Nuclear enhancement of universal dynamics of high parton densities

0.17 MB·English

by  H. Kowalski

Nuclear enhancement of universal dynamics of high parton densities
H. Kowalski,1 T. Lappi,2 and R. Venugopalan2
1Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany
2Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
We show that the enhancement of the saturation scale in large nuclei relative to the proton is
significantly influenced by the effects of quantum evolution and the impact parameter dependence
of dipole cross sections in high energy QCD. We demonstrate that there is a strong A dependence
in diffractive deeply inelastic scatteringand discuss its sensitivity to the measurement of the recoil
nucleus.
PACSnumbers: 24.85.+p,13.60.Hb
8
0
The propertiesofhadronicandnuclearwavefunctions virtual photon to produce a quark–anti-quark pair of
20 amtuhltigi-hpaernteircgleiepsraordeuocftgiorneaitnimQpCoDrt.aEncsepeinciualnlydeirnsttraingduiinngg csirzoessrse=cti|orn⊥|foarndthidds2σbdppi⊥api(rr⊥to,xs,cba⊥tt)erdeonffottehsethtaergdeitpoalet
n
a isthepossibilitythatthesmallFeynmanxcomponentsof an impact parameter b⊥. The former is well known
J thesewavefunctionsdemonstrateuniversalbehaviorthat from QED, while the latter represents the dynamics of
8 isinsensitivetothedetailsofhadronornuclearstructure QCD scattering at small x. A simple saturation model
1 in the (large x) fragmentation region. (known as the GBW model [6]) of the dipole cross sec-
h] inTQhCeDspfoeclliofiwcsnfarotumrethoef sutrnoivnegrseanlhasmncaelml exntdoyfnagmluiocns tion, parametrized as dd2σbdpi⊥p =2(1−e−r2Q2s,p(x)/4) where
Q2 (x) = (x /x)λ GeV2, gives a good qualitative fit to
p bremsstrahlung at small x leading to a rapid growth of s,p 0
- the occupationnumber of a transversemomentum mode the HERA inclusive cross section data for x0 = 3·10−4
p andλ=0.288. However,the model doesnotcontainthe
e k⊥ in the hadron or nuclear wavefunction. However, it
bremsstrahlunglimit of perturbative QCD (pQCD) that
h can maximally be of order 1/αs (where αs is the QCD
[ couplingconstant)becauseofnon-linearmulti–partonef- applies to small dipoles of size r ≪1/Qs(x).

fects such as recombination and screening which deplete In the classical effective theory of the CGC, to lead-
2
ing logarithmic accuracy,one can derive the dipole cross
v the gluon density at small x [1]. In particular, the oc-
7 cupation number is maximal for modes with k⊥ <∼ Qs, section[4]containingthe rightsmallr limit. This dipole
4 whereQ (x), appropriatelycalledthesaturationscale,is cross section can be represented (see however [7])
s
0
3 a scale generated by the multi-parton dynamics. For a dσp
5. pifreostbeinwaituhntirvaenrssavlersscealriensgolfuotrimono1f/oQb2s,ertvhaisblsecsaalesias mfuannc-- d2bdi⊥p =2 1−exp −r2F(x,r)Tp(b⊥) , (2)
70 tion of Q/Qs in a wide kinematical range in x and Q2. where Tp(b⊥) is t(cid:2)he impac(cid:0)t parameter profile(cid:1)(cid:3)function
0 In addition to the strong x dependence generated by in the proton, normalized as d2b⊥Tp(b⊥) = 1 and F
: gluon bremsstrahlung, the saturation scale Qs has a is proportional to the DGLAP evolved gluon distribu-
iv strongAdependencebecauseoftheLorentzcontraction, tion [8] R
X intheproberestframe,ofthenuclearpartondensity. For
r large enough A and smallenough x, the saturationscale F(x,r2)=π2αs µ20+4/r2 xg x,µ20+4/r2 /(2Nc).
a
is largerthan Λ , the fundamental softscale ofQCD. (3)
QCD (cid:0) (cid:1) (cid:0) (cid:1)
In this letter, we will discuss the A and x dependence The dipole cross section in Eq. (2) was implemented in
of the saturation scale and some of its ramifications for theimpactparametersaturationmodel(IPsat)[9]where
hard diffraction in nuclei. the parameters are fit to reproduce the HERA data on
A saturation scale arises naturally in the Color Glass the inclusive structure function F2.
