Which 1-loop Feynman diagrams are possible from this Lagrangian interaction term?

Question

$$ L_{int} = g \bar{L} \cdot \tilde H N $$ Where $g$ is the Yukawa coupling constant, $L$ is a lepton doublet, $H$ is the Higgs and $N$ is a right-handed neutrino. I think that at tree level only $\bar{N}+N \rightarrow\bar{L}+L$ is possible but I am interested in scattering, which I am told only occurs at 1-loop. How is this possible? I believe for scattering I am looking for a process $ \bar{L}+N \rightarrow\bar{L}+N$ but don't understand why this isn't possible at tree level by $ \bar{L}+N \rightarrow H \rightarrow \bar{L}+N$?

Answer 1

NB. I corrected your original term, by using the conjugate representation, $\tilde H$, instead of your $H^\dagger$. The term you wrote had the correct weak hypercharge neutrality, since $1=Y(\bar L)= -Y(L)=Y(H)=-Y(H^\dagger)=-Y(\tilde H)$; however, it was not weak-isospin invariant, and hence violated charge!

Given the added context of your comment, which you really should have included in your question, the relevant components in your term + h.c. are $$ g_Y(\overline{\nu_L} \phi^{*~0} \nu_R - \overline{e_L} \phi^{-} \nu_R ) +\hbox{h.c.}, $$ where, you recall, $g_Y$ is tiny, given that it likely is the origin of the small Dirac mass of the neutrinos!

So, for scattering of $\nu_R$s, (not $\overline {\nu_R}$s), off electrons (not positrons), your only option for $\nu_R ~ e_L^-\to \nu_R ~ e_L^- $ is the 1-loop box diagram:

On the upper line, you have $\nu_R \to e_L^- + \phi^+ $, the $\phi^+$ going vertically down to convert the electron of the lower line to $\nu_R$.

The second two vertices are, on the upper line, $e^-_L\to \nu_R+ \phi^-$, with the $\phi^-$ descending to convert the $\nu_R$ on the lower line to an electron.

(For completeness, you'd also have to include two neutral Higgs field exchanges and neutrinos and electrons on the top line and bottom lines of the box, respectively, but, in any case, appreciate how freakishly small this $O(g_Y^4)$ amp would be...)

As you indirectly noticed, a sterile antineutrino would, indeed, merge with an electron to a $\phi^-$ in the s channel, and would, indeed, support scattering at tree level. Check the chiralities, if you were so inclined.