I am reading a paper (arXiv:hep-ph/0002198v2) where it says the formation time of the radiation is proportional to energy and inversely proportional to the squared tranversal momentum (Eq. 3.18 in the paper). If I use just the definition of the phase in equation 3.11 (radiation spectrum), then I can get this relation between formation time and energy and momentum. But I would like to know if there is another way to get this relation.
I got the relation using the de Broglie eq. p=h/lambda and the relativistic energy and also using c=lambda x nu but i think this last equation is only validity for photon. Or can I also use it for gluons?
Is there other way to get the expression for the formation time of the gluons?
Thanks in advance!
λν=c is equivalent to ω=c|k|, i.e. E = |p|c, the energy-momentum relation for massless particles, whether they're photons or gluons. At high energies, asymptotic freedom implies that the gluons are very weakly coupled-but the non-trivial difference to photons is that it's the transverse part of the momentum that's important.
The concept of formation time t_f or formation length l_f or coherence length is used when we have an extended source. It is mainly used in Transition Radiation. A part of the source can interfere with another part. The interference phase is
Delta Phi = Delta z / l_f
If a high-energy quark emits a gluon while passing through a nucleon (q => g + q'), then t_f ~ l_f ~ 1/Q_min ~ E_q / (M^2_{q'+g}),
where Q_min is the minimum momentum transfer to the nucleus.
[publicity] You may get more familiar with l_f reading "Classical and Quantum phenomenology in radiation by relativistic electrons in matter or in external fields", Nucl. Instruments and Methods B 355 (2015) pp. 11-16 (paragraph "Longitudinal coherence")
X. A.
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