Table Of ContentCharmonium production in relativistic proton-nucleus collisions : What will we learn
from the negative x region ?
F
D. Koudelaa, C. Volpea,b
a) Institut fu¨r Theoretische Physik der Universit¨at, Philosophenweg 19, D-69120 Heidelberg, Germany
b) Groupe de Physique Th´eorique, Institut de Physique Nucl´eaire, F-91406 Orsay Cedex, France
We study the nuclear medium effects on the cc¯ time evolution and charmonium production, in
a relativistic proton-nucleus collision. In particular, we focus on the fragmentation region of the
nucleuswheretheformationlengthofthecharmoniummesonsisshorterthanthesizeofthenucleus.
Little is known on the nuclear effects in this region. We use a quantum-mechanical model which
includes arealistic potential for the cc¯system and an imaginary potential to describethecollisions
of the cc¯with the nucleons. The imaginary potential introduces a transition amplitude among the
charmonium states and produces an interference pattern on the charmonium survival probability,
3 which is particulartly important for ψ′. Our results on the suppression factors are compared with
0 data from the NA50 and E866/NuSea Collaborations. Predictions are given for the suppression of
0 J/ψ, ψ′, χc as a function of the nuclear mass and in the negative xF region, where data will be
2 available soon.
n
a
J A very exciting era in the study of strong interactions models exist at present for the production mechanisms
2 has begun with the advent of high energy heavy ion col- [4,10,11].
2
lisionexperiments. Inparticular,thishasopenedtheway Sufficiently fast cc¯can traverse the nucleus before a ψ
to the exploration of new states of matter, such as the meson is fully formed; whereas slow cc¯can traverse the
1
v colour glass condensate, corresponding to the nonlinear nucleus as fully formed ψ meson [12,13]. Experimentally
6 regimeofquantumchromodynamics,orthehightemper- onecanpinpointtothesedifferentkinematicalregionsby
8 ature phase of nuclear matter, the so-called quark-gluon measuringcharmoniumproductionatpositiveandnega-
1 plasma,whichisalsoimportantforourunderstandingof tivesmallFeynmanx and/ormakinginversekinematics
F
1
the early universe. In relativistic heavy ion collisions, measurements [4,13].
0
evidence for the formation of the quark-gluon plasma Oneoftheaspectswhichneedtobewellunderstoodis
3
0 may come from the combined information of different theeffectofthenuclearmediumonthecc¯pairevolution.
/ signals,suchasstrangeness,dilepton,photonorcharmo- Inparticular,verylittle isknownonthe nuclearmedium
h
p nium production. effectswhentheformationtimeofafullydevelopedchar-
- MatsuiandSatz[1]firstsuggestedthatthesuppression monium is smaller than the time to traverse the nucleus
p
of charmonia could signal that a phase transition has [13]. Soon,measurementsinpAandAB collisionswillbe
e
h occurred. This pioneering idea has triggered a series of performed by the HERA-B Collaboration [14] at DESY
: experimentsattheCERNSPS[2,3],whoseinterpretation andthe PHENIX CollaborationatRHIC [15]. Moreover
v
has been very controversial. It has become clear that, theE160CollaborationatSLACwillperformsimilarex-
i
X to use charmonium as a signal, all possible mechanisms periments where the charmonia are photoproduced [16].
