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The problem in Sredniki's textbook 10.5 :

For a free scalar field $\psi$, the Lagrangian is $$\cal{L}= -\frac{1}{2}\partial^\mu\psi\partial_\mu\psi-\frac{1}{2}m^2\psi^2$$ Here we use the metric $\operatorname{diag}(- + + +)$

If I make $\psi=\phi+\lambda \phi^2$, then the Lagrangian is $$\cal{L}= -\frac{1}{2}\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}m^2\phi^2-2\lambda\phi\partial^\mu\phi\partial_\mu\phi-\lambda m^2\phi^3-2\lambda^2\phi^2\partial^\mu\phi\partial_\mu\phi-\frac{1}{2}\lambda^2m^2\phi^4$$

For scattering $\phi\phi \to \phi\phi $, how do I calculate the loop correction? Since the Lagrangian is now nonrenormalizable. In loop correction we need to take ghost into consideration.

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    $\begingroup$ Since the new and old fields commute outside the lightcone and both couple to the one-particle state, the S matrix for the new and old theories will be same. $\endgroup$
    – user72382
    Feb 4, 2015 at 16:56
  • $\begingroup$ Related to physics.stackexchange.com/questions/45262/… $\endgroup$
    – 346699
    Apr 2, 2017 at 2:47

1 Answer 1

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Take your free theory, which is the kinetic term plus the mass term, plug that into a path integral, and let the rest act as a perturbation for the generating function. Then use this generating function to calculate the 4-point correlation function.

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