Feynman Rules : Propagator Question

[Hepth]

I know I should know this, but I have a quick question.
Let's say we have a diagram:

1-->----------2
          | 
          |   <- "q"
          v 
          | 
3--<----------4

Lets assume:
1 = "quark"
3 = "antiquark"
2 = W boson
4 = photon
q = same quark flavor as "3"

Time flows from left to right.

Now let's say I start writing the diagram at point 2:
$$W_\mu \bar{u}_1 \gamma^\mu (1- \gamma_5) \frac{i}{\not q - m_3} (- i e \gamma^\alpha)\epsilon^{*}_\alpha v_3$$

I think that's right.
For every dirac spinor propagator I write $$\frac{i}{\not q - m}$$

Now if I want to write out WHAT "q" is, I have to choose a direction of momentum flow. Is it convention, or by rule, that I drew the diagram from top to bottom, so the direction of "q" is up (against the direction of writing the amplitude).
so $$q = p_3- p_4$$

Where both p1 and p3 are flowing IN and p2 and p4 are flowing OUT.

Or would it be down, and if not then why? What decides which way I write the momentum flow for the propagator as if I choose differently I get a different result.

 

[blechman]

First of all, your arrows do not match your amplitude! With the arrows as you drew them, I would have written:

$$\overline{v}(p_3)(-ie\gamma^\mu)\frac{i(\not\!q+m)}{q^2-m^2}\left(ig\gamma^\nu\frac{1}{2}(1-\gamma^5)\right)u(p_1)\times\varepsilon^{\gamma*}_{\mu}(p_4) \varepsilon^{W*}_{\nu}(p_2)$$

Now you just conserve momentum at each vertex. If you are letting TIME go left ---> right, as you seem to be doing):

$$q = p_1-p_2=p_4-p_3$$

In other words: for an internal fermion line, you can always choose your momentum to flow in the direction of your spinor arrows. Then just apply the usual rules of momentum conservation at each vertex. That's all there is to it!

 

[strucka]

Blechman made it. You had it bad, because your arrows was on bad direction. Of you change direction you would have it good.

 

[Hepth]

No, the arrows are what I want. I made a mistake with the spinors, yes. But my question had more to do with, as the arrow on the propagator is now, which way should i draw the current.
The answer being that I draw it with the arrow, thus letting me use conservation of momentum at the vertices.

 

[blechman]

No, the arrows are what I want. I made a mistake with the spinors, yes. But my question had more to do with, as the arrow on the propagator is now, which way should i draw the current.
The answer being that I draw it with the arrow, thus letting me use conservation of momentum at the vertices.

If by current you actually mean MOMENTUM, then yes. When using Dirac spinors you can always chose the momenta on internal lines to be in the same direction as the spinor arrows.

 

[Hepth]

Gah, i used to be an electrical engineer... sorry :)