r whichcanaffectcharmoniumproductionneedto be well In this letter, we focus on this region, i.e. the frag-
a
understood [4,5]. mentation region of the nucleus (negative x region or
F
Inthiscontext,proton-proton(pp)andproton-nucleus slow cc¯) and study the effects of the nuclear medium on
(pA) collisions have been intensively studied [2–8]. In the cc¯ time evolution and on charmonium suppression
fact, these processes are used as a reference for the case in relativistic pA collisions [17]. Because of the kine-
of nucleus-nucleus (AB) collisions where a critical en- matical region we are interested in, we assume that the
ergy density may be eventually attained, producing the produced cc¯becomes quickly a colour singlet state (pre-
plasmaphase. Thisshouldaffectvariousobservablesand meson) and on its way through the nucleus, this pre-
in particular, it may lead to an extra (anomalous) char- meson expands to a ψ meson, while experiencing colli-
monium suppression. There have been indications re- sions with the nucleons. To describe this, we use here
cently that the plasma has been produced, but it is still a quantum-mechanical model where, contrary to previ-
too early to draw definite conclusions [9]. These stud- ous works [4,12,13,18], the cc¯ pair is bound by a real-
ieswillbe pursuedbythe runningexperimentsatRHIC, istic potential [19] and the premeson (and then meson)
and future measurements at LHC. wavefunctionisexpandedonthebasisgivenbythechar-
It takes a certain time for a cc¯produced in a collision monium eigenstates. An imaginary potential depending
to become a colour singlet state, then it expands to the on the dipole charmonium-nucleon cross section [12,20]
size of a ψ meson (the symbol ψ stands from now on for is included to describe the collisions with the nucleons.
any charmoniummeson,i.e. J/ψ, ψ and χ ). Different This introduces a transition amplitude among the char-
′ c
1
monium states. (We assume the premeson wavefunction transverse distance between c and c¯, √sψN is the en-
has a fixed angular momentum quantum number, while ergy in the center of mass of the ψN system and ρ
it has different radial components.) is the nuclear density evaluated at the position (~b,z)
We willshowthatthe interactionofthe premesonand of the cc¯ center of mass, z being the beam direction.
lateronthemesonwiththenuclearmediumproducesan For the dipole cross section we use the parametrization
interference pattern on the charmonium survival prob- σ(rT,√s) = σ0(s)(1 e−rT2/r02(s)), determined by fitting
−
ability. We will show how this affects the charmonium deep inelastic scattering data and which well reproduces
suppression both when the path in the nucleus is var- the charmonium photoproduction data [20]. Concerning
ied, by changing the mass number A, and as a function the nuclear density ρ, we presentresults obtainedwith a
of x . These effects are particularly important for ψ . Woods-Saxon profile, with parameters chosen such as to
F ′
Wecompareourresultswithexperimentaldataobtained well reproduce the nuclear radii.
at CERN SPS by the Na50 [7] and at Fermilab by the The time evolution of cc¯ wavefunction (2) in the
E866/NuSea [8] Collaborations. We present predictions nucleus is determined by solving the time dependent
for the production of J/ψ, ψ , χ in a range of small Schr¨odinger equation for H =H iW Eqs.(1) and (3).
′ c 0
−
negative x values, for different nuclei [17], for which This leads to a system of first-ordercoupled-channeldif-
F
experimental data will be avalaible soon [14,15]. ferential equations for the amplitudes c (t) :
n,l
Our cc¯ pair, produced in a pA collision, is described
(inits centerofmassframe)bythe internalHamiltonian n¯
c˙n,ℓ(t)= a ck,ℓ(t)ei(En,ℓ−Ek,ℓ)γt n,ℓσ k,ℓ (4)
− h | | i
p2 Xk=1
H = +V(r), (1)
0
m
c with a = vρ(~b,z)/2. The time t = γτ is now in the
where m is the charm quark mass. The potential V(r) laboratory frame. We have neglected in this first calcu-
c
which binds c and c¯is a realistic potential which repro- lationthehigherFockstatesthatshouldemergefromthe
duces well the properties of the charmoniumfamily [19]. Lorentz boost [20]. We see from (4) that the imaginary
The spin-dependent terms are neglected. Only transi- potential (3) introduces a transition amplitude among
tions between charmonium states with different n quan- the charmonium eigenstates.
