I want to compute, using the generating functional method, the 4-point correlation function $G^{(4)}(x_1,x_2,x_3,x_4)$ for the Lagrangian

$$ \mathcal{L} [\phi, \psi, \chi] = \overline{\chi} ( i \gamma^\mu \partial_\mu - m_\chi) \chi + \overline{\psi} (i \gamma^\mu \partial_\mu - m_\psi) \psi - g \overline{\chi} \chi \phi - g \overline{\psi} \psi \phi + \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} \mu^2 \phi^2 + ... \tag{1} $$

considering a scattering$^1$ $\phi \psi \rightarrow \phi \psi$. So far, I've just tried to glue together the results that my professor derived in his lecture notes for the generating functionals for the scalar field $\phi$ and the fermionic field $\psi$, they are

$$ Z[J] = N \int \mathcal{D} \phi e^{i \int d^4x \left\{ \mathcal{L}_0[\phi] + \mathcal{L}_i[\phi] + J(x) \phi(x) \right\}} $$


$$ Z[\eta, \overline{\eta}] = \int \mathcal{D} \psi \int \mathcal{D} \overline{\psi} e^{i \int d^4x \left\{ \overline{\psi} (i \gamma^\mu \partial_\mu - m) \psi + \overline{\eta} (x) \psi (x) + \overline{\psi}(x) \eta(x)\right\}}.$$

So that, by combining then, I've got:

$$ Z[J,\overline{\eta}, \eta] = N \int \mathcal{D} \phi \int \mathcal{D} \psi \int \mathcal{D} \overline{\psi} e^{i \int d^4x \left\{\mathcal{L}_0[\phi, \psi]' + j(x) \phi(x) + \overline{\eta}(x) \psi(x) + \overline{\psi} (x) \eta(x) - g \overline{\psi} (x) \psi (x) \phi (x) \right\}} $$

Where $$\mathcal{L}_0[\phi, \psi] := \mathcal{L}_0 [\phi ] + \overline{\psi} (i \gamma^\mu \partial_\mu - m) \psi = \frac{1}{2} \left[ \partial_\mu \phi \partial^\mu \phi - \mu^2 \phi^2 \right] + \overline{\psi} (i \gamma^\mu \partial_\mu - m) \psi.$$ Then, noting that the 4-point correlation function is define as

$$ G^{(4)}(x_1,x_2,x_3,x_4) \doteq \frac{(-i)^4}{Z[0,0,0]} \frac{\delta^4}{\delta J(x_1) \delta \eta(x_2) \delta J(x_3) \delta \overline{\eta} (x_4)} Z[J, \eta, \overline{\eta}] \bigg|_{J, \eta, \overline{\eta} = 0}.$$

I've got, by computing the functional derivatives, that:

$$\boxed{G^{(4)}(x_1,x_2,x_3,x_4) = \frac{(-i)^4}{Z[0,0,0]} Z_0[J, \eta, \overline{\eta}] e^{-i \int d^4x g \overline{\psi} (x) \psi (x) \phi (x)} \phi(x_1) \overline{\psi} (x_2) \phi(x_3) \psi(x_4)}.$$

And then my confusion starts:

  1. Do the order of the derivatives matters? I mean, I've computed them in the following order: $$ \frac{\delta^4}{\delta J(x_1) \delta \eta(x_2) \delta J(x_3) \delta \overline{\eta} (x_4) } Z[J, \eta, \overline{\eta}]$$ But I don't know if they are the correct order or even if they are the correct derivatives (e.g. if the $x_i$ arguments are right).

  2. The ellipsis/dots in the Lagrangian (1) corresponds to terms involving the real scalar field $\phi$. Should I need to take that into account?

  3. This is a scattering that, in total, has two fields $\phi$ and two $\psi$, but in my correlation function a $\overline{\psi}$ appears. Do this make sense? And, if so, why?

I'm still a bit confused in using this formalism in anything that goes beyond a simple scalar field interacting with itself, like a $\phi^4$ theory, and with zero confidence in my calculations. Any help would be appreciated!

$^1$I've read in this same site that this also known as a "Meson-Nucleon Scattering", although my professor didn't mentioned this name even once)


1 Answer 1

  1. Order of Grassmann-odd fields (and their derivatives) matters since they anticommute rather than commute.

  2. Yes, correlation functions depend on interaction terms. They are often treated perturbatively.

  3. The field $\overline{\psi}=\psi^{\ast}\gamma^0$ is not an independent field, but can often be treated as such, cf. e.g. this Phys.SE post.


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