Table Of ContentCharmonium absorption by nucleons
A. Sibirtsev1, K. Tsushima2, A. W. Thomas2
1
Institut fu¨r Theoretische Physik, Universit¨at Giessen,
D-35392 Giessen, Germany
2
Special Research Center for the Subatomic Structure of Matter (CSSM)
and Department of Physics and Mathematical Physics,
University of Adelaide, SA 5005, Australia
in the available calculations [2,3,5–7,9,10] it enters as a
J/Ψ dissociation in collisions with nucleons is studied
constant. Furthermore, to a certain extent, both of the
withinabosonexchangemodelandtheenergydependenceof
cross sections for J/Ψ dissociation on nucleons and co-
thedissociation crosssectioniscalculated fromthethreshold
moversweretakenasfreeparameters,finallyadjustedto
1 forΛcD¯ productiontohighenergies. Weillustratetheagree- the heavy ion data.
0 ment of ourresults with calculations based on short distance
Moreover, the J/Ψ+N cross section was evaluated
0 QCDandReggetheory. Thecompatibilitybetweenourcalcu-
from experimental data on J/Ψ meson production in
2 lationsandthedataonJ/Ψphotoproductiononanucleonis
γ+A and p+A reactions. The analysis of the J/Ψ
discussed. WeevaluatetheelasticJ/Ψ+N crosssectionusing
n
photoproduction from nuclei at a mean photon energy
a a dispersion relation and demonstrate the overall agreement
of 17 GeV [11] indicates a J/Ψ+N cross section of
J with the predictions from QCD sum rules. Our results are
7 compatible with thephenomenological dissociation cross sec- 3.5±0.9 mb. The combined analysis [12] of experimen-
1 tionevaluatedfromtheexperimentaldataonJ/Ψproduction taldataonJ/Ψproductionfromp+Acollisionsatbeam
from γ+A, p+A and A+A collisions. energies from 200 to 800 GeV provides a J/Ψ+N cross
2 section of 7.3 0.6 mb. On the other hand, the J/Ψ+N
PACS: 14.40.Lb, 12.38.Mh, 12.40.Nn, 12.40.Vv, 25.20.Lj ±
v cross section evaluated by the vector dominance model
1 fromtheγ+N J/Ψ+N dataisabout1mb[13]forJ/Ψ
4 →
energies in the nucleon rest frame from 8 to 250 GeV.
0
In principle, this ambiguity in the J/Ψ dissociation
5
cross section does not allow one to claim a consistent
0
0 I. INTRODUCTION interpretation of the NA50 results. Although the exper-
0 imental data might be well reproduced by the more re-
/ cent calculations [2–4] considering the hadronic dissoci-
h Very recently the NA50 Collaboration reported [1]
ation of the J/Ψ meson, the J/Ψ+N cross section was
t measurements of J/Ψ suppression in Pb+Pb collisions
- introduced therein as a free parameter. In Ref. [4] we
l at the CERN-SPS. It was claimed in Ref. [1] that these
c provided results for J/Ψ dissociation on comovers and
u new experimental data ruled out conventional hadronic
motivated the large rate of this process [2,3] as arising
n modelsofJ/Ψsuppressionandthusindicatedtheforma-
fromthe in-medium modificationof the dissociationam-
: tion of a deconfined state of quarks and gluons, namely
v plitude, which is obviously different from that given in
the quark gluon plasma (QGP). Presently,there are cal-
i
X culations [2–4] using two different methods, namely the free space [14,15]. Most recent results from the different
modelsonJ/Ψdissosiationonlighthadronsaregivenin
r cascade [5,6] and the Glauber type model [7], that re-
a Refs. [16–19]In the presentstudy we apply the hadronic
produce the new NA50 data reasonably well up to the
model to J/Ψ+N dissociation.
highest transverse energies [1] based on hadronic J/Ψ
dissociationalone. Inview ofthis, the evidence forQGP
formationbasedonanomalousJ/Ψsuppression[8]isnot
II. LAGRANGIAN DENSITIES, COUPLING
obvious and an interpretation of the NA50 data can be
CONSTANTS AND FORM FACTORS
further considered in terms of the hadronic dissociation
of the J/Ψ meson by the nucleons and comovers.
