I am studying the basics of renormalisation. In 10.3 of Weinberg QFT book, motivated by the renomalisation of field, he express the bare Lagrangian as two part
L=L0+L1,
L0=−12∂μΦ∂μΦ−12m2Φ2,
L1=−12(Z−1)(∂μΦ∂μΦ+m2Φ2)+12Zδm2Φ2−V(Φ),
where
V(Φ)≡V(√ZΦB)
After expressing the corrected propagator as a geometric series of uncorrected propagator times the sum of one-particle-irreducible graphs Π∗, the only thing need to calculate is the Π∗, but he states:
In calculating Π∗, we encounter a tree graph arising from a single insertion of vertices corresponding to the terms in L1 proportional to ∂μΦ∂μΦ and Φ2, plus a term Π∗LOOP arising from loop graphs like that in Figure 10.4(a):
Π∗(q2)=−(Z−1)[q2+m2]+Zδm2+Π∗LOOP(q2).
What's the tree graph arising from a single insertion of vertices that corresponds to the terms in L1 ? And why they are proportional?