The black-disk limit in high-energy ep and pp scattering

0.05 MB·English

by  M. Strikman

The black–disk limit in high–energy
ep and pp scattering1
M. Strikman and C. Weiss†
∗
DepartmentofPhysics,PennsylvaniaStateUniversity,UniversityPark,PA16802,USA
∗
6 †TheoryCenter,JeffersonLab,NewportNews,VA23606,USA
0
0
2 Abstract. Wereviewarecentlyproposedframeworkforstudyingtheapproachtotheunitaritylimit
insmall–xprocessesinepand ppscatteringwithintheDGLAPapproximation.Ourformulationis
n
basedonthecorrespondencebetweenthestandardQCDfactorizationtheoremsforhardprocesses
a
J andthedipolepictureofsmall–xscatteringintheleadinglogarithmicapproximation.Itallowsus
to incorporateinformationaboutthe transverse spatialdistribution of gluonsin the proton(GPD)
1
3 fromexclusivevectormesonproductionatHERA.Weshowthattheinteractionofsmall–sizecolor
singletconfigurationswiththeprotonapproachesthe“black–disklimit”(BDL)duetothegrowthof

1 theDGLAPgluondensityatsmallx.Thisnewdynamicalregimeismarginallyvisibleindiffractive
v DISatHERA,andwillbefullyreachedincentral ppcollisionsatLHC.
9
Keywords: Quantumchromodynamics,high–energyscattering,generalizedpartondistributions
5
PACS: 12.38.-t,13.85.-t,14.80.Bn,25.75.Nq
2
1
0
6 INTRODUCTION
0
/
h At TeV energies, strong interactions enter a regime in which hadron–hadron and
p
electron/photon–hadron cross sections can become comparable to the “geometric size”
-
p of hadrons, and unitarity becomes an essential feature of the dynamics. In QCD, the
e
h approachtotheunitarityregimeisdrivenbytheincreaseofthegluondensityinhadrons
: at small x. Theoretical studies so far have mostly focused on incorporating unitarity
v
i effects in QCD evolution equations based on the large log(1/x) approximation (BFKL
X
evolution) [1, 2, 3]. However, such calculations are of practical relevance only in
r
a situations where the amount of glue produced by log(1/x)–enhanced radiation exceeds
byfar thatoriginatingfrom othersources.
It is known that there is a large non-perturbative gluon density in the proton at a low
scale (Q2 4GeV2). Its physical origin may be seen in the spontaneous breaking of
∼
chiral symmetry, which implies that a significant fraction of the nucleon’s momentum
shouldbecarriedbygluonfields.Thislargenon-perturbativegluondensityisthereason
behind the success of the DGLAP evolution equations — based on the large logQ2
approximation and retaining only first (LO) or second (NLO) powers of log(1/x) — in
1 Notice:ThismanuscripthasbeenauthoredbyTheSoutheasternUniversitiesResearchAssociation,Inc.
underContractNo.DE-AC05-84150withtheU.S.DepartmentofEnergy.TheUnitedStatesGovernment
retains and the publisher, by accepting the article for publication, acknowledges that the United States
Governmentretainsanon-exclusive,paid-up,irrevocable,worldwidelicensetopublishorreproducethe
publishedformofthismanuscript,orallowotherstodoso,forUnitedStatesGovernmentpurposes.
Theblack–disklimit February2,2008 1
describingtheHERADISdata.Theoreticalstudiesindicatethatlog(1/x)–resummation
effects are small down to x 10 5 [4]. At the same time, DGLAP evolution, starting
−
∼
from the non-perturbative gluon density at the low scale, leads to a rapid growth of the
gluon density at small x. This strongly suggests that unitarity effects become important
longbeforetheregionoflog(1/x)dominanceis reached.
Recently, we have proposed a simple framework for studying the approach to the
unitarity limit within the DGLAP approximation [5, 6]. Our formulation makes use of
thecorrespondencebetweenthestandardQCDfactorizationtheoremsforhardprocesses
and the dipole picture of small–x scattering in the leading logarithmic approximation.
