Second order diagram for the "scalar graviton"

AI Thread Summary
The discussion focuses on deriving the second-order diagram for the scalar graviton, starting with the equations for the first-order perturbation, h_0 and h_1. The author substitutes h = h_0 + h_1 + h_2 into the equation of motion, leading to the result for h_2 as ##\Box h_2 = 2 \lambda h_0 h_1##. The factor of 2 indicates that the proposed last diagram is indeed correct. This analysis contributes to understanding the behavior of scalar gravitons in perturbative gravity.
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Homework Statement
Write down the next-order diagrams. Check the answer using Green's function method.
Relevant Equations
Equation of motion: ##\Box h - \lambda h^2 -J =0##
It has been shown in the text that ##h_0 = \frac 1 {\Box} J## with the diagram
1709130278067.png

and that ##h_1 = \lambda \frac 1 {\Box} (h_0 h_0) = \lambda \frac 1 {\Box} [( \frac 1 {\Box} J)( \frac 1 {\Box}J)]## with the diagram
1709130451437.png


I was not sure if the next order diagram is
1709130608327.png

or rather
1709130745770.png

Thus, I substitute ##h=h_0+h_1+h_2## in the equation of motion and calculate to the ##\mathcal O(\lambda^2)##. I get ##\Box h_2 = 2 \lambda h_0 h_1##.
I understand that the factor 2 means that the last diagram above is correct.
Is it so?
 
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