The Weinberg operator is the dimension-five Standard Model operator $(LH)(LH)/\Lambda$. After the <Higgs field> condenses, it gives a left-handed neutrino Majorana mass of order $v^2/\Lambda$.

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 Incoming links: Weinberg operator