tum numbers are considered. Adifficultchoiceisthatoftheinitialconditionscn,ℓ(0)
By solving the static Schr¨odinger equation with H for (4), related to the mechanism of hadroproduction of
0
Eq.(1), one gets the eigenenergies E and the corre- charmonia, which is still badly known [4,5,10,11]. We
n,ℓ
spondingwavefunctions n,ℓ (tosimplifythenotations, use:
| i
we do not write explicitly the dependence on magnetic
quantum number m). The 1S, 2S states correspond in φcc¯,ℓ(0)=cℓfℓ(rT)e−12β2(rT2+z2) (5)
our model to J/ψ, ψ respectively and 1P to χ where
′ c where c is a normalization constant. We describe the
the different total spin state of χ are not split in en- ℓ
c conversionofa gluoninto a cc¯throughagaussianmulti-
ergy. (We consider that our χ is produced through a
c plied by the function f (r )=r2 (or ~r ) to account for
two-gluon mechanism). ℓ T T T
thetwo(one)supplementarygluonsnecessarytoproduce
The time dependent wave function for the cc¯ in its
J/ψ,ψ (χ ). The initial wave function depends on one
rest frame is expanded on the basis of eigenstates of H ′ c
0 parameter only, β. This parameter has been determined
Eq.(1):
by fixing the ratio c /c 2 to the experimental ratio
2,0 1,0
| |
of ψ over J/ψ, produced in pp collisions at 450 GeV,
′
cc¯,ℓ (τ)= ∞ cn,ℓ(τ)e−iEn,ℓτ n,ℓ . (2) i.e. Bψ′ µµσpp→ψ′/BJ/ψ µµσpp→J/ψ = (1.60 0.04)%
| i nX=0 | i [21]. w→here Bψ µµ is th→e branching ratio to±dimuon
production and→σpp ψ the pp reaction cross section.
→
(h¯,c=1). In practice, one truncates the sum to a num-
This leads to the ratio c /c 2(0) = 0.22 0.03 and
2,0 1,0
ber n¯ of eigenstates, for a given ℓ value. As we will dis- | | ±
β = 1.33 0.5 GeV (the same β has been used for the
cuss, we have chosen n¯ large enough so that the results ±
χ states).
c
we present are not sensitive to this truncation.
The number n¯ of states in (2), to be included in the
To model the interaction of the cc¯ with the nuclear
coupled channel equations (4), has to be large enough
medium we add an imaginary part to Eq.(1) :
that the results do not depend on the truncation. We
γv have checked how the results depend on the inclusion of
iW =i 2 σ(~rT,√sψN)ρ(~b,z) (3) extra states. Up to 4 channels have been used for the S-
statesandupto2fortheP-states. Thetruncationalways
where v is the speed of the nucleus with respect to the
affectsalittlethelaststateincluded;whereastheresults
cc¯frame and σ is the dipole cross section associated to
for the lowest energy states are practically unchanged.
the interaction of the cc¯ with a nucleon N, r is the
T
2
Let us now come to the results, obtained in infinite of χ as a function of the nuclear mass, for different val-
c
nuclear matter first, i.e. by taking in (4) a constant nu- ues of γ. The largest suppression is observed when γ is
clear density ρ=ρ . In fig.1 we show the time evolution the lowest, corresponding to a longer time spent in the
0
of the probabilities associated to J/ψ and ψ produc- nucleus.
′
tion obtained by solving (4) with the initial conditions Let us now look at the x dependence. The sup-
F
(5), both neglecting and including the transition ampli- pression factor is often parametrized as SAψ(xF,√spp) =
tudesbetweendifferenteigenstates(non-diagonalterms). Aαψ(xF,√spp,A)−1 [4]. Infig.4theαvaluesfor “J/ψ”and
While inthe formercasethe probabilitiesdecreaseexpo- ψ are compared to experimental data on pA collisions,
′
nentially, in the latter they present an oscillation. The obtained by the E866/NuSea Collaboration[8]. The im-
interferencepatternismorepronouncedinthecaseofψ′ pinging proton energy is 800 GeV and the label A is the
whereas for J/ψ the oscillations stays very close to the ratio of the atomic numbers of tungsten over beryllium.