Within the boson exchange model we consider the re-
However, the long term puzzle of the problem is the
actions depicted in Fig. 1 and use the interaction La-
strength of the hadronic dissociation of the J/Ψ meson.
grangiandensities:
Thevariouscalculations[2,3,5–7,9,10]ofJ/Ψproduction
fromheavyioncollisionsinpracticeadoptdifferentdisso-
=ig Jµ D¯(∂ D) (∂ D¯)D , (1)
ciation cross sections. For instance, the cross section for LJDD JDD µ − µ
J/Ψ+N dissociation used in these models ranges from LDNΛc =igDNΛc N¯(cid:2)γ5ΛcD+D¯Λ¯cγ5N (cid:3), (2)
3im.0enutpaltdoat6a.7omnbJ/wΨhepnroddeuacltiniognwfritohmtAhe+sAamcoelleisxiponers-. LJD∗D = gJmD∗D ε(cid:0)αβµν(∂αJβ) (cid:1)
J
Moreover, it is presently accepted that the dissociation [(∂µD¯∗ν)D+D¯(∂µD∗ν)], (3)
cross section does not depend on the J/Ψ energy and ×
LD∗NΛc =−gD∗NΛc N¯γµΛcD∗µ+D¯∗µΛ¯cγµN , (4)
(cid:0) (cid:1)
1
where, N = (np ), N¯ = N†γ0, D ≡ (DD+0 ) (creation of dTihcteatfoerdmbyfatchtoerexmtaeyndbeed essttriumctautreedotfhrtohueghhadthroenvse[c3t0o]r.
the mesonstates), D¯ =D†,andsimilarnotationsforthe dominance model (VDM) [31]. The VDM predicts an
D∗ and D¯∗ should be understood. increase of the form factor with increasing mass of the
vector meson and a purely phenomenological prediction
provides Λ=m . In the following we will explicitly use
J
the cutoff parameter Λ=3.1 GeV for the form factors at
the JDD and JDD∗ vertices.
Furthermore, one should also introduce form factors
at the DNΛ and D∗NΛ vertices. Again, we use the
c c
monopole form, but there are no direct ways to evaluate
therelevantcutoffparameter. InRefs.[30,32,33]thecut-
offparameterswereadjustedtofittheempiricalnucleon-
nucleondata and rangefrom 1.3 to 2 GeV depending on
the exchange meson coupled to the NN system. Based
ontheseresults,inthefollowingweuseamonopoleform
factorwithΛ=2GeVfortheDNΛ andD∗NΛ vertices.
c c
In principle, the coupling constants and the cutoff pa-
rameters in the form factors discussed in this section
can be adjusted to the experimental data on the total
J/Ψ+N crosssectionevaluatedfromdifferentnuclearre-
actions, as γ+A J/Ψ+X and p+A J/Ψ+X. On the
→ →
other hand, these parameters can be also fixed by com-
parisontothe shortrangeQCDorReggetheorycalcula-
tions at high energies. As will be shown later, the set of
couplingconstantsandcutoffparametersproposedabove
are rather well fitted to the relevant experimental data
and the results from theoretical calculations with other
models.
FIG.1. The hadronic diagrams for J/Ψ+N dissociation. III. THE J/Ψ+N→ΛC+D¯ AND J/Ψ+N→ΛC+D¯∗
REACTIONS
The JDD coupling constant was derived in Refs. The diagrams for J/Ψ dissociation by a nucleon with
[14,20] and in the following calculations we adopt the production of the Λ +D¯ and Λ +D¯∗ final states are
c c
g =7.64. FollowingtheSU(4)relationsfromRef.[21] shown in Fig. 1a)-c). They involve the J/Ψ D+D¯ and
JDD
the DNΛ coupling constant was taken equal the KNΛ J/Ψ D+D¯∗ vertices and are OZI allowed.→These reac-
c
→
coupling constant. The analysis [22–26] of experimen- tions are endothermic, since the total mass of the final
tal data provides 13.2 gKNΛ 15.7 and in the following states is larger than the total mass of the initial J/Ψ-
≤ ≤
we use gDNΛc=14.8. However, we notice, that the cal- meson and nucleon.
