Table Of ContentNuclear Broadening of Transverse Momentum in Drell-Yan Reactions
M.B. Johnsona, B.Z. Kopeliovichb,c,d, M.J. Leitcha, P.L. McGaugheya, J.M. Mossa
I.K. Potashnikovab,c, Ivan Schmidtb
a Los Alamos National Laboratory, MS H846, Los Alamos, NM 87545, USA
b Departamento de F´ısica, Universidad Tecnica Federico Santa Mar´ıa, Valpara´ıso, Chile
c Joint Institute for Nuclear Research, Dubna, Russia
d Institute of Theoretical Physics, Heidelberg University, Heidelberg, Germany
DataforDrell-Yan(DY)processesonnucleiarecurrentlyavailablefromfixedtargetexperiments
up to thehighest energy of √s=40GeV. The bulk of the data cover the range of short coherence
7 length, where the amplitudes of the DY reaction on different nucleons do not interfere. In this
0 regime, DY processes provide direct information about broadening of the transverse momentum of
0 the projectile parton experiencing initial-state multiple interactions. We revise a previous analysis
2 of data from the E772 experiment and perform a new analysis of broadening including data from
theE866experimentatFermilab. Weconcludethattheobservedbroadeningisabouttwiceaslarge
n
as the one found previously. This helps to settle controversies that arose from a comparison of the
a
J original determination of broadening with data from other experimentsand reactions.
4 PACS:13.85.Qk; 24.85.+p
2
2 the dipole-nucleon cross section,
v
6
dσ (r ,s)
2 C(s)= q¯q T . (2)
61 drT2 (cid:12)(cid:12)(cid:12)rT=0
INTRODUCTION (cid:12)
0
6 Thecrosssectionσq¯q(rT,s)introducedin[10]varieswith
0 Understanding how partons propagate in a nucleus transverseq¯q separation~rT and energy. This quantity is
/ reveals precious information about the early stages of difficulttocalculate,sinceitinvolvesnonperturbativeef-
h
p hadronization that is difficult to obtain by other means. fects,butitcanbefittedtodataforthephotoabsorption
- This understanding also plays a role in using partons as crosssectionandtheprotonstructurefunctionF2(Q2,x),
p
asensitiveprobeofmatterproducedinrelativisticheavy which probe a wide variety of separations and energies
e
h ioncollisions. Oneoftheimportantobservablesisbroad- [11, 12]. Since each of the scatterings is accompanied
: ening of the transversemomentum of produced particles by a loss of energy, there is a close connection between
v
i due to multiple interactions of a parton in the medium. the multiple scattering that leads to momentum broad-
X This effect can be studied for the nuclear medium in eningandinitialstategluonicenergyloss[13,14]. Itwas
r Drell-Yan reactions [1, 2, 3] and deep-inelastic scatter- found, however, that this induced energy loss effects the
a
ing [4, 5] on nuclear targets. The former is an especially DY process much less than the vacuum gluon radiation
clean tool, since the produced dilepton, which has no initiated by the interaction on the nuclear surface [15].
final state interactions, carries undisturbed information In the parton model, a single interaction of a quark
onthe transversemomentumofthe partoninitiating the propagating through the medium can be viewed as a
reaction. quark-gluon correlation in a nucleus in its infinite mo-
A parton propagating through a nuclear medium ex- mentumframe. Thecorrespondingtwist-4term[16]can-
periences Brownianmotioninthe transversemomentum not be calculated, but is rather fitted to data. This phe-
plane [6], with the increase of the mean transverse mo- nomenology has not been successful so far, with the pa-
mentum squared being proportional to the length of the rameter extracted from data varying by a factor of 20,
path times the medium density ρ(z), depending onassumptions[17]andalsodemonstratinga
strong process dependence [18].
