Cross Section: Quark-Gluon vs. Quark-Photon

[lmcelroy]

This isn't a homework problem. I am preparing for a particle physics exam and although I understand the theoretical side of field theory, I have little idea how to approach practical scattering questions like these.

THE PROBLEM:
Dark matter might be observed at the LHC with monojet and monophoton signals, which proceed via the parton processes:

Q aQ → χχϒ

Q aQ → χχg

where Q is a quark, aQ is an anti-quark, χ is the dark matter candidate, γ is a photon, g is a gluon.

Explain why the cross sections of the two processes are related by σ(Q aQ → χχϒ) = A σ(Q aQ → χχg) where A is a numerical coefficient.

EXTRA:
It would be great if someone could explain how to determine A for a particular process; e.g. σ(Q_red aQ_red → χχϒ) = A σ(Q_red aQ_red → χχϒ).

ATTEMPT:
Using QCD and QED Feynman rules to determine the Feynman amplitude, the only difference is in the quark-gluon vs. quark-photon vertex. These terms only contribute constant terms so the equations are equal except for a proportionality constant.

 

[Einj]

The difference, as you said, is in the interaction vertex. I don't know exactly what the explicit form of it is in the presence of dark matter (also because it strongly depends on the model you are considering) however, I can tell you what would happen in an ordinary process, say photon radiation vs. gluon radiation.
The difference is that in the photon vertex you basically just have (omitting annoying factors of i etc.) $e\gamma_\mu$, where e is the electric charge and $\gamma_\mu$ is the usual Dirac matrix. In the gluon case you group is SU(3) instead of U(1) and hence you vertex is $gT^a_{ij}\gamma_\mu$, where now $T^a_{ij}$ is a matrix in the fundamental representation of SU(3). When you take the absolute value squared of your amplitude, to compute the cross section, in the first case you will just have $e^2$ while in the second case you'll end up with $g^2C_F$ where $C_F$ is the square (plus a trace maybe, I don't remember) of the color matrix. If I remember correctly for the ordinary 3 color case you have $C_F=3/4$ or something like that. So I would say that your final proportionality factor A is just $A=C_Fg^2/e^2$.

I might be messing up with the actual numerical coefficients but I'm confident that this is the right path. I hope this answers your question.

 

[RGevo]

Einj is exactly right.

The question uses dark matter because it can be assumed that the photon and gluon only couple to the initial state q or Aq.

In which case the ratio of cross sections is a colour factor and coupling constant.

In this particular example, the casimir Cf (4/3) which is calculated by averaging/summing over colours - for this see an introduction to qcd or chapter 16 of peskin. (The couplings also come with scale dependence in fixed-order, but that's almost certainly irrelevant for the exam).

Good luck