
#1
Jun808, 11:58 AM

P: 8

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 



#2
Jun808, 01:11 PM

Mentor
P: 15,593

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. 



#3
Jun808, 01:26 PM

P: 2,828

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



#4
Jun808, 02:29 PM

P: 200

Gluon radiation 



#5
Jun808, 02:50 PM

P: 2,828





#6
Jun908, 02:43 AM

P: 8

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 threeparton crosssection [tex] \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}{(1x_1)(1x_2)} \end{equation} [/tex] where \sigma_0 is the leadingorder crosssection as predicted by the electroweak theory alone, and x_1, x_2 and x_3 = 2x_1x_2 are the scaled energies (2E/E_{cm}) of the quark, antiquark and gluon respectively. We note that this crosssection 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)(infrared 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 crosssection, but the sum of all O(\alpha_s) contributions remains finite for appropriately defined observables, such as the total hadronic crosssection. [tex] \begin{equation} \sigma = \sigma_0\left(1+\frac{\alpha_s}{\pi}\right) + O(\alpha_s^2) \end{equation} [/tex] 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 



#7
Jun908, 06:44 AM

P: 2,828

"OP" means "original poster". The creator of the thread if you will. You are an OP.




#8
Jun908, 08:11 AM

P: 8

yes your right the formula is not exactly what I want...
References: I found the formula in the book "ElectronPositron 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 crosssection 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 



#9
Jun908, 03:48 PM

P: 200

That threeparton 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 partonlevel 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. 



#10
Jun908, 10:32 PM

Sci Advisor
P: 1,663

Thinking about this briefly, i'd venture to guess there is indeed some heurestic jet model for the scenario as Dasch says. Otoh anything more detailed is most likely computer modeling and probably does not have a nice closed form.
The 3 parton structure eqn gives you a hint at first order analytic behaviour, but its horribly more complex in full detail. Probably the best bet is to send an email to a specialist and ask them directly. 



#11
Jun908, 10:43 PM

P: 2,828




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