L The first observation of nuclear enhancement of large
∆hp2(L)i=2C(s) dzρ(z), (1) transverse momenta was made in hadron-nucleus colli-
T Z
sions [19]. A sensitive tool to measure the transverse
0
momentum broadening of quarks propagating through
where z is the longitudinal coordinate. The energy de- nuclei is heavy dilepton production in the Drell-Yan re-
pendentdimensionlessfactorC characterizesmultiplein- action. The leptons have no interaction in the nuclear
teractions of the parton with bound nucleons. In the medium and thus carry undisturbed information about
light-cone dipole description of the broadening process the transverse momentum of the quark. Nuclear broad-
[7, 8, 9]) C is related to the small separationbehavior of ening of the mean transverse momentum squared of the
2
dilepton is defined as The same figure presents nuclear broadening in J/Ψ
and Υ production at the same energy. Comparing with
∆hp2Ti=hp2TiA−hp2TiN . (3) this data and results available at lower energies, one
can make the following observations: (i) the magni-
PerturbativeQCDpredictsdσDY/dp2T ∝1/p4T (fortrans- tude of broadening in reactions of heavy dilepton and
versely polarized virtual photons) at large pT. There- quarkoniumproduction differ by a factor of 5, while one
fore, the mean value hp2i is divergent, and the broad- would expect the Casimir factor 9/4. Indeed, heavy
T
ening of Eq. (3) might be ill defined (see discussion in quarkoniashouldbe producedvia gluons,whichinteract
[20]). Ofcourse,theexperimentimposesacutoff. Thus, in the medium more strongly than quarks. (ii) The
the experimentally observed broadening is finite, but it errors for Drell-Yan data are small compared to J/Ψ
maydependontheexperimentalacceptance. Thisdiver- production, whereas the latter is measured with much
gence, however,sometimes cancels in the broadening for higher statistics. (iii) Comparison with broadening
several reasons. First of all, perturbative QCD predicts measured at lower energies presented in Fig. 2 suggests
no nuclear effects at large pT, i.e., dσDY(pA)/dp2T → thatbroadeninginheavyquarkoniumproductionshould
AdσDY(pp)/dp2T (see,e.g.,[21]). Secondly,nuclearshad- rise with energy, whereas it is seen to be rather flat, or
owing is expected to vanish at large pT, especially at an even perhaps drop, at high energies in the Drell-Yan
energyaslowas800GeV,theenergyofourcurrentanal- reaction.
ysis. In this case, the divergent high-p part of the inte-
T
grations cancel in (3), and one can measure the nuclear
broadeningeffectwithinarestrictedintervalofp where
T 0.2
the data exhibit nuclear effects. Drell-Yan ∆<p 2> = D * [(A/2)1/3 - 1]
p + A → J/Ψ T
An analysis of broadening in the Drell-Yan reaction p + A →ϒ
waspresentedin[3]usingdatafromtheE772experiment 0.15 π- + A → J/Ψ
atFermilab. Theresultsoftheanalysisfor∆hp2i are π+ + A → J/Ψ
T DY
depicted in Fig. 1 as function of A.
D 0.1
D=0.0297+0.002226*√S
Drell-Yan (E772, 800 GeV) D=0.133 0.05
0.5 J/ψ (E866/789/771, 800 GeV) D=0.0289
ϒ (E772, 800 GeV)
∆<p2> = D * [(A/2)1/3-1]
2c) 0.4 T 00 10 20 30 40 50
2V/ √S (GeV)
e 0.3
G
2p> (T0.2 tFoIGda.t2a:fEornedrigleyptdoenpeannddenhceeavoyfqthuearpkaornaimumeteprroDduincti(o4n)ifinttpeAd
<
∆ 0.1 and πA collisions.
D=0.027
0
The latter observation is difficult to explain, since the
cross section controlling the multiple interactions in the
101 102
MASS NUMBER medium rises with energy. (iv) The dipole approach,
Eq. (2), based on a phenomenological color dipole cross
FIG. 1: Broadening of transverse momentum in production section fitted to data, predicts the factor C [8] to be
of heavy dileptons (squares), J/Ψ (circles) and Υ (triangles)
about twice as large as one suggested by the data
in pA collisions at 800GeV. The data are from the E772,
presented in Fig. 1.
