Effective cross section and spatial structure of the hadrons

6 Pages·0.1 MB·English

by  Giorgio Calucci

EFFECTIVE CROSS SECTIONS AND SPATIAL STRUCTURE
OF THE HADRONS
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G. CALUCCI, M. DAZZI
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Dipartimento di Fisica teorica, Universit`a di Trieste, Strada Costiera 11. I-34014
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and I.N.F.N. Sezione di Trieste, Italy
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n
Informationsaboutthespatialstructureofpartondistributionwithinthehadron
a
are provided by the ratios between the inclusive cross sections for a pair of jets,
J
two pairs of jets, three pairs of jets..., and so on. It results, however that these
7 ratiosdependnotonlyonthespatialdistributionbutalso,andevenmore,onthe
1 multiplicitydistributionoftheinitialpartons.

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v 1 Motivations and overall description
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1 Themultipleproductionofpairsofjetsinthehigh-energyhadron-hadroncolli-
1 sionsprovidesawaytoinvestigatethespatial,andalsononspatial,structures
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of the hadrons. A particular instance of this investigation is presented here,
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takinginto accountthe inclusivecrosssectionsintegratedoverthe momentum
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/ spectrum.
h
When the momentum transfer to the pair of jets ∆p is large enough,
p
- i.e.1/∆p << RH, whereRH isthe hadronradiusadescriptionoftheprocess
p in termofthe impactparameterb is justified. This possibility simplifies many
e
calculations and is moreover very useful in visualizing the processes one wish
h
: to study.
v
In absence of a well established non perturbative QCD, a lot of models
i
X maybe proposed,allofthemstemming fromthe originalpartonicdescription.
r Using the impact parameter language a list of features which are related to
a
the more general aspects of the experimental evidences is presented:
The distribution of the partons in the transverse plane (b-distribution).
The distribution in the fractional longitudinal momenta (x-distribution).
The correlations between transverse and longitudinal variables of the parton:
(~b,x) and among different partons: (~b ,~b ), (x ,x ).
1 2 1 2
The multiplicity distributions of the incoming partons.
Other features like spin, color, flavor distributions involve clearly finer
experimental analyses.
Thestartingpointfortheanalysiswillbe,then,theinclusivecrosssections
formultiplepairproductiondkσ/dp ...dp : integratingtheseexpressionsover
1 k
therelevantkinematicalvariableswegettheintegratedinclusivecrosssections:
1
σ =<n>σ ,···,σ =<n(n−1)···(n−k+1)>σ , (1)
1 H k H
where σ is the hard contribution to the inelastic cross section.
H
In order to connect these general definition with the model analysis that
will be presented, an expression for σ is given here, in a particularly simple
H
case
σIJ = d2β ∞ 1Γ (x ,~b )...Γ (x ,~b )e− ΓI(x,~b)dxd2b
H n=1 n! I 1 1 I n n R
R R P
× ∞ 1 Γ (x′,~b′ −β~)...Γ (x′ ,~b′ −β~)e− ΓJ(x′,~b′)dx′d2b′
m=1 m! J 1 1 J m m R
P
× 1− n m (1−σˆ ) dx d2b ...dx d2b dx′d2b′ ...dx′ d2b′ (2)
i=1 j=1 ij 1 1 n n 1 1 m m
h i
Q Q
In this first model the distribution of the partons in the hadron is Poissonian
and completely uncorrelated, different distributions will be considered in the
following. The integration over the fractional momenta must present a lower
bound in order that the parton scattering may involve a finite momentum
transfer, larger than a fixed threshold.
The expression in square parentheses of Eq.(2) represents the probability
of having at least one semi-hard partonic interaction between hadron I and
hadron J, the expression σˆ ≡ σˆ (~b −~b ;x ,x ) is the probability of having
ij ij i j i j
a semi-hard interaction of the parton i from hadron I with the parton j of
the hadron J, so it depends on x ,x , and on the difference of the transverse
i j
relativedistance~b −~b′,accordingtotheconsiderationsmadeatthebeginning
i j
the expressionwill be taken as localin~b: σˆ =σ¯ δ(~b−~b′). The crosssection
x,x′
resultsfromthesumoverallpossiblepartonicconfigurationsofthetwohadrons
followed by the integration on the overall hadronic impact parameter β.
