User:Agostino981

Self-Energy

There are different types of self-energy in quantum field theory. To illustrate what is self-energy, consider electron (lepton) self-energy,

$$S^{(2)}(e^-\rightarrow e^-)=-e^2 \int d^4 x_1 d^4 x_2 \bar{\phi}^-(x_1)\gamma^\mu iS_F(x_1-x_2)\gamma^\nu \phi^+(x_2) iD_{F\mu\nu}(x_1-x_2) $$

Please note that we treat the self-energy as a composition of two propagators, photon propagator and fermion propagator.

Similarly, we would expect the Feynman amplitude be

$$ M=ie_0 \bar{u}(p) ie_0^2\sum (p) u(p) $$

where $$ie^2_0\sum(p)$$ is the fermion self-energy part and it is given by

$$ ie^2_0 \sum(p)= \frac{(ie_0)^2}{(2\pi)^4} \int d^4 k iD_{F\mu\nu}(k)\gamma^\mu iS_F(p-k)\gamma^\nu $$

Transition is pretty much trivial, as $$|i\rangle=|f\rangle=|e^- p\rangle$$, so we will skip it here, though you may want to do it as an exercise.

Apart from electron self-energy, we will also consider photon self-energy. Photon self-energy can simply be described as the photon gives rise to a pair of electron and positron and they annihilates each other to give another photon with the same momentum as the previous one. The photon self-energy is given by

$$ S^{(2)}(\gamma \rightarrow \gamma)=-e^2\int d^4 x_1 d^4 x_2 (-1) Tr[iS_F(x_2-x_1) \gamma^\alpha A^-_\alpha(x_1) iS_F(x_1-x_2)\gamma^\beta A^+_\beta(x_2)] $$

The negative sign denotes the closed fermion loop and taking the trace is due to the hidden indices of the gamma indices. The operator can be constructed in a much simpler and implicit way in terms of normal ordering, and it will not be showed here.

In Feynman amplitude, the photon self-energy part is given by

$$ie^2_0 \Pi^{\mu\nu}(q)=\frac{(ie_0)^2}{(2\pi)^4} (-1) Tr[\int d^4 \bar{p} \gamma^\mu iS_F(\bar{}p)+q) \gamma^\nu iS_F(\bar{p})]$$

Note that this closely resemble the operator, after all, they can derive each other directly by doing some maths.

In certain cases, we need to consider radiative correction to refine our approximation, and to achieve this, we usually consider applying self-energy to the process. Consider electron-electron scattering, we expect there will be a virtual photon during the process and we can apply photon self-energy on that virtual photon. We would expect the replacement

$$iD_{F\alpha\beta}(k)\rightarrow iD_{F\alpha\beta} + iD_{F\alpha\mu}(k)ie^2_0\Pi^{\mu\nu}(k)iD_{F\nu\beta}(k)+O(e^4_0)$$

will able to describe the process.

Renormalization

There are several types of renormalization, and here, we will consider a few kinds of them.

The first one is photon self-energy, which we have considered earlier.

The second one is electron self-energy.

The fermion self-energy correction is given above.