What are the issues with the covariant partners in this amplitude calculation?

[jono90one]

Hi,
I have a question regarding pair production, regarding the amplitude that I am trying to understand.

I have attached a photograph of the feynmann diagram, which I believe to be correct - although I don't like the vertex/propogator combination as shown below: (I have split up integral so it fits onto more than 1 line)
$-iM = \int u(p_{3},s_{3}) (-ie\gamma_{\mu}) \epsilon(k_{1},\lambda_{1})$

$(i\frac{q_{\mu}\gamma^{mu}+m}{q^{2}-m^{2}})$

$\bar{u}(p_{4},s_{4}) (-ie\gamma_{\mu}) \epsilon(k_{2},\lambda_{2}) (2\pi)^{4} \delta^{4}(q-p_{3}-{k1}) (2\pi)^{4} \delta^{4}(q-p_{3}-{k1}) \frac{d^{4}q}{(2\pi)^4}$

(p = mmt, s = spin, k = 4-mmt, lambda = polorisation, integrated over all mmt space q.)

My questions are:
- The bit I don't like is the fact the \mu and \nu covariants don't have contravariant partners (just \gamma_{\mu} \gamma_{\nu}). If the propagator was a photon, these would nicely have partners. Isn't the idea it should be invariant, so isn't this an issue?

-Labels, I am doing the 1, 2, 3, 4 based on the order it happens in (this makes sense given time axis goes horizontally) - Are these correct?

- When I integrate this, will I get two terms, e.g. one for when $q = p_{3} + p_{1}$ - I guess these just add to give an overall amplitude?

-On notation, should it be u(p_{3}, s_{3}) or $\bar{u}(p_{3}, s_{3})$ - i.e. Am I saying, oh it's a positron, so I should make that known, or do I follow the feynmann digram and say it's an electron going backwards in time. I'm sure the former is true.

Many thanks for you help

 

[ChrisVer]

he bit I don't like is the fact the \mu and \nu covariants don't have contravariant partners (just \gamma_{\mu} \gamma_{\nu}). If the propagator was a photon, these would nicely have partners. Isn't the idea it should be invariant, so isn't this an issue?

What are the $\epsilon$s you have written?

-Labels, I am doing the 1, 2, 3, 4 based on the order it happens in (this makes sense given time axis goes horizontally) - Are these correct?

I don't get this... also are you sure about your delta functions? you have given the same expression for both, while (I think) you wanted to apply the conservation of energy/momentum in each vertex... eg one of the deltas should have k2 and p4 in the argument.

- When I integrate this, will I get two terms, e.g. one for when q=p3+p1q = p_{3} + p_{1} - I guess these just add to give an overall amplitude?

add up?

-On notation, should it be u(p_{3}, s_{3}) or u¯(p3,s3)\bar{u}(p_{3}, s_{3}) - i.e. Am I saying, oh it's a positron, so I should make that known, or do I follow the feynmann digram and say it's an electron going backwards in time. I'm sure the former is true.

The 4spinors are $u,v$ the $u$ is for particles and $v$ for antiparticles.