Tikz-Feynman: Alignment of multiple diagrams

Basically you are looking for the baseline key. However, I should say that I had some difficulties in precisely aligning them when keeping your original syntax, which is why now the second diagram is also of the form \begin{feynman}...\end{feynman}. Not sure what the problem was, but this seems to work:

\documentclass{article}
\usepackage{tikz-feynman}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[baseline=(b.base)]
\begin{feynman}
\diagram [horizontal=a to b, layered layout] {
  a [particle=\(h_0\)] -- [scalar] b -- [scalar] c [particle={\(h_0\)}]
};
\path (b)--++(90:0.5) coordinate (A);
\draw [dashed] (A) circle(0.5) node[left] at (1.0,0.5) {$\tilde{f}$};
\end{feynman}
\end{tikzpicture}\quad
\begin{tikzpicture}[baseline=(b.base)]
\begin{feynman}
\diagram[layered layout, horizontal=b to c] {
a [particle=\(h_0\)]-- [scalar] b
-- [scalar, half left, edge label=\(\tilde{f}\)] c
-- [scalar, half left, edge label=\(\tilde{f}\)] b,
c -- [scalar] d [particle=\(h_0\)],
};
\end{feynman}
\end{tikzpicture}
\end{figure}
\end{document}

Putting the two in one tikzpicture does the trick

\begin{tikzpicture}
\begin{feynman}
\diagram [horizontal=a to b, layered layout] {
  a [nudge=(180:5mm),particle=\(h_0\)] -- [scalar] b --
  b -- [scalar] c [nudge=(0:5mm),particle=\(h_0\)]
};
\path (b)--++(90:0.7) coordinate (A);
\draw [dashed] (A) circle(0.7) node[left] at (1.25,0.7) {$\tilde{f}$};
\diagram [xshift=5cm,layered layout, horizontal=b to c] {
a [particle=\(h_0\)]-- [scalar] b
-- [scalar, half left, edge label=\(\tilde{f}\)] c
-- [scalar, half left, edge label=\(\tilde{f}\)] b,
c -- [scalar] d [particle=\(h_0\)],
};
\end{feynman}
\end{tikzpicture}