Hi
I am searching for the equation for the radiation probability of a gluon from a quark. I need this for an explanation of the jet structure after a Z decay. Because I heard that hard gluons are radiated with small angles to the quark. This means the energy is focused in the directions of the both quark of the Z decay. Now I need the equation to show this in a talk, but it is unfortunetelly not easy to find the corresponding equation...
does anybody can help me?
best regards
florian

Staff Emeritus
2021 Award
The probability is 1.

More specifically, the probability of radiating a gluon of momentum p diverges as p goes to zero. The exact same thing happens with a charged object and photon radiation. I suspect that's why you can't find the equation you want.

The probability is 1.

More specifically, the probability of radiating a gluon of momentum p diverges as p goes to zero. The exact same thing happens with a charged object and photon radiation. I suspect that's why you can't find the equation you want.
Well, the topic of soft singulatities in QCD is indeed quite involved : there is no way to distinguish between a single quark and a collinear pair gluon-quark for instance, so we do expect collinear divergencies. Although I have no clue who your talk will be given to, I would advise you to check the literature before embarquing on that.

See for instance Quarks, partons and Quantum Chromodynamics (Jiˇr´ı Ch´yla) for a general introduction.

The probability is 1.

More specifically, the probability of radiating a gluon of momentum p diverges as p goes to zero. The exact same thing happens with a charged object and photon radiation. I suspect that's why you can't find the equation you want.

But it sounds like we're after the hard emission probability -- hard enough to affect the jet structure -- so soft divergences shouldn't be relevant. I doubt there's any fundamental equation, but should be possible to find phenomenological models or empirical data for particular energies and jet algorithms. Searching scholar.google.com for Z decay jet structure gives lots of results, some of which may be relevant.

But it sounds like we're after the hard emission probability -- hard enough to affect the jet structure -- so soft divergences shouldn't be relevant. I doubt there's any fundamental equation, but should be possible to find phenomenological models or empirical data for particular energies and jet algorithms. Searching scholar.google.com for Z decay jet structure gives lots of results, some of which may be relevant.

You are right, but I think it is relevant to mention this distinction, because the OP probably has not gone through this. There is an ambiguity at the limit between hard and soft, and this is what defines a hard gluon, and in turn, a jet. So those questions are important and should be reviewed by the OP if he really wants to go through them in his talk.

Hi
Thanks for all replies...
I found something in a book... it concerns to the O(\alpha_s) terms, means the one gluon radiations...
The evaluation of these Feynman diagrams leads to the following form for the three-parton cross-section
$$\begin{equation} \frac{d^2\sigma}{dx_1dx_2} = \sigma_0\frac{\alpha_s}{2\pi}C_F\frac{x_1^2+x_2^2}{(1-x_1)(1-x_2)} \end{equation}$$
where \sigma_0 is the leading-order cross-section as predicted by the electroweak theory alone, and x_1, x_2 and x_3 = 2-x_1-x_2 are the scaled energies (2E/E_{cm}) of the quark, antiquark and gluon respectively. We note that this cross-section is divergent in two cases:

\begin{itemize}
\item As the energy of the gluon tends to zero (x_3 \rightarrow 0, hence x_1 \rightarrow 1, x_2 \rightarrow 1)(infra-red divergence);
\item As the gluon becomes collinear with the quark or antiquark (in the former case, for example, we have x_1 + x_2 + x_3 = 2 from energy conservation and x_1 + x_3 = x_2 from momentum conservation, hence x_2 \rightarrow 1)(collinear divergence).
\end{itemize}

The loop diagram also yields a divergent cross-section, but the sum of all O(\alpha_s) contributions remains finite for appropriately defined observables, such as the total hadronic cross-section.
$$\begin{equation} \sigma = \sigma_0\left(1+\frac{\alpha_s}{\pi}\right) + O(\alpha_s^2) \end{equation}$$

What do you think...
and one question... who is OP? Am I OP? What does it mean?
best regards
florian
p.s. I had a few problems with including latex code, therefore it looks so ugly... sorry

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"OP" means "original poster". The creator of the thread if you will. You are an OP. What do you think...
I think your formula looks correct, but I have not checked. Do you have a reference for your book ? Do you know what terms where included in the calculation ? The reason I ask is that in you first message, you seemed to be concerned with the emission of a real gluon producing and additional jet, but I think other terms have been included as well in your formula, and some divergencies probably cancel each other. I'm saying that on the top of my head, it may be wrong.

hi again

yes your right the formula is not exactly what I want...
References: I found the formula in the book "Electron-Positron Physics at the Z" from several authors (M.G.Green, S.L.Lloyd, P.N.Ratoff and D.R.Ward) the derivation of the equation is in "QCD and Collider Physics" Cambridge University Press (1996) from Ellis R. K., Stirling W.J and Webber B.R.
Unfortunately the article of Ellis in not free available, at least I can not find a free version of the paper in the web (therefore I never read the derivation, the reference above is just given in the book of Green)...
The Formula describes the Feynman graphs with the one gluon radiation. There is also another feynman graph (which contributes with O(\alpha_s)), where one final state quark radiates a gluon, and the other quark absorbs it. Together with this graph the cross-section is finite.
I would like to have a equation with an angle, or at least a equation which I understood completely...
I want to explain, why there is a preferred energy direction, means a jet structure. The sentence, that the hard gluon (gluons with much energy) are radiated with small angle is written in each paper, but I can not find a equation for this process...
best regards and thanks for your replies
florian

That three-parton cross section looks familiar to me, but it doesn't have all that much to say about jets. At the parton level, a hard gluon radiated from a quark has a significant proportion of the quark's original momentum, and will therefore need to be emitted at a relatively small angle to conserve momentum -- the harder the emission the smaller the angle. This suggests that both quark and gluon will end up in the same jet after hadronization (and the precise angle of emission will never be known).

If it's the angle between a quark jet and a gluon jet that you want to know (my initial impression) I don't think the parton-level cross section will be of much use, which is why I suggested looking at phenomenological models of jet structure. However, reading things again, it sounds more like you want to justify the formation of relatively collimated jets in the first place. In this case, I think the momentum conservation argument is reliable, and easier for the audience to appreciate than an equation would be.

Haelfix