exponential. This effect directly influences the suppres- In this region of small positive and negative x values,
F
sionfactorsinpAcollisionsaswewillsee. Thedominant our calculations are in good agreement with the experi-
oscillation frequency in Eq.(4) is ω = (E2,ℓ − E1,ℓ)/γ. mentalvaluesfor“J/ψ”,whereasforψ′ theyoverpredict
Wehavealsoseenthatthe deviationfromtheuncoupled the data. In fact, in the region of small positive x , the
F
solution becomes stronger the higher γ is. cc¯ can traverse the nucleus before developing into a ψ
Let us now consider a cc¯which is produced in a rela- meson. In this kinematical regime, other effects missing
tivisticpAcollisionandevolvesaccordingtoEqs.(4). In- in our model, like for example gluon shadowing [22], can
tegratingoverallpossiblepathstheratio cψ( )/cψ(0)2, play a role.
| ∞ |
which gives the survival probability of the ψ in the nu- Finally,we presentpredictionsfortheα values(Fig.5)
cleus, one gets the pA ψ reaction cross section : andfor the suppressionfactors (Fig.6)for “J/ψ”,ψ and
→ ′
χ in the negative x region, where experimental data
c (t(~b,z)) 2 c F
σpA ψ = d~bdzρ(~b,z)σpN ψ ψ (6) from the HERA-B Collaboration [14] at DESY and the
→ →
Z (cid:12) cψ(0) (cid:12) PHENIX Collaboration at RHIC [15] will soon be avail-
(cid:12) (cid:12)
(cid:12) (cid:12) able. As we cansee,both“J/ψ” andχ presentarather
c
where t(~b,z) is the time necessary to a cc¯produced at a flat behavior, whereas ψ is very sensitive to the x val-
′ F
point(~b,z)totraversethenucleus. IfσpA→ψ =AσpN→ψ, ues. Besides, the results on ψ′ also show a strongervari-
the suppression factor, defined as ation with the mass of the nucleus. This can be under-
stoodfromthefactthattheeffectofthecouplingbetween
σpA ψ
Sψ → , (7) thecharmoniumeigenstatesismuchstrongerforψ′ than
A ≡ AσpN→ψ for J/ψ (and χc), as one can see from Fig.1.
Fig.5alsoshowsthattheαparametrizationofthesup-
becomes Sψ =1.
A pression (often used with α independent of A) is espe-
Before making predictions, let us compare our calcu-
cially not good in the negative x region, since there is
lations to existing data. For J/ψ, we have to take into F
a strong dependence of the α values on the nuclei.
accountthatthedimuonsmeasuredinthedetectorcome
In summary, we have studied the production of char-
from the decay of the directly produced J/ψ’s as well as
monium (J/ψ, ψ , χ ) in relativistic proton-nucleuscol-
those produced by the decays of ψ ’s and χ ’s. We will ′ c
′ c lisions. We focussedonthe nucleareffectsonthe cc¯time
denote by “J/ψ” the total number of produced J/ψ, i.e.
evolution and on the charmonium production when the
S“J/ψ” =0.62SJ/ψ+0.30Sχc +0.08Sψ′ [6] . timeforacc¯todeveloptoaψmesonisshortcomparedto
pA pA pA pA
Let us first discuss charmonium production as a func- thetimenecessarytotraversethenucleus. Thiskinemat-
tion of the nuclear mass. The cc¯pair travels along dif- icalregionmaybe exploredexperimentallybylookingat
ferent lenghts, spending different time intervals in the slow enough cc¯ and therefore charmonium produced in
nuclei. Therefore, experiments with different A explore the negative x region. The quantum-mechanicalmodel
F
the time dependence of the cn,ℓ amplitudes (Fig.1). We used here includes a realistic potential binding the cc¯
present results on “J/ψ” and ψ′ (Fig.2) in comparison pair and an imaginary potential, describing the collision
withthe experimentalvaluesby theNA50Collaboration of the cc¯with the nucleons and depending on the dipole
[7], obtained for pA collisions with an impinging pro- charmonium-nucleon cross section. The imaginary po-
ton energy of 450 GeV. The data are given as branching tential introduces transition amplitudes among charmo-
ratio times cross section divided by the nuclear mass, nium eigenstates. We have shown that this produces an
which differs from the suppression by a constant. (For interference pattern on the charmonium survival proba-
the comparison, we normalize the data by this constant bility,particularlyimportantforψ ,andaffectsthechar-
′
determined from the average of the ratio experimen- monium suppression factors. The results are compared
tal/calculated values.) We can see that our results are to experimental data from the NA50 and E866/NuSea
in good agreement with the experiment both for “J/ψ” Collaborations. We have made predictions on the sup-
and ψ . In Fig.3 we give predictions for the suppression
′
3
pression factors of J/ψ, ψ′ and χc, for different nuclei [17] D. Koudela, Diplomarbeit, University of Heidelberg,
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F
and B. Kopeliovich, Phys. Rev. Lett. 76, 192 (1996); J.
will bring new information on the charmonium-nucleon
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c (1981).