Condensate (CGC) [2] description of universal proper- Ingeneral,thedipolecrosssectioncanrangefrom0in
tiesofhadronandnuclearwavefunctionsatsmallx. The ther →0colortransparencylimitto2,themaximaluni-
CGC,whenappliedtoDeeplyInelasticScattering(DIS), tarity bound. The saturation scale Qs characterizes the
results [3, 4], at leading order in α , in the dipole pic- qualitative change between these regimes; we shall here
s
tureofDIS[5],wheretheinclusivevirtualphotonhadron defineQs asthesolutionof dd2σbdi⊥p(x,r2 =1/Q2s(x,b⊥))=
cross section is 2(1−e−1/4) [24].
σLγ∗,Tp =Z d2r⊥Z01dz(cid:12)ΨγL∗,T(cid:12)2Z d2b⊥ dd2σbdpi⊥p. (1) binlegTwhloehgeaInPristlaheatmddsinipignollxeog.carrAoitsthsvmseesrcytiniosmnQa2ilnldxEom,q.qinu(a2at)netiuosmvaeprepvleloiacldua---
Here ΨγL∗,T(r⊥,z,Q) 2 rep(cid:12)(cid:12)resent(cid:12)(cid:12)s the probability for a tliimonitinoftlhineeCarGsCma[2ll] xdeesvcorilbuitniognbaosthwtehlleabsrneomnslsintreaahrluRnGg
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2
1 Q2 (Ca) [GeV2] the initial scale µ20. A stronger x-dependence also for
0.2 2 largedipoles, suchas in the in the GBW or bCGC mod-
els, gives a stronger x-dependence of shadowing at fixed
9
0. Q2. As shown in Fig. 1 (center), both the IPsat and
p
F2 bCGC modelspredictstrongQ2-dependence(atfixedx)
A
AF/20.8 filoyrrsehsapdoonwsiibnlge.foIrtthisetshhiasdloawttienrgeeffffeecctt wseheinchinisthperNimMaCr-
Ca, IPsat
Ca, bCGC NMC Sn/C Ca, IPsat data. Precision measurements of FA/AFp would shed
NMC Ca IPsat Ca, bCGC 2 2
0.7 CCN,,M IbPCCs GaCtC bCGC PPbb,, IbPCsGatC morelightonthe relativeimportance ofQ2 andxevolu-
tion in this regime.
-3 -2 2 4 8 -4 -3
10 10 10 10 WenowturntoadiscussionoftheAandxdependence
x Q2 [GeV2] x
of the saturation scale. In a simple GBW type model,
insertingaθ-functionimpactparameterdependenceinto
FdaIGta..1:CeLnetfetr::Pprerdedicitcitoinonssfofrorsh1a2dFow2Snin/g11c8oFm2CpacroemdptaoreNdMtCo Eq. (5) yields the estimate Q2s,A ≈ A1/3R2pRA22/3Q2s,p ≈
NMC data at x=0.0125. Right: Likewise for Q2 = 5 GeV2 0.26A1/3Q2 for 2πR2 ≈ 20 mb and R ≈ 1.A1A1/3 fm.
as a function of x. s,p p A
The smallness of Q2 /Q2 , due to the constant factor
s,A s,p
∼ 0.26 has sometimes been interpreted [9, 15, 16] as a
evolution at high parton densities, combined with a re- weak nuclear enhancement of Qs. We will argue here
alistic b-dependence, is better captured in the bCGC that detailed considerations of QCD evolution and the
model [10, 11]. Both the IPsat model and the bCGC b-dependence of the dipole cross section result in a sig-
model provide excellent fits to a wide range of HERA nificantly larger nuclear enhancement of Qs.
data for x ≤ 0.01 [11, 12]. We will now discuss the pos- The effect of QCD evolution on Qs,A in the IPsat nu-
sibility that DIS off nuclei can distinguish respectively cleardipolecrosssectionisfromtheDGLAP-likegrowth
betweenthese“classicalCGC”and“quantumCGC”mo- of the gluondistribution. The increasein the gluonden-
tivated models. sitywithincreasingQ2anddecreasing(dominant)dipole
A straightforward generalization of the dipole formal- radius r causes Qs grow even faster as a function of A.
