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The massive gravity propagator goes like $\sim \frac{p^2}{m^4}$ at high energies and in this case we cannot apply Weinberg's standard power counting arguments.

I have read something like that in an article but didn't understand what is Weinberg's power counting arguments and how to use them. The original article of Weinberg, with the title "High-Energy Behavior in Quantum Field Theory", is too complicated for me. Can anybody explain this method in a more simpler way?

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  • $\begingroup$ "apply Weinberg's standard power counting arguments" to do what ? $\endgroup$
    – picop
    Apr 22, 2016 at 14:41

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Let's look at the analog of the massive photon. The propagator for this is $$ \frac{ g^{\mu\nu} - \frac{p^\mu p^\nu}{m^2} }{ p^2 + m^2 } $$ At large energies, this scales like $\frac{p^\mu p^\nu }{m^2 p^2} \sim \frac{1}{m^2}$.

In the same way, the massive graviton propagator takes the general form $$ \frac{g^{\mu\alpha} g^{\nu\beta} + g^{\mu\beta} g^{\nu\alpha} - \# g^{\mu\nu} g^{\alpha\beta} + \# \frac{ p^\mu p^\alpha }{ m^2 } g^{\nu\beta} + \# \frac{ p^\nu p^\beta }{ m^2 } g^{\mu\alpha} + \# \frac{ p^\mu p^\nu }{ m^2 } g^{\alpha\beta} + \# \frac{ p^\alpha p^\beta }{ m^2 } g^{\mu\nu} + \# \frac{p^\mu p^\nu p^\alpha p^\beta }{ m^4 } }{p^2 + m^2 } $$ where $\#$ are numbers whose values I cannot recall off the top of my head. They are fixed by requirements that when contracted with $p_\mu$ or $p_\nu$ or $p_\alpha$ or $p_\beta$ or $g_{\alpha\beta}$ or $g_{\mu\nu}$ it vanishes.

At high energies, this propagator scales like $$ \frac{p^\mu p^\nu p^\alpha p^\beta }{ p^2 m ^4 } \sim \frac{p^2 }{ m^4 } ~. $$

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  • $\begingroup$ The q title mentions a theorem? $\endgroup$
    – innisfree
    Apr 22, 2016 at 16:35
  • $\begingroup$ The question is not about the behavior of the propagators at high energies, what I wonder is how can we apply Weinberg's power counting theorem and deduce if a theory is renormalizable or not. $\endgroup$
    – sahin
    Apr 24, 2016 at 20:36

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