E789, and E866 experiments at Fermilab [2, 22, 23, 34]. The
curves are results of a fit with the parametrization ∆ p2 =
D (A/2)1/3 1 . h Ti In this paper we revise the resultof the previousanal-
−
(cid:2) (cid:3) ysis [3] of the E772 data for Drell-Yan nuclear ratios,
combining them with the brand-new data [25] from the
Following Eq. (1), one can parametrize the A-
E866 experiment for Drell-Yan cross section in pD colli-
dependence as
sions. The improvement in the analysis arises from the
factthattheE866experimentprovidesmoreaccuratepD
∆hp2Ti=D (A/2)1/3−1 (4) cross section data at small pT than that available from
h i
E772. Additionally, it was assumed with no good justi-
A fit to the data points for the Drell-Yan reaction in fication that the same simple parametrizationof the p -
T
Fig. 1 results in D = 0.027GeV2, which corresponds to dependence isvalidforbothpD andpAcollisions,which
C =2 in (1). dictated a specific shape for the ratios and led to a sub-
3
stantial underestimate of the errors for p -broadening. interpretation of a heavy dilepton production is not ap-
T
Our new analysis is less model-biased, but it results in plicable in target rest frame. Indeed, in order to satisfy
somewhat larger errors. We also perform here an anal- thekinematicsannihilatingwiththeprojectile,thetarget
ysis of new data from the E866 experiment, which leads parton must move with momentum p ∼ p /x , where
L T 2
to a broadening about twice as large as in [3], although x in standard notation is the Bjorken x of the target
2
with larger experimental errors. This result resolves the parton. At small x , x ≪ 1, this would mean presence
2 2
problems stated earlier. of highly energetic partons in the stationary target. The
properspace-timeinterpretationinthiscaseisanalogous
to the Weitz¨acker-Williams mechanism of electromag-
LIGHT-CONE DIPOLE DESCRIPTION OF neticbremsstrahlung. Namely,abeamquark(antiquark)
NUCLEAR BROADENING developsafluctuation,q →qγ∗ (withγ∗ →¯ll),whichin-
teracts with the target freeing the dilepton[31, 32].
What looked like overlap of parton clouds at small x
When a projectile scatters from a nucleus, one can 2
in the infinite momentum frame, now takes the form of
treat the target nucleons as independent with some de-
interferencebetweentheamplitudesofapartoninteract-
greeofaccuracyifviewedintherestframeofthenucleus,
ing with different nucleons. On the same footing it can
since the bound nucleons are sufficiently well separated
alsobeinterpretedasthelifetimeofthe|qγ∗ifluctuation,
in that frame. However, when boosted to the infinite
given by the uncertainty principle
momentum frame the same reactionlooks different since
the nucleons may start communicating via their parton 1 2E
q
l = = , (6)
clouds at small Bjorken x. Indeed the Lorentz contrac- c qL Mq2γ∗ −m2q
tion is proportionalto 1/x, therefore it is much stronger
at large, than at small x. As a result, the parton clouds where q is the longitudinal momentum transfer in the
L
arestillseparatedforx>x ,butoverlapandinteractat q → qγ∗ fluctuation. The effective mass squared of the
c
smaller x, x < xc. The border line xc can be estimated fluctuationisMq2γ∗ =Mγ2∗/α+m2q/(1−α)+kT2/α(1−α),
as, whereαisthefractionofthelight-conemomentumofthe
parent quark taken by the dilepton (γ∗), and k is the
T
1 γ∗ −q relative transverse momentum. Hereafter we do
x ∼ , (5)
c
mNRA not discriminate between coherence time and length, as-
suming that the quark has the speed of light. Evidently,
where R is the nuclear radius. Note that this is only
A the same length ∼ 1/q defines the maximal longitu-
L
an order of magnitude estimate. There is a numerical
dinal distance between two scattering centers that can
factorin (5) thatvariesconsiderablyfordifferent species
interfere. This is whythe distance, Eq.(6), is also called
ofpartonsandthe waythe partondistributionis probed
coherence length.
