How to Determine Time Ordering in Phi-3 Theory for a 2-to-3 Particle Process?

[tayyaba aftab]

i have been given a problem for writing s matrix in second order perturbative theory for an interaction hamiltonian with phi 4 and phi 3 contributions.
it is also given that our initial state is of 2 particles and final state is of three particles.
now in solving that i have to take time ordering of the hamiltonian at two different points.
problem is that i am unable to guess the right term of time ordering for the cpndition of two in coming and three outgoing particles.
can some body please help me in writing that.
i have to submit my assignment after vacations
so its urgent


tayyaba

 

[SergejVictorov]

You don't have to guess here.
I will just cover the phi cubed case.
Are you allowed to use Feynman's rules?
If yes, just draw the 2+3 external lines and two vertices with three outgoing lines each and draw all topologically distinct diagrams. After that, you can write down the terms very easily.

If not, use Wick's theorem to transform the time-ordered into a normal-ordered product:

$$T\left[\phi^3(x)\phi^3(y)\right]=N\left[\phi^3(x)\phi^3(y)+\mathrm{all\, possible\, contractions}\right]=N\left[\phi^3(x)\phi^3(y)\right]+9D_F(x-y)N\left[\phi^2(x)\phi^2(y)\right]+18D_F^2(x-y)N\left[\phi(x)\phi(y)\right]+6D_F^3(x-y)$$

where $$D_F$$ denotes the Feynman propagator. Please check the combinatorial factors. After that, write out the normal-ordered products to see which terms remain after contracting them with the creation and annihilation operators.