Partons distribution functions

Ignorance of particle physics at high-energies, e.g. parton distribution functions, additional field degrees of freedom above Planckian energies. 

In this chapter the theoretical concepts relevant to describe the physics of heavy quarks at the LHC are introduced. The main ideas of Quantum Chromodynamics are reviewed, before their appUcation to high-energy hadron-hadron collisions is discussed. This includes the factorization ansatz, the evolution of the parton distribution functions, the partonic processes important for beauty quark production and the phenomenological treatment of heavy quark fragmentation. A further section is dedicated to the description of the decay of -hadrons via the weak interaction. The Monte Carlo event generators which are used in this analysis to generate full hadronic events within the QCD framework are presented in the last section. 

Fig. 5.4. Calculated parton model cross-sections for the production of W bosons in pp and pp collisions. Solid and dashed lines correspond to different assumed parton distribution functions. (Prom Quigg, 1977.)...

Note that despite the fact that F2 x) depends on parton distribution functions in the form xq x) there is no doubt that F2 x) / 0 as x 0. The implication that, as x — 0,... 

Several fits have been made to the data from proton and neutron targets to try to determine the shape of the individual parton distribution functions and to distinguish between the sea and valence contributions. Typical shapes are shown in Fig. 17.17 wherein was used u x) = uv(x) + Usea(x) and Usea(a ) = dsea, x) etc. The importance of the sea at small x only is clearly shown. 

All this is a nice confirmation of the general picture. If the parton distribution functions in the nucleon are assumed known from deep inelastic scattering, one can use the Drell-Yan process to learn about the q and q distributions inside the meson information which otherwise would be very difficult to come by. 

For each type (flavour) of quark-parton or antiquark-antiparton we now allow a separate distribution function u(a ), d(x), s x) and u(a ), d x),s x) giving the nmnber densities of such objects with momentum fraction be-... 

We have already remarked in Section 16.3 that uW2 does not decrease as u —> 00. Since u oo corresponds to a 0 we see that the electromagnetic data are not incompatible with 2(0 ) —> constant as a —> 0. The distributions shown in Figs. 16.4 and 17.5 support this. Taken literally this implies that the parton or antiparton distribution functions diverge as X —> 0, i.e. 

What is needed to build a smooth fiat curve out of 5-functions is clearly a continuous distribution of masses. We thus introduce /j(x )dx as the parton densities, i.e. as the number of partons of t5rpe j with mass x ruN with 0 < x < 1. The efiective mass of the constituents thus varies between 0 and rriN and is not a fixed number. 

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