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In quantum mechanics, we can define the scattering amplitude $f_k(\theta)$ for two particles as the magnitude of an outgoing spherical wave. More precisely, the asymptotic behaviour (when $r\rightarrow\infty$) of a wave function of two scattering particles, interacting with some short range potential, is given by

$\psi(r)=e^{ikz} + \frac{f_k(\theta)}{r}e^{ikr}$

where the incoming wave is the plane wave $e^{ikz}$ (the coordinates are the relative coordinates between the two particles). The full Hamiltonian is given by

$H=\frac{1}{2m}p_1^2+\frac{1}{2m}p_2^2+V(r_{12})$

The low energy limit can be obtained by expanding the scattering amplitude in partial waves and only include the lowest partial wave.

However, we can also compute this in effective field theory. In the low energy limit, the effective lagrangian is

$L=\psi^\dagger\left(i\frac{\partial}{\partial t}+\frac{1}{2}\nabla^2\right)\psi-\frac{g_2}{4}(\psi^\dagger \psi)^2$

We can then define the four point greens function as $\langle0|T\psi\psi\psi^\dagger\psi^\dagger|0\rangle$. We can then define the scattering amplitude A as, and I quote (see later for reference) "It is obtained by subtracting the disconnected terms that have the factored form $\langle0|T\psi\psi^\dagger|0\rangle\langle0|T\psi\psi^\dagger|0\rangle$, Fourier transforming in all coordinates, factoring out an overall energy-momentum conserving delta function, and also factoring out propagators associated with each of the four external legs". For two particles, the amplitude A only depends on the total energy E. The claim is then that we have

$f_k(\theta)=\frac{1}{8\pi}A(E=k^2)$

My question is, although I understand that it is reasonable that there is a relation between these two quantities, I have no idea how to prove this and how to get the numerical factors right etc. So basically, what is the exact link between doing scattering computations in QM vs QFT? How can one show that the observables we are looking at is the same quantity?

The paper I am following is http://arxiv.org/abs/cond-mat/0410417 . The Effective field theory part I am referring to starts at page 135 , especially the relation (295). The above quote of the definition of A is given on page 139. The definition of $f_k$ is given on page 10-11.

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