Integral on phase space

Question

I'd like to integrate the square amplitude in the phase space( in $d^3k/(2\pi)^3 2E_k$ and $d^3k'/(2\pi)^3 2E_k$) where $p$ and $p'$ are the 4-momentum of the input particles, $k$ and $k'$ 4-momentum of output particles and $q$ the 4-momentum of the exchange particle. Have any suggestions?

$|A|^2 \propto \frac{1}{q^4}[-q\cdot k' q\cdot p' k\cdot p - 2 q \cdot k' q \cdot p k \cdot p'$ + ... the difficulty I find is in the product $q \cdot k'$ (for example). How do I integrate in $d^3k'$ ? What useful relations can I use to simplify the calculations?

Answer 1

As you have tagged the standard model and scattering I assume what you are talking about is scatting in the $\phi^\dagger \psi \phi$ theory and you are familiar with concepts such as the propagator. There are plenty of resources that cover this material such as this one.

You might want to read about the the Feynman rules to calculate the amplitudes. In your case the Feynman rules can be summarized as

• Each vertex contributes a factor $-ig$
• For each internal line carrying momentum $q$ you include a propator $\frac{i}{q^2-\mu^2+i\epsilon}$
• Integrate over all undetermined momenta. In your case you don't integrate over $q$ since it is completely determined by momentum conservation.
• Incoming and outgoing lines contribute a factor 1
• Symmetry factors are 1
• Include an overall energy-momentum conserving delta function for each diagram.

This means the amplitude for the second-order scattering diagrams is

$$ \mathcal{A}^{(2)}=(-ig)^2\frac{i}{q^2-\mu^2+i\epsilon}(2\pi)^4 \delta^{(4)}\left(\sum p_f -\sum p_i\right). $$ For the three channels we have the following amplitudes

t-channel $$ \mathcal{A_t} = (-ig)^2\frac{i}{t-\mu^2+i\epsilon}(2\pi)^4 \delta^{(4)}\left(p'+k'-p-k\right), \quad q^2=(p'-p)^2=t $$ u-channel $$ \mathcal{A_u} = (-ig)^2\frac{i}{u-\mu^2+i\epsilon}(2\pi)^4 \delta^{(4)}\left(p'+k'-p-k\right), \quad q^2=(p'-k)^2=u $$ s-channel $$ \mathcal{A_s} = (-ig)^2\frac{i}{s-\mu^2+i\epsilon}(2\pi)^4 \delta^{(4)}\left(k+k'-p-p'\right), \quad q^2=(p+p')^2=s. $$