[27].
In the case of a nuclear target, the coherence length is
The interaction between partons at x < xc leads to a of great importance. If this time interval is small com-
modification of their transversemomentum distribution: pared to the mean spacing of nucleons in the nucleus,
suppression of small momenta, enhancement at interme- dileptons are radiated in the Bethe-Heitler regime, i.e.,
diate, and no effect at large transverse momenta. This with no interference between the amplitudes of radia-
phenomenon is known as the saturation effect [28], or tion from different nucleons. At the another extreme,
color glass condensate [29]. One should keep in mind, l ≫R ,theLandau-Pomeranchukphenomenongoverns
c A
however,thatonsetofthiseffecthappensonlyatx<xc. the radiation. Namely, interferences suppress radiation
The partonic interpretation of hard reactions at small withsmalltransversemomenta,enhance it atintermedi-
Bjorken x is known to be Lorentz non-invariant. Deep- ate values of few GeV, and make no changes at higher
inelastic scattering (DIS) looks like absorption of the momentum transfer. In both cases, the linear depen-
virtual photon by a quark (antiquark) that belongs to dence on the path length, or nuclear thickness, Eq. (1),
the target in the infinite momentum frame of the latter. holds, although with different factors C [20]. The bor-
The reaction looks, however, quite different in the tar- derline between the two regimes, l = R , corresponds
c A
get rest frame, as the interaction with a q¯q fluctuation to the relation for x , Eq. (5).
c
of the high-energy virtual photon. Of course, all observ-
ables, including the structure function F (x,Q2), which
2
is proportional to the total photon-target cross section, Short coherence length
are Lorentz invariant.
The Drell-Yan reaction is usually interpreted as an- Themechanismofnuclearbroadeningdependsonhow
nihilation of a quark with an antiquark belonging to the longthecoherencelengthiscomparedtothenuclearsize.
beamandtargetrespectively(orviceversa),q¯q →¯ll[30]. In the regime of short coherence length, l ≪ R , the
c A
Whilethisiscorrectintherestframeofthedilepton,this dilepton fluctuation appears for a very short time deep
4
inside the nucleus and is immediately released on mass
shellbytheinteraction,whichisassumedtobethesame 1.2 C/2H Ca/2H
as on a free nucleon. Therefore, this interaction does
not lead to any change in the transverse momentum de- 1
pendence, i.e., it provides no broadening. However, the
initialstateinteractionsoftheprojectilequarkaffectthe 0.8 E772 E772
o
transverse momentum distribution of the quark, which
i
t
acquires the shape [8] a
R 1.2 Fe/2H W/2H
dN
q = d2r d2r eip~T(~r1−~r2)Ωq (~r ,~r ) n
d2p Z 1 2 in 1 2 a 1
T
Y
× e−12σqN¯q(~r1−~r2,x)TA(b) . (7) -
l 0.8 E772 E772
l
e
Here Ωq (~r ,~r ) is the density matrix describing the im- r
in 1 2 D
pact parameter distribution of the quark in the incident 1.2 Fe/Be W/Be
hadron. We assume a Gaussian shape,
1
Ωqin(~r1,~r2)= hpπ20ie−12(r12+r22)hp20i , (8)
0.8 E866 E866
wherehp2iisthemeanvalueoftheprimordialtransverse
0 0 1 2 3 1 2 3 4
momentum squared of the quark.