One will notice that in Eq.(2) all possible interactions between partons
of hadron I and partons of hadron J are taken into account, so that also all
possible hard elastic rescatterings are included.
A relevant simplification is obtained by neglecting every rescattering, i.e.
by saying that a given parton will interact only once[1]. In that case the
expression in the square parentheses in eq.(2) is simplified to:
n m
1 1
1− (1−σˆ )≈ σˆ − σˆ σˆ + σˆ σˆ σˆ −··· (3)
ij ij ij i′j′ ij i′j′ i′′j′′
2 3!
YY X X X
i=1j=1 i,j i6=i′ i6=i′6=i′′
j6=j′ j6=j′6=j′′
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Inthesamewaytheinclusivecrosssectionforproductionofk pairsofjets
with momentum transfer p ,...,p is given by
1 k
dk σ/dp ...dp = d2β ∞ 1 n Γ (x ,~b )...Γ (x ,~b )e− ΓI(x,~b)dxd2b
1 k n=1 n! k I 1 1 I n n R
R R P (cid:0) (cid:1)
× ∞ 1 m Γ (x′,~b′ −β~)...Γ (x′ ,~b′ −β~)e− ΓJ(x′,~b′)dx′d2b′
m=1 m! k J 1 1 J m m R
P (cid:0) (cid:1)
×k! dσˆ/dp ···dσˆ/dp dx d2b ...dx d2b dx′ d2b′ ...dx′ d2b′ ,(4)
1 k k+1 k+1 n n k+1 k+1 m m
i.e.k partons from the hadronI, k partons from the hadronJ are chosenand
connected in all the k! ways with the elementary cross sections dσˆ/dp the re-
mainingvariablesareintegratedwithoutanyconstraint. Afurtherintegration
over the kinematical variables p ,...,p gives the integrated inclusive cross
1 k
section, in this particular case:
1 k
σ = d2β Γ (~b,x)σ¯ Γ (~b−β~,x′)d2bdxdx′ . (5)
k k!Z hZ I x,x′ J i
The effective cross section is introduced in the usual way[2]:
σ2
σ = 1 (6)
eff
2σ
2
and the generalizations for higher integrated inclusive cross sections may be
defined in term of dimensionless parameters τ
k
(σ )k
1
σ = . (7)
k k!(σ )k−1τ
eff k
In this simplified treatment it is clear that σ is mainly connected with the
eff
geometrical properties of the hadron, in fact if σ¯ is multiplied by a constant
then σ remains unaffected, this property holds also for the parameters τ .
eff k
The relevance of the effective cross section σ has been discussed in an-
eff
other talk[3], here the attention is concentrated on the parameters τ .
k
2 Examples
The population of partons, whichever may be its detailed shape, certainly
increases with decreasing x. When the total energy is so high that hard scat-
terings can occur even between low-x partons, these processes are more likely
thanthoseinvolvingthefewvalencequarks. Thissuggestsafurthersimplifica-
tion obtained by performing an integration in x of the distribution, assuming
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thatthe kinematicalconstraintsoverthexvariablesarenotveryrelevantpre-
cisely because the small-x processesare the dominantones. The expressionin
eq.(5) is substituted*a with:
1 k
σ = d2β Γ¯ (~b)σ¯Γ¯ (~b−β~)d2b . (8)
k k!Z hZ I J i
Itisnoweasytoproceedwiththeactualcalculationchoosingsomedefinite
forms for Γ. Two choices, easy to treat and different enough to allow a first
exploration, are:
1 1
Γ =ρ exp[−b2/R2] , Γ =ρ θ(R2−b2). (9)
G πR2 D πR2
Interms ofthese choicesthe correspondingvaluesof τ , τ arecomputed.
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The parton population that has been consideredtill now completely lacks
correlations among the partons.
