Why is there an additional prefactor in equation (12.52) of Peskin's QFT book?

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Thread starter thatboi
Start date Apr 2, 2023
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Peskin Qft

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Apr 2, 2023

thatboi

Hey all,
I am currently having trouble understanding equation (12.52) in Peskin's QFT book. Specifically the term for external leg corrections, in which they tack on an additional prefactor of ##(-ig)##. Normally with external leg prefactors, we don't see the coupling constant multiplied onto the field renormalization counterterm so why is such a prefactor included here?
Thanks.

1. Why is there an additional prefactor in equation (12.52) of Peskin's QFT book?

The additional prefactor in equation (12.52) of Peskin's QFT book is known as the "symmetry factor" or "group factor". It accounts for the number of ways a particular Feynman diagram can be drawn, taking into consideration the symmetries of the underlying theory. This factor is necessary for the correct calculation of scattering amplitudes in quantum field theory.

2. What is the purpose of the symmetry factor in equation (12.52)?

The symmetry factor ensures that the calculated scattering amplitudes are gauge-invariant and consistent with the underlying symmetries of the theory. Without this factor, the results would be incorrect and would not reflect the true nature of the physical system being studied.

3. How is the symmetry factor determined in equation (12.52)?

The symmetry factor is determined by considering the symmetries of the underlying theory, such as gauge symmetries and Lorentz invariance. It is also influenced by the number of identical particles involved in the scattering process. The specific calculation of the symmetry factor can be quite complex and may require advanced mathematical techniques.

4. Can the symmetry factor be ignored in equation (12.52)?

No, the symmetry factor cannot be ignored in equation (12.52) or any other calculation in quantum field theory. It is a crucial component in obtaining accurate and physically meaningful results. Ignoring the symmetry factor would lead to incorrect predictions and would not reflect the true nature of the physical system being studied.

5. Are there any alternative methods to determine the symmetry factor in equation (12.52)?

Yes, there are alternative methods for determining the symmetry factor in equation (12.52). One approach is to use symmetry arguments and group theory to calculate the factor. Another approach is to use combinatorial techniques to count the number of ways a Feynman diagram can be drawn. Both methods require a deep understanding of the underlying theory and can be quite challenging.