Fluctuations, Saturation, and Diffractive Excitation at low x

0.05 MB·English

by  Christoffer Flensburg

Fluctuations, Saturation, and Diffractive
Excitation in High Energy Collisions1
Christoffer Flensburg2
Dept.ofTheoreticalPhysics,Sölvegatan14A,S-22362Lund,Sweden
1
1
0 Abstract. Diffractive excitation is usually described by the Good–Walker formalism for low
2 masses, and by the triple-Regge formalism for high masses. In the Good–Walker formalism the

n crosssection isdeterminedbythe fluctuationsin the interaction.By takingthe fluctuationsin the
a BFKLladderintoaccount,itispossibletodescribebothlowandhighmassexcitationintheGood–
J Walkerformalism.Inhighenergy ppcollisionsthefluctuationsarestronglysuppressedbysatura-

7 tion,whichimpliesthatpomeronexchangedoesnotfactorisebetweenDISand ppcollisions.The
Dipole Cascade Model reproducesthe expected triple-Regge form for the bare pomeron,and the

] triple-pomeroncouplingisestimated.
h
p Keywords: Small-xphysics,Saturation,Diffraction,DipoleModel,DIS
- PACS: 13.85.Hd,13.85.Lg
p
e
h
[
Introduction

1
v
4 Diffractive excitation represents large fractions of the cross sections in pp collisions
0 or DIS. In most analyses of pp collisions low mass excitation is described by the
4
Good–Walker formalism [1], while high mass excitation is described by a triple-Regge
1
. formula [2, 3]. In the Good–Walker formalism the fluctuations in the pomeron ladder
1
0 are normally not included, which is what limits the application to low masses. In the
1 dipolecascademodel[4,5,6,7,8]thefluctuationsintheladdersaretakenintoaccount,
1
allowingtheGood–Walkerformalismtodescribediffractiveexcitationat allmasses.
:
v It turns out that saturation plays a very important role in suppressing diffractive
i
X excitation. This means that pp collisions will have a much lower fraction of diffractive
r excitation than without saturation, specially at higher energies. For DIS, saturation is
a
a smaller effect, and diffractive excitation can be expected to be stronger, which is
confirmed by experiments. The impact parameter profile for diffractive excitation in
high energy pp is found to be in the shape of a ring, due to the approaching black disc
limitat lowb.
Triple-Regge without saturation predicts powerlike growth with energy of the total,
elasticanddiffractiveexcitationcrosssections.By removingsaturationeffects fromthe
dipole cascade model, also that gives a powerlike energy growth, just like the triple-
Reggemodels.Theintercept,slopeandtriple-Pomeroncouplingscanbeextractedfrom
theenergy dependencies.
1 WorksupportedinpartbytheMarieCurieRTN“MCnet”(contractnumberMRTN-CT-2006-035606).
2 IncollaborationwithGöstaGustafsonandLeifLönnblad
100000 0.4 2.5
b=6 DIPSY 220 GeV DIPSY 2000 GeV DIPSY 2000 GeV
10000 AF-p + cutoff 0.35 AFpe-aF AFpe-aF
2
0.3
b=4
1000
0.25 1.5
F) 100 b=2 F) 0.2 b=0 T) b=6 b=3
P( P( P(
0.15 b=3 1 b=0
10 b=9
0.1 b=6 b=9
0.5
1
0.05
b=9
0.1 0 0
1e-05 0.0001 0.001 0.01 0.1 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1
F F T
FIGURE1. ThedistributionofinteractionamplitudesforDIS(left),unsaturatedpp(mid)andsaturated
pp(right).
The dipole cascade model
In the Good–Walker formalism the incoming mass eigenstates are not necessarily
eigenstates of diffractive interaction. However, the mass eigenstates Y are linear com-
k
binationsof the diffraction eigenstates F with eigenvalues T and coefficients c . The
n n kn
diffractivecrosssectionoftheincomingmasseigenstateY canthenbewrittenbysum-
0
mingoveroutgoingmass eigenstates:
ds /d2b=(cid:229) ( Y T Y )2 = Y T2 Y = T2
diff 0 k 0 0
h | | i h | | i h i
k
where the last average over T is with the weights from Y . By subtracting the elastic
0
part,onefinds thediffractiveexcitation:
ds /d2b=ds /d2b ds /d2b= T2 T 2 V .
diffex diff el T
− h i−h i ≡
It turns out that it is the fluctuations in the interaction amplitude that gives the
diffractiveexcitations.
Inourmodel,theeigenstatesofinteractionarecascadesofcolourdipolesintransverse
space. Mueller showed that the cascade was equivalent with leading logarithm BFKL
[9, 10, 11] and has since been enhanced with several non-leading order effects such as
energy-momentumconservation,confinement,runninga andimprovedsaturation.The