Thetransversemomentumoftheradiateddileptonre- p (GeV/c)
flects the broadening acquired by the quark in the nu- T
clearmedium. However,the effectis weakerthan forthe
FIG. 3: The nucleus-to-deuterium and -beryllium ratios of
quark,since the transversemomenta of the dilepton and
dilepton production cross sections as function of transverse
quark are connected by the relation, momentum. The data are from the E772 [2] and E866 [34]
experimentsat 800GeV.
q¯ll =αpq , (9)
T T
where α < 1 is the fractional momentum of the ¯ll. This
seems to be compensated by a stronger enhancement at
relation clearly demonstrates that the effect of broad-
largerp inaccordancewiththesumruleEq.(10). Data
T
ening cannot be translated into a modification of the
in the small p < 1−2GeV domain, where the main
T
quark distribution function (as is frequently done), in
partofthe crosssectioncomesfrom,aremostimportant
the regime of short coherence length. Thus, it confirms
for broadening. In this region the E866 data in Fig. 3
the breakdownofk -factorizationimposed by the initial
T demonstrate stronger nuclear suppressionthan the E772
state interaction, as suggested in [33].
data. The observed deviation from unity at small p ,
T
which is responsible for broadening, is about twice as
large in E866 as in E772 data.
No-shadowing sum rule
Notice that the p -dependence of the ratios in Fig. 3
T
has the typical shape predicted for the regime of satura-
To the extent that the coherence length for the Drell-
tion (of quarks), or color glass condensate [29]. There-
Yanreactionisshortcomparedto the internucleonspac-
fore, it might be tempting to interpret such data as an
ing in the nucleus, no shadowingis possible. This means
observation of this phenomenon. We warn again that
thatinspiteofthenuclearmodificationofthetransverse
this would be a wrong conclusion, since coherence and
momentum dependence of Drell-Yan cross section, the
saturation are impossible in the short coherence length
number of quarks is conserved. This fact can be repre-
regime;initialstateinteractioneffectsimitatesaturation.
sented in the form of a sum rule,
dσ (pA) dσ (pN)
d2p DY =A d2p DY . (10) Long coherence length
Z T d2p Z T d2p
T T
The nuclear ratios measured in the E772 and E866 ex- In this regime, l ≫ R , the amplitudes γ∗ → ¯ll for
c A
periments at 800GeV depicted inFig. 3 havea tendency radiatingaheavyphotonfromdifferentnucleonsinterfere
to satisfy this constraint. Indeed, the ratios are below with each other. This interference, which modifies the
one at small p , but rise above one at larger momen- p distribution of radiation, is known as the Landau-
T T
tum transfers. The weak nuclear suppression observed Pomeranchukeffect andis a genuine effect ofsaturation.
atsmallp ,wheretheabsolutecrosssectionismaximal, Alternatively, one may say that a long-lived fluctua-
T
5
tion q →γ∗q that propagates through the entire nucleus on data for p -dependent ratios depicted in Fig. 3.
T
may be freed as a result of the multiple interactions, in Theseincludethenucleus-to-deuteriumratiosRA/Dfrom
contrasttowhathappensintheregimeofshortl ,where the E772[2]experiment, andnucleus-to-berylliumratios
c
the fluctuation is freed after an interaction with a single RA/Be from the E866 experiment [34], both at 800GeV.
nucleon. These multiple interactions also lead to a dif- These results are quite reliable since they are least af-
ferent type of broadening mechanism. Its origin can be fected by systematic uncertainties. The ratios R are
i
understood from the fact that most of the q →γ∗q fluc- availablein7 bins inp ,i=1−7. Correspondingly,one
T
tuations in the quarkareinvisible in the sense thattheir can calculate the broadening, Eq. (3), as
interactions with the target are so weak that the target
hasrestrictedresolvingpowerorabilitytodiscriminatea ∆hp2i= iσiRip2i − iσip2i , (12)
hard qγ∗ fluctuation, with large intrinsic transverse mo- T P σ R P σ
i i i i i
mentumk ,fromabarequark. However,becauseofthe P P
T
largermomentum transfer allowedby the multiple inter- where for each i-th bin
actionsinthenuclearmedium,theresolutionprovidedby
(pT)mi ax
the nucleus is greater than that of a free nucleon. The dσN
σ = d2p DY , (13)
nucleus canthereforefree harderfluctuations thana free i Z T d2p
T
nucleon. This is the source of nuclear broadening in the (pT)mi in
large coherence length regime.