This framework allows one to approach the question of unitarity at small x on the
basis of the vast amount of data on hard processes in ep scattering at HERA (inclusive,
diffractive,andexclusive)and p¯pscatteringattheTevatron.Inparticular,weincorporate
information about the transverse spatial distribution of gluons in the proton, obtained
from studies of exclusive vector meson (J/y ,r ) production at HERA (generalized
parton distributions). Based on this information, we show that the interaction of small–
size color singlet configurations with the proton approaches the “black–disk limit”
(BDL) at small x. This new dynamical regime is marginally visible in diffractive DIS
at HERA, and will be fully reached in central pp collisions at LHC, where it will have
numerous qualitative implications for the hadronic final state. In these proceedings we
briefly summarize the theoretical arguments leading to the BDL, and the evidence for
it in presently available small–x data (HERA, Tevatron). For details, as well as for a
discussionoftheimplicationsfor pp collisionsat LHC, werefer toRefs. [5, 6].
DIPOLE PICTURE FROM QCD FACTORIZATION
Atsmallx, g p scatteringwithlongitudinalpolarizationcan beviewedas thescattering
∗
ofsmall–sizecolor–singletquark/gluonconfigurationsinthephotonwavefunctionfrom
theproton.Intuitively,thiscanbeunderstoodfromthespace–timeevolutioninthetarget
rest frame, in which the virtual photon typically fluctuates into a qq¯ pair of transverse
sized 1/Qalongtime,t 1/(2m x),beforehittingthetarget.Formally,itcanbe
coh N
shown∼thattheleadinglogQ2 a∼pproximationfortheDIScrosssection,s ,isequivalent
L
tothescatteringofaqq¯dipolefrom theprotonwithcrosssection[7, 8]
p 2
s dp = F2 d2 a (Q2 )xG(x,Q2 ), (1)
4 s eff eff
where F2 = 4/3 (for gg dipoles, F2 =3). The cross section, which vanishes for small
dipoles (“color transparency”), is proportional to the gluon density in the proton, eval-
uated at an effective scale Q2 (p /d)2. We stress that, by the derivation from the
eff ≈
factorized DIS cross section, the gluon density in Eq. (1) is unambiguouslydetermined
as the leading–twist gluon density in the leading logQ2 approximation. In particular, it
is subject to DGLAP evolution, which produces a strong rise at small x, and implies a
rapid growth of the dipole cross section with increasing energy. The crucial question is
atwhich energies unitarityleads toa breakdownofthisapproximation.
Though restricted to the leading–log approximation, the dipole picture with Eq. (1)
is useful for discussing qualitative features of the HERA inclusive DIS data, such as
Theblack–disklimit February2,2008 2
the breakdown of the leading–twist approximation at small x for Q2 .4GeV2, and the
difference between s and s [6]. The extension to diffractive final states requires one
L T
to explicitly consider qq¯g...g configurations in the photon wave function. Diffractive
DIS thus allows one to probe the interaction of small–size quark/gluon configurations
withhadronicmatterinmuchmoredetailthan inclusiveDIS.
TRANSVERSE SPATIAL DISTRIBUTION OF GLUONS
Tostudytheroleofunitarityinhard processes at smallx, we need toknownot onlythe
total density of gluons in the proton, but also their spatial distribution in the transverse
plane. Thisinformationcomes from studiesofexclusivevectormeson production(V =
J/y ,r )ing pscatteringatsmallx.OngroundsofageneralQCDfactorizationtheorem,
L∗
the amplitudes for these processes can be expressed in terms of the gluon generalized
partondistribution(GPD)intheproton(hereintheapproximationofzero “skewness”),
G(x,Q2;t) = G(x,Q2)F (x,Q2;t), F (x,Q2;t =0) = 1. (2)
g g
F is the normalized two–gluon formfactor, which can directly be extracted from the
g
t–dependence of the differential cross section, ds (g p Vp)/dt (cid:181) F2(t). Its Fourier
L∗ g
transformwithrespect tothetransversemomentumtrans→ferto theproton,DDD ,
⊥
d2D
Fg(x,Q2;b) ≡ Z (2p )⊥2 ei(DDD ⊥bbb) Fg(x,Q2;t) (t =−DDD 2), (3)
⊥
describesthedistributionofgluons(withlongitudinalmomentumfractionx)withregard
totransverseposition,bbb, normalizedaccording to d2bF (x,Q2;b)=1.
