Second order diagram for the "scalar graviton"

Thread starter Hill
Start date Feb 28, 2024
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Diagram Graviton Scalar

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SUMMARY

The discussion focuses on the calculation of second-order diagrams in the context of scalar graviton interactions. It establishes that the equation ##h_0 = \frac{1}{\Box} J## leads to the first-order term ##h_1 = \lambda \frac{1}{\Box} (h_0 h_0)##. The participant confirms the validity of the second-order term by substituting ##h = h_0 + h_1 + h_2## into the equation of motion, resulting in ##\Box h_2 = 2 \lambda h_0 h_1##, which validates the correctness of the last diagram presented.

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Understanding of scalar field theory
Familiarity with perturbation theory in quantum field theory
Knowledge of the d'Alembert operator (##\Box##)
Basic grasp of Feynman diagrams and their interpretation

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Study perturbative expansions in quantum field theory
Learn about Feynman rules for scalar fields
Explore the implications of the d'Alembert operator in field equations
Investigate higher-order corrections in quantum gravity theories

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The discussion is beneficial for theoretical physicists, graduate students in quantum field theory, and researchers focusing on gravitational interactions and perturbative methods.

Feb 28, 2024

Hill

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

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

I was not sure if the next order diagram is

or rather

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?