naturally small because it arises at the one-loop level.

An extension of the Higgs Triplet Model {#sec:1-loop}
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Since we try to generate the $\mu$-term in the HTM radiatively, the term must be forbidden at the tree level. The simplest way would be to impose lepton number conservation to the Lagrangian. The conservation is assumed to be broken by the VEV of a new scalar field $s_1^0$ which is singlet under the SM gauge symmetry. Notice that $s_1^0$ \[$= (s_{1r}^0 + i s_{1i}^0)/\sqrt{2}$\] is a complex (“charged”) field with non-zero lepton number although it is electrically neutral. One might think that the VEV of $s_1^0$ could be generated by using soft breaking terms of $L\#$. However, the $\mu$-term is also a soft breaking term. Therefore lepton number must be broken spontaneously in our scenario. One may worry about Nambu-Goldstone boson corresponds to the spontaneous breaking of the lepton number conservation (the so-called Majoron, $J^0$). However the Majoron which comes from gauge singlet field can evade experimental searches (constraints) because it interacts very weakly with matter fields [@majoron]. It is also possible to make it absorbed by a gauge boson by introducing the $\text{U}(1)_{{\text{B}-\text{L}}}$ gauge symmetry to the model (See, e.g., Ref. [@BL]). In this paper we just accept the Majoron without assuming the $\text{U}(1)_{{\text{B}-\text{L}}}$ gauge symmetry for simplicity.

If $s_1^0$ has $L\# = -2$, we can have a dimension-4 operator $\lambda\, s_1^0\, \Phi^T i\sigma_2 \Delta^\dagger \Phi$. This gives a trivial result $\mu = \lambda \langle s_1^0 \rangle$ at the tree level. Although the dim.-4 operator could be forbidden by some extra global symmetries with extra scalars to break them, we do not take such a possibility in this paper. We just assume the $s_1^0$ has $L\#=-1$. Then the lepton number conserving operator which results in the $\mu$-term is of dimension-5 as $$\begin{aligned}
(s_1^0)^2 \Phi^T i\sigma_2 \Delta^\dagger \Phi .
\label{