Saturday, 11 October 2014

quantum field theory - $phi^{4}$ Propagator - Feynman Diagram: internal vertex that loops back to itself


In all that follows I'll be dealing with everything massless.


The free, massless propagator ($\mathcal{L} = \int d^{4}x \left(\partial \phi(x) \right)^{2} $) is supposedly given by $G_{0}(x,y) = c (x-y)^{-2}$, where I believe $c = \frac{1}{4\pi^{2}}$.



I'm trying to calculate the propagator in $\phi^{4}$-theory, specifically the contribution due to this diagram: enter image description here


Using the Feynman rules in position space, I believe that I should be getting a contribution of the form: $$ C(x_{1},x_{2}) = -i\lambda \int d^{4}u\ G_{0}(x_{1}, u) G_{0}(u,u) G_{0}(u,x_{2}) $$


However, here is my question: why do I get $G_{0}(u,u) = c (u-u)^{-2} = \mathrm{undefined}$? There's no way I can see to evaluate this integral.


How do I deal with this? Maybe I've got the wrong order of variables? I'm new to these kinds of calculations.




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