Generating Functional for Yukawa Interactions

Phileas.Fogg

Hello,

I want to derive the connected two point function for the interacting boson-fermion theory.

I know that the generating functional is

$$Z(J, \overline{\eta}, \eta) = N \; exp \left( \int d^4 z \; L_{int} \left(-i \frac{\delta}{\delta J(z)} \right) \left(-i \frac{\delta}{\delta \overline{\eta}(z)} \right) \left(-i \frac{\delta}{\delta \eta (z)} \right) \right)$$ $$exp \left( -\int d^4 x d^4 y \left[ \frac{1}{2} J(x) i \Delta_F (x - y) J(y) + \overline{\eta}(x) i S_F (x-y) \eta(y) \right] \right)$$

The connected correlation function is

$$G_C^2 (x_1 , x_2) = \left( -i \frac{\delta}{\delta J(x_1)} \right) \left( -i \frac{\delta}{\delta J(x_2)} \right) i W(J) \quad |_{J=0}$$

I found the identity

$$\left( -i \frac{\delta}{\delta J(x_1)} \right) \left( -i \frac{\delta}{\delta J(x_2)} \right) i W(J) = \frac{1}{Z} \left( -i \frac{\delta}{\delta J(x_1)} \right) \left( -i \frac{\delta }{\delta J(x_2)} \right) Z - \frac{1}{Z^2} \left( -i \frac{\delta Z}{\delta J(x_1)} \right) \left( -i \frac{\delta Z}{\delta J(x_2)} \right)$$

But I don't know, how to use it with the given Z.

How do I compute the correlation function $$G_C^2 (x_1 , x_2)$$ ?

Regards,
Mr. Fogg

rogerluther

I'm interested in this too. How do I find the two point function for the scalar to scalar process, with a fermion virtual loop?

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