g
Extensive studies of J/y photo/electroproductRion at HERA have demonstrated the
applicabilityoftheQCD factorizationformulae,withcorrectionsduetothefinitetrans-
versesizeofthevectormeson[6].TogetherwithJ/y photoproductiondatafromfixed–
a¢
hard
0.4 Pion
cloud
]
2
m HERA
f
[
>
2 0.2
b Fixed-
<
target
exp.
0
10-4 10-3 10-2 10-1 1
x
FIGURE 1. Summary of the information on the proton’s average transverse gluonic size, b2
d2bb2 F (x,Q2;b), from J/y photoproduction (Q2 3GeV2). The increase between x 10h1 ian≡d
g −
≈ ∼
R10 2canbeattributedtothepioncloud[10].FordetailsseeRefs.[5,6].
−
Theblack–disklimit February2,2008 3
target experiments at lower energies, these data have produced a detailed picture of the
transverse spatial distribution of gluons at Q2 3GeV2 over a wide range of x. The
≈
proton’s average transverse gluonic size increases with decreasing x, see Fig 1. Various
dynamical mechanisms have been identified, which contribute to this growth in differ-
ent regions of x [9, 10]. At higher Q2, the growth with decreasing x is slower, because
DGLAP evolution with increasing Q2 effectively probes higher and higher x values in
theinputdistributionat thestartingscale.
An important observation is that the average transverse radius of the distribution of
gluons with significant momentum fraction, x&10 2, is considerablysmaller than the
−
radius of soft hadronic interactions in high–energy pp collisions. This implies that pp
eventswithhardprocessesinvolvingtwopartonswithx &10 2(e.g.,dijetproduction)
1,2 −
originatefrommuchmorecentral collisionsthanminimumbiasevents[5].
BLACK–DISK LIMIT OF DIPOLE–PROTON SCATTERING
On the basis of the dipole picture in the leading logQ2 approximation, cf. Eq. (1),
and the information about the transverse spatial distribution of gluons, we can study
the approach to the unitarity regime in hard processes at small x. The problem can
be formulated in the language of an optical model for hadron–hadron scattering. We
considertheelasticscatteringamplitudeforasmall–sizecolordipolefromtheprotonin
impactparameterrepresentation(seeFig. 2)
is
Adp(s,t) = d2be i(DDD bbb) G dp(s,b) (t = DDD 2 ), (4)
4p Z − ⊥ −
⊥
where G dp(s,b) is the profile function. Using the optical theorem, the inelastic (total
minuselastic)crosssectioncan beexpressedintermsoftheprofilefunctionas
s (s) = d2bs (s,b), s (s,b) 1 1 G dp(s,b) 2. (5)
in in in
Z ≡ −| − |
d
b
FIGURE2. Dipole–protonscatteringinimpactparameterrepresentation
Theblack–disklimit February2,2008 4
G G G G
qq gg qq gg
1 1
−2
x =10
−3
10
−4
10
−5
10
0.5 1 0.5 1
0.5 0.5
b[fm] b[fm]
0 0
0 0.5 1 0 0.5 1
d = 0.1 fm d = 0.3 fm
FIGURE3. Theprofilefunctionfordipole–protonscattering,G dp(s,b), Eq.(4),asobtainedfromthe
LODGLAPgluondensity,cf.Eq.(1),andaphenomenologicalparametrizationofthetransversespatial
distributionofgluons[5].Shownaretheresultsfordipolesofsized=0.1and0.3fm,andvariousvalues
ofx=p 2/(d2s).Theleftaxesrefertoqq¯dipoles,therightaxestoggdipoles.
The function s (s,b) can be interpreted as the probability for inelastic interaction in
in
dipole–protonscattering at agivenimpactparameter, b. It tendstounityif
G dp 1. (6)
→
In optics, this limit corresponds to the scattering from a black disk, whence Eq. (6) is
referred toas the“black–disklimit”(BDL)ofdipole–hadronscattering.