The same effect can be interpreted within the color with p2 the mean value of p2 within the i-th bin. In
i T
dipole approach in terms of color filtering. The nucleus addition to the seven pT-intervals in data depicted in
filters out projectile dipoles of smaller size than is pos- Fig.3, we have included an eighth interval in Eq. (12)
sible on a free nucleon, leading to larger transverse mo- covering (pT)m7 ax < pT < ∞ and assumed that R8 = 1
menta. Inthiscase,insteadofEq.(7),thetransversemo- in accordance with perturbative QCD expectations [21]
mentumdistributionofdileptonsradiatedbyaquarkin- (see, however,next section). Since the cross section falls
teracting with a nucleus is givenby the factorizeddipole steeply at large pT, this point should not substantially
formula [32], affect the results.
ThedifferentialcrosssectionfortheDrell-Yanreaction
dσ(qA→¯llX) on a free nucleon, dσN /d2p , is more affected by sys-
= d2b d2r d2r eiq~T(~r1−~r2) DY T
d2q Z Z 1 2 tematic uncertainties than the ratio is. We rely on the
T
× Ψ†qγ∗(~r1,α)Ψqγ∗(~r2,α)h1−e−21σqN¯q(r1,x2)TA(b) rdeiffceenretlnytipaulbclriosshsedsercetsiounltsinofppthaenEd8p6D6ecxoplleisriiomnesn[t2f5o].rWthee
− e−21σqN¯q(r2,x2)TA(b)+e−21σqN¯q(~r1−~r2,x2)TA(b) . (11) usethisdatain(13)fortheanalysesofalldatapresented
i in Fig. 3.
We parametrize the differential cross section of the
Here Ψqγ∗(~r,α) is the pQCD calculated light-cone dis-
Drell-Yan reaction as
tribution amplitude, which describes the dependence of
ftrhaect|qioγn∗ailFmocokmceonmtupmo.nenTthoennturcalnesavrertsheicskenpeasrsatfiuonnctainond dσ =N (cid:16)1+ pλ2T21 e−p2T/λ22(cid:17) . (14)
at impact parameter b is given by the integral of the dp2 (1+p2/λ2)n
T T 3
nuclear density along the quark trajectory, T (b) =
A
∞ dzρ (b,z). The factor in the numerator is introduced to describe a
−∞ A
R Theratioofthenucleartonucleoncrosssectionsofthe possible forward minimum in the cross section indicated
Drell-Yan reaction calculated in [32] has a shape typical by data [25].
for the Cronin effect. Such a modification of the trans- We found that the shape of the pT dependence does
verse momentum dependence is a genuine effect of satu- not vary with dilepton mass and x2, as one can see in
ration (or its onset). In spite of the similarity with the Fig. 4. A global fit to Drell-Yan data from pD colli-
analogousresults in the short l regime,the mechanisms sions measured by the E866 experiment for different M
c
andtheir formulationsinEqs.(7)-(11)aredifferent. Ad- and x2 bins [25] with common parameters n, λi and dif-
ditionally,the magnitude ofthe Cronineffect inthe long ferent normalization factors N, led to quite a good de-
lc regime is smaller [20]. scription, χ2 = 83.3 with Nd.o.f. = 52. The values of
the parameters are λ = 0.78±0.08; λ = 0.65±0.04;
1 2
λ = 3.31±0.21; n = 7.00±0.66. The results of the
3
ANALYSIS OF E772/866 DATA FOR DRELL-YAN fit and the data are depicted in Fig. 4. The error bars
REACTION include both statistical and systematic errors added in
quadrature.
