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.

q q

1~ qx

1

~11

~

~

2

~

xqxx

b

xqxx

b

e

qxxA

xqx

e

qxAxxxq

2Tp

2Q

1

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

iii

ii

ii

nEnNnN !!ln

2

The Stirling formula

Nn

en

NNeNN

i

i

N

N

i

22! neglecting

With α and β 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

implies

En ii

3

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

One may write

zfzzffT

f

zfzz

f

ffT

zffz

f

ffTfz

fz

zz

pppMxmp

x

pMpp

pmp

pxp

pmp

p

EpMpMpE

222

1

22

2

2

222

222

22

222

which implies

22

2

22

~1

Tx

mp

xx

dpeexdx

M

ffT

x 2

2Tpx

2, Tpx

ffT xp 22

2

22

x

mp

xx ffT

ee

4

For the deep inelastic the situation is better defined for the constraints

xpmpxpE

PMPE

Tzp

zzP

2

222

2

So for the parton distributions we have

2222

22

,

1,

Mdxdppxpxp

dxdppxxp

iTTi

T

iTTi

Which implies

By integrating on we find

Where R(x) is the Rieman function

ixpxxTTi pYR

e

xfxdppxpi

~, ~

222

xxy eedy

1ln10

2Tp

1

1

1,

~2

~2

2

2

iTi pYxp

xpxxTi

ee

xfpxp

5

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 K=4 in order to get equal to the fit

ii pxkpY ~~

2Tp ipx~

q qx~/1

0

0xdxu

qx~

qxkqY eeqYR~~~

ipx~

2Tp

6

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

1

1

1

,~~

2

2

22

sxxmp

xsxxT

sT

ee

xfpxs

2Tpand by integrating on we find

2Tp

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

xu xs xd

2

2

~

2~

1

xmsxkR

e

xxf s

xsxx

7

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

1

1

1,

2

22

xp

xxTDIF

T

ee

xfpxp

By integrating on one gets2Tp

1

2ln2

xx

e

xf

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

2ln

~2

~

bxAxf

1

1

1 2

2

xp

xxG

T

ee

xfxG

2Tp

8

xpT2

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 px=py=0, I don’t know x and y.

The partons, being coloured, cannot walk to much, since they are confined, which implies the desired lower limit on

2Tp

2Tp

Conclusions

Non-diffractive parton light quarks ***

‘ ‘ parton light anti-quarks **

Natural cut on ***

Heavier flavours, gluons and diffractive *contributions

2Tp

9