Color in matrix element calculation

[kaksmet]

Hello!

I am trying to calculate the matrix element of the process where an incoming upp quark and gluon interact via an upquark and emit a photon and an upquark.

---u(p1)->--~~gamma(p3)~~

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u(q)​

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~~g(p2)~~~-------u(p4)-------

from the gluon vertex I get a color factor $$t^a$$ which I am not sure how to handle. My matrix element then looks like this
$$\bar{u}^s(p_4)ig_s \gamma^{mu}t^a\epsilon_{\mu}(p_2)\frac{\gamma^{\alpha}q_{\alpha} + m}{q^2-m^2}\epsilon^*_{\nu}(p_3)iQ_ue\gamma^{\nu}u^s(p_1)$$

But there must be something that I am missing, since $$t^a$$ is a matrix it seems like my matrix element has the matrix dimensions of $$t^a$$. Should I include a color part of the u quark? And if so, how does that work? Any reference where I could read more about it?

Thanks a lot for any imput!

 

[tom.stoer]

Hello!

I am trying to calculate the matrix element of the process where an incoming upp quark and gluon interact via an upquark and emit a photon and an upquark.

---u(p1)->--~~gamma(p3)~~

|​

|​

u(q)​

|​

|​

~~g(p2)~~~-------u(p4)-------

from the gluon vertex I get a color factor $$t^a$$ which I am not sure how to handle. My matrix element then looks like this
$$\bar{u}^s(p_4)ig_s \gamma^{mu}t^a\epsilon_{\mu}(p_2)\frac{\gamma^{\alpha}q_{\alpha} + m}{q^2-m^2}\epsilon^*_{\nu}(p_3)iQ_ue\gamma^{\nu}u^s(p_1)$$

But there must be something that I am missing, since $$t^a$$ is a matrix it seems like my matrix element has the matrix dimensions of $$t^a$$. Should I include a color part of the u quark? And if so, how does that work? Any reference where I could read more about it?

Thanks a lot for any imput!

The quarks are carrying color indices as well:
$$\bar{u}_i \ldots (t^a)_{ik} \ldots u_k$$

But you diagram looks strange; I am not sure if you can couple the quark-photon in the same representation as the quark-gluon part.

 

[kaksmet]

The quarks are carrying color indices as well:
$$\bar{u}_i \ldots (t^a)_{ik} \ldots u_k$$

But you diagram looks strange; I am not sure if you can couple the quark-photon in the same representation as the quark-gluon part.

Thanks for the help with the color part, will try do continue with it now.

However, could you elaborate a bit more about why the diagram would be wrong? I do not quite understand what you mean with "in the same representation"? How would you otherwise draw a diagram where a quark first emits a photon and then absorbs a gluon (or the other way around)?

 

[tom.stoer]

I do not say that it's wrong, but that the process you are trying to calculate looks rather strange. Usually you do not have external gluon lines.

"In the same representation" is misleading, sorry for that. What I mean is the following (I hope I remember correctly): usually QCD processes go from color singlet to color singlet; but a quark plus a gluon couple to 3*8 = 3*(3*3') which does not yield a singlet but a triplet as smallest rep. This is afaik due to the 3' which is the conjugate rep. of 3 (in SU(2) the doublet 2 and the conjugate doublet 2' are identical, in SU(3) this is no longer true).

In SU(3) you can couple 3*3*3 = 1 + higher reps. but for 3*3*3' this does not work. This is what I mean by "in the other representation".

Enforcing a color singlet condition means that your incoming state is unphysical. So your process has to be part of something more complex which I do not understand.

 

[kaksmet]

My process is a part of a pp collision, where a quark and a gluon from respective proton interact.

 

[tom.stoer]

So you now both the four-momentum of the quark and of the gluon e.g. from the structure functions of the proton?

 

[kaksmet]

Well, either that or alternatively that I will integrate over the momentum fractions in a later part of the calculation.