Planck-constant loop counting with source normalization (${\hslash }^{L-1}$)

In a scalar field path integral with weight $\exp [i(S+\int J\varphi )/\hslash ]$, each internal propagator contributes $\hslash$, each interaction vertex contributes ${\hslash }^{-1}$, and each source insertion contributes ${\hslash }^{-1}$. A connected graph with $I$ internal lines and $V$ interaction vertices has loop order $L=I-V+1$. Its amputated amplitude, or its coefficient in $\log Z[J]$ with the source insertions included, is proportional to ${\hslash }^{I-V}={\hslash }^{L-1}$. An ordinary unamputated connected $n$-field correlation function instead has the additional $n$ external propagator factors and scales as ${\hslash }^{L+n-1}$. Multiplication by $-i\hslash$ to form the connected generating functional shifts the source-kernel counting to ${\hslash }^L$. These are compatible normalization conventions.