Diapositiva%201

1
1
Extension to the transverse degrees of freedom of
the statistical parton distributions C. Bourrely,
F. Buccella and J. Soffer Frascati, 8 June 2005
The non-diffractive part of the light and
parton distributions contain the
spurious factors An usual
criticism to the use of quantum statistical
distributions is that the opening of the
-phase space proportional to brings to a
dilution which implies the Boltzman
limit. Sometimes in physics it is better to have
two problems than just one.
2
2
The purpose of this talk is to extend the
analysis to the transverse degrees of freedom
with the hope to account for these factors. Let
us first recall the general method of statistical
mechanics to find the most probable distribution
of occupation numbers for energy levels when the
total energy of N particles is E One looks with
the Lagrange multipliers method for the maximum
of
3
3
The Stirling formula
neglecting
implies
With a and ß to be determined by the constraints
The exponential form, which for fermions and
bosons, is modified into Fermi-Dirac and
Bose-Einstein functions applies also to different
quantities to be divided by constituents (like
the total proton momentum between partons) For
instance statistical considerations have been
applied to multi-hadron production in very high
energy nucleon-nucleon scattering. Take the
example of p-p one may think that by crossing
each other the two particles lose part of their
momentum and by energy conservation create
particles
4
4
One may write
which implies
with and to be determined by the
constraints We get for the
distributions which has intriguing properties
as and harder and behaviour as
experimentally found
5
5
For the deep inelastic the situation is better
defined for the constraints
So for the parton distributions we have
Which implies
By integrating on we find
Where R(x) is the Rieman function
6
6
R(x) for large positive x ? x and for large
negative x ? e-x If we assume which implies
that the broader in x distributions have also, at
given x, broader we find the
proportionality to assumed ad hoc! Every
time you account for something you put in your
formula unwillingly to explain data, seems
good. An intriguing aspect is that one finds the
factor , by considering the transverse
degrees of freedom, where a cut on comes
from the energy sum rule. For s we inserted
arbitrarily the factors which do
not coincide with but both are decreasing
functions and we have adjusted K4 in
order to get equal to the fit
7
7
The extension to the transverse degree of freedom
opens a way to consider the heavier partons
beginning from the strange one, for which an
authomatical reduction with respect to light see
quarks comes from the mass
and by integrating on we find
So the reduction decreases with x and one expects
harder distributions as or A
similar property holds for kaons with harder x
and distributions than pions
8
8
The extension to the transverses degrees of
freedom may be performed also for the diffractive
and the gluon distributions. By taking vanishing
potential also for the transverse degree of
freedom
By integrating on one gets
So with We recover the expression
proposed At small x we have an infrared
catastrophy in 1/ in place of th one in 1/x
9
9
Also the contribution of the diffractive part to
diverges
Both these unpleasants features may be taken care
by a lower limit on related to the
uncertainty principle. If I know pxpy0, I dont
know x and y. The partons, being coloured, cannot
walk to much, since they are confined, which
implies the desired lower limit on
Conclusions
Non-diffractive parton light quarks
parton light anti-quarks
Natural cut on Heavier
flavours, gluons and diffractive contributions