In theleading–twist(LT)approximation,thedistributions (s,b)isgivenbyEq.(1),
in
with the total gluon density replaced by the local density in transverse space. While
the LT formula is meaningful only as long as s (s,b) 1, we can use it to study the
in
≪
breakdownoftheLTapproximationandtheonsetoftheBDLregime.Figure3showsthe
resultforG dp obtainedinthisapproximation,forbothqq¯(leftscale)andgg(rightscale)
dipoles.Onesees thattheprofilefunctionapproaches theBDL, Eq. (6), atsmallimpact
parameters, even for small dipole sizes, once x becomes sufficiently small. The reason
isthegrowthofthegluondensityin theprotonat smallxdueto DGLAPevolution.
BLACK–DISK LIMIT IN ep SCATTERING AT HERA
Inclusive g p scattering in LO corresponds to the scattering of a qq¯ dipole from the
L∗
proton. One sees that the BDL regime becomes marginally relevant at the upper end of
the HERA energy range (x & 10 4) for Q2 4GeV2 (corresponding to d = 0.3fm).
−
We emphasize that in the dipole picture on∼ly the longitudinal cross section, s , is
L
dominated by small–size dipoles. The transverse cross section, s , receives significant
T
contributions from large–size configurations in the virtual photon even for substantial
Q2, and isthusnotasuitableobservableforprobingtheBDL inhard interactions.
More sensitive to the onset of the BDL regime are diffractive processes in g p
∗
scattering, which probe the interaction of qq¯g...g dipoles with the target. The fact that
Theblack–disklimit February2,2008 5
theprobabilityforgluon–induceddiffraction,P ,isexperimentallyclosetoitsmaximum
g
value, 1/2, at the upper end of the HERA energy range indicates that the interaction of
suchdipoleswiththeprotonis closeto theBDL [6].
BLACK–DISK LIMIT IN pp SCATTERING AT LHC
Muchsmallervalues ofx than in ep scattering can be probed in hard processes induced
byleadingpartonsinhigh–energy ppcollisions.AtLHC(√s=14TeV),apartoninthe
“projectile” proton with momentumfraction x 10 1, in a hard collision producing a
1 −
∼
finalstatewithtransversemomentum p =2GeV,resolvespartonsinthe“target”with
⊥
x = 4p2 /(x s) 10 6. (7)
1 −
∼
⊥
Inthedipolepicture,suchparton–partonprocessescanpredominantlybeassociatedwith
scattering of gg dipoles from the target (for which the cross section is 9/4 times larger
than for qq¯ dipoles), with Q2 = (p /d)2 = 4p2. Under these conditions, the dipole–
protoninteractionforcentral collisionsis deep i⊥nsidetheBDL regime,seeFigure3.
The approach to the BDL qualitativelychanges the interactions of leading partons in
central pp collisions. They acquire substantial transverse momenta through their inter-
action with the dense medium of small–x gluons, p 4 5GeV for x 10 1
,BDL 1 −
in the estimate of Ref. [5]. As a result, one expects s⊥ignific≈antl−y larger transver∼se mo-
mentaand lowermultiplicityin forward particleproduction,as wellas otherqualitative
changesinthehadronicfinalstate[5].Theseeffectscanbeobservedbyselectingcentral
ppcollisionswithatriggeronharddijetproductionatcentralrapidities(seeabove)[5].
Such measurementsarefeasiblewiththeCMSand TOTEMdetectorsat LHC.
Evidence for the BDL in existing pp/p¯p data is the observation that the pp elastic
scattering amplitude above the Tevatron energy (√s=1.8TeV) seems to be “black” at
central impact parameters, G pp(b = 0) 1 [11]. This is consistent with the fact that
≈
in the BDL all leading (valence) quarks acquire large transverse momenta due to hard
interactions,makingitimpossiblefortheprotonwavefunctionstoremaincoherent[12].
Acknowledgments. Theresearch ofM. S. issupportedby DOE.
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