A simple but efficient way of introducing correlation, that allows also a
model interpretation is to build up the parton populations in terms of two
clusters, having their centers spread over the hadron size*b To be definite a
term of this distribution is written as:
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d2B′d2B′′f(B′)f(B′′)Γ(~b′−B′)...Γ(~b′ −B′)Γ(~b′′−B′′)...Γ(~b′′ −B′′)
n′!n′′!Z 1 n′ 1 n′′
(10)
with d2Bf(B) = 1. In so doing the Poissonian character of the integrated
distribRutions is preserved but correlations in the impact parameter are intro-
duced. The actual calculation is performed by choosing:
1 1
Γ=ρ exp[−(~b−B~)2/r2] , f = exp[−B2/R2] (11)
πr2 πR2
One verifiesthat correlations,in form ofdependence on~b′−~b′′ are introduced
by the integration over B. The values of τ , τ are explicitly computed, they
3 4
depend on the ratio u=(R/r)2.
A definite way of departing from the Poissondistribution is to change the
original weights of the multiplicities, the general term of the non correlated
distribution:
C
N(C ) nΓ(~b )...Γ(~b ) (12)
j 1 n
n!
aThe expression Γ¯I represents the effect of an integration in dx, the ”bar” willbe omitted
inthefollowing.
bThisdescriptionhassomesimilaritieswiththevalonmodelofR.C.Hwa[4];howeverthemain
attention isdirectedtheretothelongitudinalvariables,heretothetransversevariables.
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requiressomemanageablechoiceofthecoefficientsC ,inparticularanexplicit
j
formofN(C ) is needed. A possible choice is a negativebinomialdistribution
j
for the initial partons*c ; it gives for the coefficients andfor the normalization
term:
ν
C =(ν) ≡ν(ν+1)···(ν+n−1) , N(C )= 1− Γ(~b)d2b (13)
n n j h Z i
ThePoissondistributionisreachedinthelimitρ→ρ/ν andthenν →∞.
In this way it is possible to measure how much the results deviate from the
previous ones when the distribution deviates from the Poissonian form. The
shape in~b of the parton distribution enters in a way which is independent of
the choice of the coefficients C , so different choices, e.g. the ones of eq. (11),
n
are possible. The calculation of the quantities τ is a bit more laborious than
in the Poissoniancase, anyhow it can be carried out explicitly.
3 Numerical results and conclusions
The numerical results for the quantities τ are presented in Table 1.
Table1: Valuesofτ fordifferentpartonpopulations
A A B C
1 2
τ 3 0.80 3 ·F (u) 3 ν+1 2
3 4 4 3 4hν+2i
τ 1 0.56 1 ·F (u) 1 (ν+1)2 2
4 2 2 4 2h(ν+2)(ν+3)i
The column A corresponds to an uncorrelated Poissonian distribution and
1
Gaussian shape Γ in eq.(9).
G
The column A corresponds to an uncorrelated Poissonian distribution and
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rigid disk shape Γ in eq.(9).
D
The column B corresponds to a Poissonian distribution with correlations and
Gaussian shape, eqs. (10,11).
The column C correspond to an uncorrelated negative binomial distribution
and Gaussian shape Γ , eqs. (12,13).
G
The functions F ,F which appear in column B are rational functions of
3 4
u; both are equal to 1 when u = 0 and when u → ∞, moreover it results
cThiskindofdistributionwasproposed,inadifferentcontext[5] alongtimeago.
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F (1)=1.09, F (1)=0.93,itmaybe verifiedingeneralthattheydonotvary
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verymuch;forthis reasonthe morenaturalcasewiththreeclusters[4]wasnot
worked out. The square parentheses in column C, which are always less than
1, may differ strongly from unity for small values of ν, i.e. for distributions
that differ much from the Poissonian.
From this preliminary analysis it results that the higher order integrated
inclusive cross sections feel, obviously, all the characteristics of the parton
distribution,buttheyaremainlyaffectedbythemultiplicitydistributionofthe
incomingpartonsandlessbythespatialshapeorbythethespatialcorrelations
of the parton distribution.
Acknowledgments
This workis partofa widerinvestigationperformedtogetherwithD.Treleani.
DiscussionswithM.A.BraunandR.C.HwaduringISMD99areacknowledged.
This work has been partially supportedby the ItalianMinistry of Univer-
sity and ofScientific and TechnologicalResearchby means of the Fondi per la
Ricerca scientifica - Universita` di Trieste .
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