ism to nuclei is to introduce the coordinates of the indi- This is seen qualitatively for two different nuclei, A and
vidual nucleons {b⊥i}. One obtains in the IPsat model, B (with A > B), in a “smooth nucleus” approximation
of Eq. (4) whereby Ai=1Tp(b⊥ − b⊥i) is replaced by
dd2σbdAi⊥p =2h1−e−r2F(x,r)PAi=1Tp(b⊥−b⊥i)i, (4) ATQA2s,(Ab⊥=).AWTeAo(bbt⊥a)inPF(x,Q2s,A) ∼ A1/3 F(x,Q2s,A). (6)
where F is defined in Eq. (3). The positions of Q2s,B BTB(b⊥)F(x,Q2s,B) B1/3F(x,Q2s,B)
the nucleons {b⊥i} are distributed according to the
aWvoeroadgse-Saoxfoanndoisbtsreirbvuatbiolen OTAo(bve⊥ri){.b⊥Wi}ebdyenhOotieNth≡e bRTehecfeas.us[sc9ea,lti1hn7eg],ivntihcoreleaagtsrioeonwosfthiFnowfFQitihsmiQpsl2fyaisstthfeaarsttt,ehraasnfoorAbs1sm/e3ra.vlleAedrlsxion,,
Ai=1 d2b⊥iTA(b⊥i)O({b⊥i}). The average differen- theA-dependenceofQ isstrongerforhigherenergies. In
tial dipole cross section is well approximated by[9] s
R Q contrast,the dipole crosssectioninthebCGC modelde-
dσdAip ≈2 1− 1− TA(b⊥)σp A (5) prQend(xs)ownliythoonutthDeG“gLeAomPestcraiclianlgscvailoilnagt”iocnosmabnidnatthieornef[o2r6e]
*d2b⊥+N " (cid:18) 2 dip(cid:19) # doess not have this particular nuclear enhancement [27].
PreciseextractionoftheAdependenceofQ willplayan
s
where, for large A, the expression in parenthesis can be
important role in distinguishing between “classical” and
replaced by exp −ATA2(b⊥)σdpip [13]. All parameters of “quantum” evolution in the CGC.
the model come(cid:16)from either fits(cid:17)of the model to ep-data A careful evaluation shows that because the density
orfromtheWoods-Saxondistributions;noadditionalpa- profileinanucleusis moreuniformthanthatofthe pro-
rameters are introduced for eA collisions. The same ex- ton, the saturation scales in nuclei decrease more slowly
ercise is repeated for the bCGC model. with b than in the proton. The dependence of the satu-
In Fig. 1 (left), we compare the prediction of the IP- rationscaleonthe impactparameterisplottedinFig.2.
sat and bCGC models with the experimental data [25] The saturationscaleingoldnucleiatthe medianimpact
on nuclear DIS from the NMC collaboration [14]). Fig- parameter for the total cross section b is about 70%
med.
ure 1 (right) shows that the x dependence of shadow- of the value at b = 0; in contrast, Q2 (b ) is only
s,p med.
ing for fixed Q2 in the IPsat model is very flat. This is ∼35% of the value at b=0.
because the best fit to ep-data in DGLAP-based dipole The A dependence of the saturation scale for various
models [8, 9] is given by a very weak x-dependence at x is shown in Fig. 3, for the IPsat model on the left and
3
1 Eq. (7) over t, reads [29]
p
=0) 0.8 Ca 2
2Q(bs0.6 Au σLD,T = 14 d2r⊥dz ΨγL∗,T 2 d2b⊥ dd2σbdi⊥p . (8)
b)/ 0.4 Z (cid:12) (cid:12) Z (cid:18) (cid:19)
2(Qs0.2 The diffractive slope(cid:12)(cid:12)at t (cid:12)(cid:12)= 0 depends on the size of
the system. For small t ∼ −1/R2 one expects a very
0 A
0 0.5 1 1.5 2 2.5 steep t-dependence ∼ exp{DtR2} (with D ∼ 1). In our
b/b A
med pictureofthenucleusasa“lumpy”collectionofpartially
overlapping nucleons (Eqs. (4) and (5)), an interesting
FIG.2: Impactparameterdependenceofthesaturationscale
for p, Ca and Au at x=0.001 and Q2 =1 GeV2. Details in questioniswhetherthislumpinessshowsupasaproton-
text. like tail ∼exp{D′tRp2)} of the t-distribution.
