Computing the scattering amplitude from the S-matrix

[leo.]

Homework Statement

Consider two real scalar fields $\phi,\psi$ with masses $m$ and $\mu$ respectively interacting via the Hamiltonian $$\mathcal{H}_{\mathrm{int}}(x)=\dfrac{\lambda}{4}\phi^2(x)\psi^2(x).$$

Using the definition of the S-matrix and Wick's contraction find the $O(\lambda)$ contribution to the scattering amplitude for the process $\phi(p_1)+\psi(p_2)\to \phi(p_1')+\psi(p_2')$, being $p_i$ the initial momenta and $p_i'$ the final momenta.

Homework Equations

First I would consider the definition of the S-matrix from the book: $$\langle f | S | i \rangle_{\mathrm{Heisenberg}}= \langle f; \infty | i; -\infty\rangle_{\mathrm{Schrödinger}},$$ secondly I believe Wick's theorem would be needed: $$T\{\phi_1\cdots\phi_n\}=:\phi_1\cdots\phi_n + \text{all contractions}:$$

Finaly I believe Dyson's series can help here $$U(t,t_0)= T\exp \left[ i \int _{t_0}^t H_I(t')dt'\right]$$ being $U(t,t_0)$ the interaction picture time evolution operator and $H_I$ the interaction Hamiltonian at the interaction picture.

The Attempt at a Solution

[/B]
I wasn't able to do much really. My first guess was to use Dyson's series to expand the time evolution operator of the interaction picture to first order in $\lambda$, that is, being $H_I$ the interaction Hamiltonian in the interaction picture defined by $$H_I (t) = e^{iH_0(t-t_0)} H_\mathrm{int} e^{-i H_0(t-t_0)}$$ we have Dyson's series

$$U(t,t_0)= T\exp\left[-i \int_{t_0}^t H_I(t') dt'\right] \approx T\left\{1 - i \int_{t_0}^t H_I(t')dt' - \dfrac{1}{2}\int_{t_0}^t \int_{t_0}^t H_I(t')H_I(t'')dt'dt'' \right\},$$

now $H_I$ is just the same as $H_{\mathrm{int}}$ changing the fields by their interaction picture counterparts, which can be expanded in terms of creation and annihilation operators, so that
$$H_I(t)= \dfrac{\lambda}{4}\int d^3x \phi(x)^2 \psi(x)^2$$

but I don't know how this can be used. The definition of the S-matrix as I understand works as follows: given $S(t,t_0)$ the time evolution operator associated to the full Hamiltonian $H= H_0+H_\mathrm{int}$ the S-matrix is defined as
$$\langle f | S | i \rangle = \lim_{t_{\pm}\to \pm\infty} \langle f | S(t_+,t_-) | i\rangle$$

so we pick the state $|i\rangle$ at $t=-\infty$ and evolve it to $t=+\infty$ according to the full Hamiltonian, and project onto the final state.

The issue is that $S(t,t_0)= e^{i H_0(t-t_0)}U(t,t_0)$ so I can't use $U$ directly.

By the problem statement I guess I can't use Feynman rules derived from the LSZ formula, just the definition of the S-matrix.

I'm really stuck with this. How can I proceed?

 

[mark726]

First, let's rewrite the interaction Hamiltonian in terms of creation and annihilation operators:
H_{\mathrm{int}} = \dfrac{\lambda}{4}\int d^3x \phi(x)^2 \psi(x)^2 = \dfrac{\lambda}{4}\int d^3x \left(a^\dagger(x)a^\dagger(x)a(x)a(x) + a^\dagger(x)a(x)a^\dagger(x)a(x)\right)

where a(x) and a^\dagger(x) are the annihilation and creation operators for the fields \phi and \psi respectively. Using Wick's theorem, we can simplify this to:
H_{\mathrm{int}} = \dfrac{\lambda}{4}\int d^3x \left(:\phi^2(x)\psi^2(x): + \phi(x)\psi(x)\phi(x)\psi(x) \right)

where :: indicates normal ordering. Now, using the definition of the S-matrix and Wick's contraction, we can write the O(\lambda) contribution to the scattering amplitude as:
\langle f | S | i \rangle = \lim_{t_{\pm}\to \pm\infty} \langle f | S(t_+,t_-) | i\rangle = \langle f | T\left\{e^{-i \int_{-\infty}^\infty H_{\mathrm{int}}(t)dt}\right\} | i \rangle

Expanding the time evolution operator to first order in \lambda, we have:
\langle f | S | i \rangle = \langle f | T\left\{1 - i \int_{-\infty}^\infty H_{\mathrm{int}}(t)dt\right\} | i \rangle

Using the expression for H_{\mathrm{int}}, we can write:
\langle f | S | i \rangle = \langle f | T\left\{1 - i \dfrac{\lambda}{4}\int_{-\infty}^\infty \int d^3x :\phi^2(x)\psi^2(x):dt\right\} | i \rangle

Now, we can use Wick's theorem to evaluate the contractions and obtain:
\langle f | S | i \rangle = \langle f | T\left\{1 - i \dfrac{\