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We are very grateful to J¨org Hu¨fner for having inter-
ested us in this problem. We thank him for his useful
suggestionsandcommentsallalongtherealizationofthis 0
wWvYeoverrayknt,ohuavasnenffdkourlfoBhdroiishrsicikssuinsvKsdeiorohpyneescllpaiao.rnveOidfcunhlceoraemonafmdduiensSngt(teCsop,f.hVtaah.s)neewamcePkallneniouagwssncl´eeYridupfgorteri. 2c(t)/c(0)|)1,01,0 −−21 J/ψ
all the people at the Institut fu¨r Theoretische Physik of n(|
l −3
theUniversityofHeidelberg,forthe verywarmhospital-
ψ’
ity during her stay. 20)|) −2
c(2,0
c(t)/2,0 −5
n(|
l −8
0 10 20 30 40 50
t/fm
FIG. 1. Probability of surviving in the nuclear medium
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5.5
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[7] P. Cortese et al., NA50 Collaboration, Proceedings to nb)
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ψ/
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2 3 4 5 6
CERN 2001-2002 Workshop on “Hard Probes in Heavy
ln(A)
Ion Collisions at the LHC”, hep-ph/0209365. FIG. 2. Results for the ψ production cross section times
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3466 (1991). “J/ψ” (top) and ψ′ (bottom) produced in pA collisions with
[13] D.KharzeevandH.Satz,Phys.Lett.B356,365(1995). Ep = 450 GeV as a function of the nuclear mass. The ex-
[14] http://www-hera-b.desy.de/general/info. perimental data obtained by the NA50 Collaboration [7] are
[15] Jen-Chieh Peng, privatecommunication. shown for comparison. (The curves are only given to guide
[16] Keith Griffioen, privatecommunication. theeye).
4
FIG. 5. Predictions for “J/ψ” (top), ψ′ (middle) and χc
χ (bottom),on thedependenceof theαvaluesas a function of
−0.1 c xF,obtainedwithEp =800GeV.Wepresentresultsobtained
for tungsten (triangles) and beryllium (squares).
−0.2
)
χc
A
S 1
n((cid:5) −0.3
l
−0.4 0.9
−0.52 3 4 5 6 SBe0.8
/W
ln(A) S
FIG.3. Predictionsonthesuppressionfactoroftheχc me- 0.7
son,producedinrelativisticpAcollisions,asafunctionofthe
mass ofthenucleusA. Wepresent resultsfor γ =5(circles),
16 (squares), 21 (triangles). 0.6
−0.4 −0.3 −0.2 −0.1 0
x
F
1
’’J/ψ’’ FIG. 6. Predictions for the suppression factors for “J/ψ”
′
0.95 (circles), ψ (squares) and χc (triangles) as a function of
α the negative xF region, obtained in a pA collision with
0.9 Ep =800 GeV and A being the ratio of tungsten over beryl-
lium.
1
ψ’
0.95
α
0.9
0.85
−0.1 −0.05 0 0.05 0.1 0.15 0.2
x
F
FIG.4. Comparison between calculated and experimental
′
α values for “J/ψ” (top) and ψ (bottom) as a function of
Feynman xF obtained for pA collisions with Ep = 800 GeV.
Here the α values are relative to the ratio W/Be. The data
are from theE866/NuSea Collaboration [8].
1
’’J/ψ’’
0.95
0.9
1
ψ’
0.95
α
0.9 Be
W
1
χ
0.95 c
0.9
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3
x
F
5