Ifonerequiresthatthenucleusstayscompletelyintact,
2-1/30.3 2[GeV]A (x/0.001)Qs0.20.40.60.8 xxxxxx ====== 000000......000000100100,10,10 b,1m,1 =b,m, e=bm0de=0de0d ←←←←←← 0.20.40.60.8 xxxxxx ====== 000000......000000100100,10,10 b,1m,1 =b,m, e=bm0de=0de0d ←←←←←← Mtlplteehievvneereegailmisa,npuvaegrrenno.irntdcaaegCglsdletsoyσhnhDaces·atii/idNnnadettrabfmaufceabuttlulsloerstrbeeopctbffoehaeliiyelvnscpeenitrceduryarocflirlnoneairunipmosesindiveagdeltchynotstaalusltsiincd∼whttehohraeeex[tr2shpaem3e{m]DatdiphlsltileffiRcttrhuA2neaadux}cle----.
cleus breaks up into color neutral constituents without
01 50 100 150 200 01 50 100 150 200
A A filling the rapidity gap between the qq¯ dipole and the
nuclearfragmentationregion. Suchevents correspondto
FIG. 3: Saturation scale at b = 0 (open symbols) and b =
performing the average h·i over the cross section [22],
bmed. (filled symbols) as a function of A for different x. The N
saturation scale for theproton is shown at A=1 and by the Eq.(8),insteadoftheamplitude. Thedifferencebetween
arrowsontheright. Left: IPsatmodel. Right: bCGCmodel. the two averaging procedures can be significant with in-
creasingvaluesoft;theresultforcalciumnucleiisshown
in Fig. 4. The t dependence of the proton is shown as
the bCGC model on the right. Note that in the IPsat well; as expected, the “break up” cross section for cal-
model, at small x, Q2s,A(bmed.) for gold nuclei is nearly cium converges to A times the proton cross section with
identicaltoA1/3 timesthevaluefortheproton. Thecor- increasing t.
respondingenhancementforb=0issignificantlysmaller The difference between “no breakup” and“breakup”
as anticipated. The nuclear enhancement in the bCGC integrated cross sections can be seen in Fig. 5 where we
model is nearly as large, showing that it owes, for the plot, as a function of A for fixed x and Q2, the double
kinematic range studied, much more to the relative im- ratio RA , defined as the ratio (σD +σD)/(σγ∗ +σγ∗)
diff. T L T L
pactparameterprofiles(seeFig.2)thantodifferencesin from Eqs. (1) and (8) for a nucleus divided by the same
QCDevolution. Nevertheless,thestrongerAdependence ratio for a proton. For light nuclei (A < 40), RA < 1
of Q2s,A(bmed.) in the IPsat model relative to the bCGC beforegoingwellaboveunityforlargeA. Thisisdbieffc.ause
model, especially at the smallest x values, clearly illus- the diffractive qq¯ cross section is dominated by smaller
trates the differences in quantum evolution between the impact parameters than the inclusive cross section; at
models. The factor of 2001/3 ≈ 6 gives a huge “oomph” smallimpact parameters,the matter density in a proton
in the parton density of a nucleus relative to that of a is larger than in light nuclei. For large nuclei, especially
proton; one requires a center of mass energy ∼14 times in the “break up” case, the nuclear enhancement of the
larger in an e+p collider relative to an e+Au collider to fraction of diffractive events can be quite significant, up
obtain the same Q2s,A(bmed.)(x). toa100%enhancementrelativetothefractionforapro-
We will now focus on some interesting qualitative fea- ton.
turesofharddiffractionoffnuclei(seealso[20]). Forsim-
We thank A. Caldwell, C. Marquet and L. Motyka for
plicity, we will consider only the IPsat model here. The
veryusefuldiscussions. Thismanuscripthasbeenautho-
contributionof qq¯dipoles [28] to the inclusive diffractive
rizedunderContractNo. DE-AC02-98CH10886withthe
cross section can be expressed as
U.S. Department of Energy.
dσdLDt,T = 161π d2r⊥dz ΨγL∗,T 2σd2ip(x,r,∆⊥), (7)
Z (cid:12) (cid:12)
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