Superficial degree of divergence for scalar theories

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CAF123
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I have a few questions regarding the derivation of the degree of divergence for feynman diagrams. The result is $$D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]$$ (following notation in Srednicki, ##P118##)

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?
 

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nrqed
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I have a few questions regarding the derivation of the degree of divergence for feynman diagrams. The result is $$D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]$$ (following notation in Srednicki, ##P118##)

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?
Hi,

Well, I would not say that we are "summing over all possible scalar theories", that's misleading. We are consider one diagram in a single theory. Now, this diagram has a certain number of external lines, let's call it "k", and ##[g_E]## is the dimension of the coefficient of the term with k powers of the scalar field in the potential. For example, if there are 7 external lines, ##[g_E]## is the dimension of the coefficient of ##\phi^7##. Then the sum is over all the vertices in the diagram.
 
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CAF123
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Hi nqred,

Srednicki begins by writing down $$\mathcal L = - \frac{1}{2} Z_{\phi} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}Z_m m^2 \phi^2 - \sum_{n=3}^{\infty} \frac{1}{n!} Z_n g_n \phi^n$$ which seems to me to be a summation over all possible scalar theories, i.e in such a theory we have a ##\phi^3, \phi^4...## etc interaction. Is it not? Suppose we have a diagram with E external legs. Then in ##D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]##, the first term ##[g_E]## corresponds to the tree level local interaction vertex that occurs when ##E=i## for some i in the interval [3,infinity]. So we can write $$D = [g_E] - V_i[g_i] - \sum_{n \neq i}^{\infty} V_n [g_n] = (1-V_i)[g_E] - \sum_{n \neq i}^{\infty} V_n [g_n]$$ Each of the ##[g_n]## have a different mass dimensionality in a fixed number of space time dimensions.

Am I thinking about this wrongly? Edit: The last equality I wrote must be wrong because it doesn't reproduce the correct D for some diagrams. I was thinking of the one loop box correction to the ##\phi^3## local vertex interaction. In d=6, the coupling has null mass dimension and we only have one theory (phi^3) so Srednicki's formula reduces to ##D = [g_E] - V_3 [g_3]##. In this case, ##V_3 = 4## and ##[g_E]## = 0. But this doesn't recover the D=-2 you get by power counting.

Thanks!
 
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nrqed
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Hi nqred,

Srednicki begins by writing down $$\mathcal L = - \frac{1}{2} Z_{\phi} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2}Z_m m^2 \phi^2 - \sum_{n=3}^{\infty} \frac{1}{n!} Z_n g_n \phi^n$$ which seems to me to be a summation over all possible scalar theories, i.e in such a theory we have a ##\phi^3, \phi^4...## etc interaction. Is it not?
Hi again!

Well, it is just a question of terminology. The way I see it is that we are considering one theory (because we are dealing with a single lagrangian) but this theory may contain an infinite number of terms. Even if there is an infinite number of terms it is still one theory.



Suppose we have a diagram with E external legs. Then in ##D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]##, the first term ##[g_E]## corresponds to the tree level local interaction vertex that occurs when ##E=i## for some i in the interval [3,infinity]. So we can write $$D = [g_E] - V_i[g_i] - \sum_{n \neq i}^{\infty} V_n [g_n] = (1-V_i)[g_E] - \sum_{n \neq i}^{\infty} V_n [g_n]$$ Each of the ##[g_n]## have a different mass dimensionality in a fixed number of space time dimensions.

Am I thinking about this wrongly?
Yes, we can write this.

Edit: The last equality I wrote must be wrong because it doesn't reproduce the correct D for some diagrams. I was thinking of the one loop box correction to the ##\phi^3## local vertex interaction. In d=6, the coupling has null mass dimension and we only have one theory (phi^3)
Not necessarily! One can also have ##\phi^4,\phi^5 \ldots ## terms! The theory will be nonrenormalizable but it is still perfectly well defined as an effective field theory

so Srednicki's formula reduces to ##D = [g_E] - V_3 [g_3]##. In this case, ##V_3 = 4## and ##[g_E]## = 0. But this doesn't recover the D=-2 you get by power counting.

Thanks!
Watch out. This box diagram has four external lines. Therefore ##[g_E]## is the dimension of the coefficient of an ##\phi^4## interaction would have (note that it does not matter if there is such an interaction in the Lagrangian or not, ##[g_E]## is the dimension of the coefficient of such an interaction if it was present) and for a ##\phi^4## interaction in 6 dimensions, the coefficient would have dimension -2. So for your diagram, ##[g_E]=-2## and therefore D=-2.
 
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