Here we determine the nuclear broadening ∆hp2i of Note that at large p the pD cross section in Eq. (14)
T T
dileptons produced on different nuclear targets relying isfallingasp−14,muchsteeperthanonemayexpectfrom
T
6
5 < M < 6 (GeV) 6 < M < 7 (GeV)
10
DY E772
DY E866
1 0.4 J/ψ
)
-1 Y
2
V 10
e 2)
G -2 V
/ 10 e
b -3 G
p
10 (
p ( 10 7 < M < 8 (GeV) 8 < M < 9 (GeV) 2> T
3 P 0.2
d
<
/
1 ∆
σ
d
-1
E 10
-2
10
-3
10 0.0
0 1 2 3 4 0 1 2 3 4 2
10 10
p (GeV)
T A
FIG. 4: Differential cross section of Drell-Yan reaction in
FIG. 5: Results of thepresent analysis for broadening in the
proton-deuteron collisions. Data are from the E866 exper-
Drell-Yan reaction on different nuclei. Closed squares corre-
iment [34] at 800GeV for different intervals of dilepton ef-
spond to E772 data for C, Ca, Fe and W. Open squares
fective mass. Curves show the result of the global fit with
correspond to E866 data for Fe and W. For comparison,
common parameters, Eq. (14), controlling the shape of the
broadening for J/Ψ and Υ, the same as in Fig. 1, are also
cross section.
shown.
perturbative QCD. This happens due to restrictions im-
carbontodeuteriumratiodepictedinFig.3iscompatible
posedbyfiniteness ofenergy,namely,bothx andx rise
1 2
withnonucleareffectsatall. IfwesimplyfixR =1,
with p towards the kinematic limit causing an addi- Be/D
T
the broadeninggiveninEq.(12)willbe underestimated.
tional suppression of the cross section [20]. Addition-
However,one can do better by relying on a theoretically
ally, available data cover a rather restricted interval of
predicted value for R [26] with Eq. (7). Of course,
p <4GeV. Be/D
T
this introduces an uncertainty related to model depen-
With the cross section in Eq. (14) we performed an
dence,butthisuncertaintyshouldnotexceedfewpercent
analysis of data from both experiments, E772 and E866.
of the observed broadening. The results of this analysis
While the ratios measured in E772 are free from major
oftheE866dataaredepictedinFig.5byemptysquares.
systematic uncertainties, the absolute cross sections are
Noticethatthecrosssectionratiosareaffectedbysys-
substantiallyless reliable. Indeed, comparisonofthe dif-
tematic uncertaities much less than the absolute cross
ferentialcrosssectionsoftheDrell-Yanprocessmeasured
sections. The ratios measured in both experiments have
for p−D collisions in the two experiments reveals sub-
mainlyanoverallsystematicuncertainty1%inE866[34],
stantialdifferences. Therefore,werelyonthep−Dcross
and 2% in E772 [2] measurements. One can see from
section from the E866 experiment. The results of such
Eq. (12) that an overall variation of all R leaves the
a combined analysis of the E772 ratios are depicted in i
broadeningunchanged. Thereforewecoulddisregardthe
Fig. 5 by closed squares.
systematic errors of the ratios.
The ratios measured in the E866 experiment are for a
IfwefitthedatainFig.5assumingalineardependence
nucleus relative to beryllium. We use the chain relation
on A1/3, in accordance with (4), and do this separately
between the two ratios,
for the E772 and E866 data, we obtain,
R =R R ; (15)
A/D A/Be Be/D D(E772)=(0.029±0.008)(GeV/c)2 ;
however, the beryllium-to-deuterium ratio is unfortu- D(E866)=(0.059±0.009)(GeV/c)2 . (16)
nately lacking. Nevertheless, this cannotbe a significant
correction,sinceberylliumisalightnucleus. Indeed,the Thevalue ofD(E772)isclosetothe resultofprevious
7
analysis [3] depicted in Fig. 1, although with a consid- BROADENING VERSUS COHERENCE LENGTH
erably larger error. However, the E866 data suggest a
much stronger broadening. Nevertheless, one may con- The mechanism of broadening is expected to correlate
sider the results Eq. (16) as nearly consistent since they with the coherence length [8, 20, 21] as described above.
