# Figuring out higher loop corrections

1. Jul 20, 2011

### Jim Kata

Let's say I'm trying to calculate the vacuum polarization to the two loop level for QED, how do I do that from a path integral formalism? As in how do I know which Feynman diagrams to calculate?

I would have a path integral like this.

$$- i\Delta _{\mu x,\tau y} = \left\langle {T\{ A_\mu (x)A_\tau (y)} \right\rangle = \frac{{\int {\prod\limits_{z,m} {da_m (z)a_\mu (x)a_\tau (y)\exp \left( {i - \frac{1} {2}\int {d^4 zd^4 wa^\xi (z)a^\zeta (w)D_{z\xi ,w\zeta } } } \right)\exp \left( {\sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }} {n}} Tr(F^{ - 1} G)^n } \right)} } }} {{\int {\prod\limits_{z,m} {da_m (z)\exp \left( {i - \frac{1} {2}\int {d^4 zd^4 wa^\xi (z)a^\zeta (w)D_{z\xi ,w\zeta } } } \right)\exp \left( {\sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }} {n}} Tr(F^{ - 1} G)^n } \right)} } }}$$

where

$$F^{ - 1} (x,y) = \int {\frac{{d^4 k}} {{(2\pi )^4 }}} \frac{{ - \gamma ^0 }} {{i\gamma ^\mu k_\mu + m - i\varepsilon }}e^{ik \cdot (x - y)}$$

$$D_{x\mu ,y\nu } = \left[ {\eta _{\mu \nu } \frac{{\partial ^2 }} {{\partial x^\rho \partial x_\rho }}\delta ^4 (x - y) + i\varepsilon } \right]$$

$$G(x,y) = ie\gamma ^0 \gamma ^\mu a_\mu (x)\delta ^4 (x - y)$$

At the one loop level I just take the quadratic term from this sum below and that's my 1PI.

$$\sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }} {n}}Tr(F^{ - 1} G)^n$$

According to Itzyskon and Zuber there are 8 diagrams (including renormalization graphs) that must be calculated at the 2 loop level to calculate the vacuum polarization. My question how do you determine which graphs to calculate? It should follow from this functional integral, but how?