Problem, exercise, anomalous magnetic moment interaction

[rayohauno]

I need to solve an exercise on Quantum Field Theory that reads as follows:

Supousse that the electron has an anomalous magnetic moment, which makes the QED Lagrangian
(density) to have an additional term:

$$L'_I(x) = \frac{2ie}{m} \bar{\psi}(x) \sigma^{\alpha\beta} \psi(x) F_{\alpha\beta}(x)$$

where:

$$F_{\alpha\beta} = \partial_{\alpha}A_{\beta}(x) - \partial_{\beta}A_{\alpha}(x)$$

1. Find the matrix element (Feynman amplitude) for Moller's dispersion, having in account this additional
term.

*****************

I found that

$$\partial_{\alpha}\beta = \sum_k (\mp i k_{\alpha}) A^{\pm}_{\beta}(x)$$

from where, I found that:

$$2i\sigma^{\alpha\beta}F_{\alpha\beta}(x) = -i2[\check{K},\check{A}^+ - \check{A}^-]$$

where:

$$\check{K},\check{A}^+,\check{A}^-$$

denotes slash operators ! (I couldn't find how to draw slash operators here). The problem that I have
as from here is, that I don't know how to calculate Time Contractions to the operator:

$$A^+ - A^-$$

for example, I don't know how to calculate the Feynman propagator to the Time Contraction:

$$T\{ A(x)[A^+(y) - A^-(y)] \} = T\{ [A^+(x) + A^-(x) ][A^+(y) - A^-(y)] \}$$Does anyone knows how to proceed in this case??

best regards

Rayo

 

[Jhero]

Dear Rayo,

Thank you for your question. This is a complex problem that involves both Quantum Field Theory and Feynman diagrams. To calculate the Feynman amplitude for Moller's dispersion with the additional term in the QED Lagrangian, you will need to use the Feynman rules for QED, which can be found in any standard textbook on the subject.

To begin, you will need to use the Feynman propagator for the electron, which includes the additional term in the Lagrangian. This propagator can be written as:

G(x,y) = <T{ψ(x)ψ(y)}> = <T{ψ(x)ψ(y)e^{iS_I(x)}e^{-iS_I(y)}}>

where S_I is the interaction term in the Lagrangian. To calculate the Feynman amplitude, you will need to use this propagator for both the initial and final electron states.

Next, you will need to construct the Feynman diagram for Moller's dispersion, which involves two electron lines and two photon lines. The additional term in the Lagrangian will contribute to the vertex where the photon attaches to the electron line. This vertex will have a factor of 2iσ^{\alpha\beta}F_{\alpha\beta}(x), as you have correctly found.

To calculate the Feynman amplitude, you will need to use the Feynman rules for QED, which involve summing over all possible diagrams and taking into account the appropriate factors for each vertex, propagator, and loop. This can be a tedious process, but it is essential for obtaining the correct result.

I suggest consulting a textbook or seeking guidance from a professor or colleague who is familiar with QED calculations. Good luck with your exercise!