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I am trying to understand what ##[g_E]## is here? Since in this set up we are summing over all possible scalar theories ##n \in [3, \infty)##, we will have a tree level local interaction diagram corresponding to the case where ##E=i## where ##i## is some element of the set ##[3,\infty)##. So is it right to say that ##[g_E]## denotes a particular element of the set ##[3,\infty)##?

If that is correct, then in the theory $$\mathcal L = \frac{1}{2} Z_{\phi} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}Z_m m^2 \phi^2 - \frac{1}{k!} Z_g g \phi^k$$ (i.e a theory where we now include only one of the elements in the above set) the formula for ##D## now becomes ##D = [g_E] - v_k [g]## and in this case ##[g_E] = [g]##, with ##v_k## the number of times a vertex shows up in some diagram? Is that correct understanding?