lie within twostandarddeviations. Ifthe broadeningac- The coherence length of the Drell-Yan process is con-
cording to the E866 data were actually about twice as trolled by the value of x , l ∼ 1/m x [15]. Thus, at
2 c N 2
large as that given by the previous analysis [3] of the large x the dilepton is produced inside the nucleus and
2
E772data,thiswouldsettle thecontradictionwiththeo- onlyhalfofthenuclearthicknesscontributestobroaden-
reticalexpectationsandconclusionsdrawnfromthedata ing. However, at small x the Drell-Yan pair is radiated
2
for the broadening of J/Ψ and Υ mentioned above. outside the nucleus and the entire nuclear thickness is
effective. It would be a spectacular observation to see a
manifestation of this effect in data.
Using recent data from the E866 experiment for the
BEYOND THE DATA p -dependentratiosbinnedinx [34,35]wecanalsotest
T 2
the predicted x -independence of broadening. We have
2
performed a similar analysis of the Fe/Be and W/Be
Although QCD predicts no nuclear effects at high p ,
T
ratios in different x bins and the results are depicted in
the data in Fig. 3 show considerable nuclear enhance- 2
Fig. 6
ment at intermediate values of p . This is the well-
T
known Cronin effect first observed for hadronic beams
[19]. It is not clear from the data how far in p this
T
effect can propagate. The assumption made above for
the last unobserved p interval, R = 1, is probably in-
T 8
correct, therefore the broadening effect shown in Fig. 5
could be be underestimated. The ad hoc shape of R(p )
T
used in [3] is difficult to justify and does not suggest a
solution to the problem.
In order to get a hint about how much of the effect
might have been missed, one may make use of a theo-
retically calculated value for R . We rely on the dipole
8
formalism, Eqs. (7)-(9), from which one is able to cal-
culate the p dependence of the Drell-Yan reaction on
T
proton and nuclear targets in a parameter-free way and
explain the data in Fig. 3 well with no adjustment of
parameters. The dipole approach also provided the first
quantitativeexplanationofthe Cronineffectin available
FIG. 6: Broadening of the mean transverse momentum
dataforhadron-nucleuscollisionsandcorrectlypredicted
squared in theDrell-Yanreaction measured [34,35] on tung-
the effectforRHIC [21]. Thereforewebelievethatacal- sten andiron targets relativeto beryllium,∆ p2 = p2
culatedvalueofR8 shouldberatherreliable. Doingthis, hp2TiBe. h Ti h TiA−
Eq. (12) leads to somewhat larger values for the broad-
ening.
Unfortunately,withintheratherlargestatisticalerrors
The fit of Eq. (4) to this data leads to new values of one cannot draw a definite conclusion regarding a possi-
parameter D, ble variation of broadening with x .
2
D˜(E772)=(0.038±0.008)(GeV/c)2 ;
CONCLUSIONS
D˜(E866)=(0.069±0.009)(GeV/c)2 . (17)
Analysisofdatafromthe E866experimentforthe DY
This procedure involving theoretical calculations may reaction on nuclear targets has resulted in a determina-
be considered as a means to estimate the systematic un- tionofthe magnitudeofthe nuclearbroadeningparame-
certainty. The sign (positive) we obtain for the uncer- terD(E866)givenin(16)and(17),whichisabouttwice
taintyiscertainlycorrect,anditsmagnitudeisnotlarge. as large as the one suggested by the E772 data. This
Note also that these corrections are correlated for the result helps to resolve the controversies that were a par-
twosetsofdata,andthereforeonemustapplythemcon- tial motivation for this analysis. One of these was the
sistently to both the the results of the E772 and E866 expectation that the observed broadening in DY would
experiments. be a factor of 4/3less thanthat observedin J/Ψ andΥ,
8
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