s
saturationintheinteractionisincludedthroughunitarisationT =1 e F,whereF isthe
−
−
Bornamplitude.Inthecascade,asaturating2 2“dipoleswing”isincluded,providing
→
asaturationinthecascadeequivalentto theinteractionup toafew percent.
This model has proven to describe a wide range of total, elastic and diffractive cross
sectionsin both pp collisionsand DIS. Thisis describedin detailin[6, 7, 8].
Fluctuations and saturation
Inthissectionwewilluseourmodeltostudyhowsaturationaffectsthefluctuationsin
theinteraction amplitude,and thus diffractiveexcitation.We want to study theeffect of
saturation,andwillseparatebetweentheBorn-levelamplitudeF,andthefullysaturated
1
W = 100 GeV W = 2000 GeV<<TT>>2 W = 14000 GeV
0.8 VT
0.6
0.4
0.2
0
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
b
FIGURE 2. Impactparameterdistributionsfromthe MC for T =(ds /d2b)/2, T 2 =ds /d2b,
tot el
andV =ds /d2bin ppcollisionsatW =100,2000,and14h00i0GeV.bisinunitsohfiGeV 1.
T diffex −
amplitude T. Both these are calculated from our model with a Monte Carlo simulation
program called DIPSY. A large number of colliding dipole cascades are generated and
collidedatfiximpactparameter,andthefrequenciesofinteractionamplitudes,P(F)and
P(T), arestudied.
In the left plot of figure 1 is the distribution of Born amplitudes for g ⋆p at W =
220GeV.ItisseenthatthedistributionbehavesroughlyasapowerofF,givingarather
wide distribution and large fluctuations. This corresponds to a large cross section for
diffractive excitation in DIS. Since F is well below unity, T F and saturation is a
≈
smalleffect.
The distribution of the Born amplitudes of pp at √s = 2000 GeV is shown in the
middle plot of figure 1. This distribution behaves as a gamma function, which very
wide and corresponds to a very large cross section for diffractive excitation. However,
since F is not smaller than 1, unitarity is important. The distribution in the saturated
amplitudeT is shown in the rightmostplot. The shape for the large b distributionsdoes
not change much, but the low b distributions that previously were very wide are now
sharplypeakedjustbelowT =1. T approaching1correspondstotheblackdisclimit,
h i
andisseentostronglysuppressthefluctuations,andthusthediffractiveexcitations.This
isclearlyseenintheimpactparameterprofileinfigure2wherethecentralcollisionsget
suppressedfluctuationsasenergyincrease.Thus,thediffractiveexcitationsliveinaring
whereT 0.5.
≈
Comparison between Good–Walker and triple-Regge
In unsaturatedReggeformalism,thecross sectionsare
s = b 2(0)sa (0) 1 s pp¯se ,
tot − ≡ 0
ds 1
el = b 4(t)s2(a (t) 1),
dt 16p −
M2 ds SD = 1 b 2(t)b (0)g (t) s 2(a (t)−1) M2 e . (1)
Xdtd(M2) 16p 3P (cid:18)M2(cid:19) X
X X (cid:0) (cid:1)
1000
b)
m
( 100
s
total
elastic
single diffractive
10
100 1000 10000
(cid:214) s (GeV)
FIGURE3. Thetotal, elastic andsingle diffractivecrosssectionsin theone-pomeronapproximation.
The crossesare fromthe dipole cascade modelwithoutsaturation,and the linesare froma tunedtriple
Reggeparametrisation.
Here a (t)=1+e +a t is the pomeron trajectory, and b (t) and g (t) are the proton-
′ 3P
pomeron and triple-pomeron couplings respectively. These cross sections are in most
modelsincreasingmuchfasterthanthemeasuredones,andsaturationindifferentforms
are added to fit with experiments. To make a comparison between our model in the
Good–Walker formalism and the Regge models, it is better to compare the unsaturated
models,to avoidthemodeldependence inthesaturationscheme.
Running the DIPSY Monte Carlo without saturation, it turns out (figure 3) that the
energy dependence of the total, elastic and diffractive cross sections fit perfectly to the
Reggeparametrisationswith
a (0) = 1+e =1.21, a =0.2GeV 2,
′ −
s pp¯ = b 2(0)=12.6mb, b =8GeV 2, g (t)=const.=0.3GeV 1. (2)
0 0,el − 3P −
This is not a trivial result. For example without the logarithmic corrections in 1 the
fit would have been significantly worse. Similarly, without the confinement, energy
conservation and running a , that is just leading logarithm BFKL, the increase with
s
energy would have been too strong. The NLL corrections are necessary for the two
approaches toagree thiswell.
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