culations [27] by the QCD sum rules suggest a smaller The J/Ψ+N Λ +D¯ reaction via D meson exchange
c
sctoaunptlinmgigghDtNbΛec≃a6g.a7i±n2r.e1l.ateTdhetoDt∗hNe ΛKc∗NcoΛupclionngstcaonnt-. itsudsehowwitnhianntahme→pFliitgu.d1eaa)dadnedditnheorcdoerrretosppornedseinrgveagmaupglie-
The K∗NΛ constant is given in Ref. [28] within the invariance in the limit, m 0, may be given by:
J
→
range –18.8 gK∗NΛ –21.4, while it varies between –
≤ ≤ ≃
5.9 and –22 in Ref. [29]. In the following calculations we Ma =2igJDDgDNΛc(ǫJ ·pD¯)
adopt gD∗NΛc=–19 [28]. Furthermore, we assume that 1 1
gJDTDhe=fgoJrDmD∗fa[c2t0o]r.s associated with the interaction ver- ×(cid:18)q2−m2D + 2pJ ·pD¯(cid:19) u¯Λc(pΛc)γ5uN(pN), (6)
tfoicrems were parameterized in a conventional monopole wthheerDe∗qm=espoJn−epxDc¯h(a=ngpeΛcis−sphNow),nwihnilFeigth.e1ba)mapnlidtucdaenvbiae
written as
Λ2
F(t)= , (5)
Λ2−t Mb = gJD∗DmgD∗NΛc q2 1m2
where t is the four-momentum transfer and Λ is the cut- J − D∗
off parameter. The introduction of the form factors is ×εαβµνpαJǫβJqµu¯Λc(pΛc)γνuN(pN). (7)
2
Furthermore,theamplitudefortheJ/Ψ+N Λ +D¯∗ re- Fig. 2b). Both reactions substantially contribute at low
c
→
actionduetoDmesonexchangeisshowninFig.1c),and energies, close to the respective thresholds, while their
is given by contributiontotheJ/Ψ+N dissociationcrosssectionde-
creases with increasing √s.
= igJD∗DgDNΛc 1
Mc m q2 m2
J − D
×εαβµνpαJǫβJpµD¯∗ǫ∗D¯ν∗u¯Λc(pΛc)γ5uN(pN). (8)
In Eqs. (6)- (8) ǫJ and ǫD¯∗ are respectively the polar-
ization vector of the J/Ψ meson and D¯∗ meson, and
q=pΛc −pN denotes the four-momentum transfer. We
notice that there arises no interference term among the
amplitudes, , and , because of the Dirac
a b c
M M M
structure, when spin components are not specified and
summed over all spins for the N and Λ , and the differ-
c
ent final states.
In our normalization the corresponding differential
cross sections in the center of mass frame of the J/Ψ
meson and nucleon system are given by:
dσ 1
= 2
dΩa,b 64π2s|Ma,b|
[(mΛc +mD)2−s][(mΛc −mD)2−s] 1/2, (9)
×(cid:18) [(m +m )2 s][(m m )2 s] (cid:19)
N J N J
− − −
dσ 1
= 2
dΩc 64π2s|Mc|
[(mΛc +mD∗)2−s][(mΛc −mD∗)2−s] 1/2, (10)
×(cid:18) [(m +m )2 s][(m m )2 s] (cid:19)
N J N J
− − −
where s=(p +p )2 is the squared invariant collision en-
N J
ergy and 2 are the corresponding amplitudes
a,b,c
|M |
squared, averaged over the initial and summed over the
final spins. Explicitly, they are given by:
FIG.2. The cross section for J/Ψ dissociation by the nu-
2 = 8gJ2DDgD2NΛc 1 + 1 2 cleonwithΛc+D¯ (a)andΛc+D¯∗(b)production,asafunction
|Ma| 3m2J (cid:18)q2−m2D 2pJ ·pD¯(cid:19) tohfeincvaalcriualnatticoonlslisiniovnolevninegrgDy,∗√mse.sTonheexdcahsahnegde,liwnehiilneath)eshsoolwids
×(pN ·pΛc −mNmΛc)[(pJ ·pD¯)2−m2Jm2D], (11) lineina)indicatesthecontributionfromtheDexchange. The
2 = gJ2D∗DgD2∗NΛc 1 resultsareshownforcalculationswithformfactorsintroduced
|Mb| 3m2 (q2 m2 )2 at theinteraction vertices.
J − D∗
[ m2(p2q2 (m2 m2 )2)
× J − Λc − N
+2(p p)(p q)(m2 m2 ) IV. THE J/Ψ+N N+D+D¯ REACTION
J · J · Λc − N →
p2(p q)2 q2(p p)2
J J
− · − · The diagram for J/Ψ dissociation by a nucleon with
−4(pN ·pΛc −mNmΛc)(m2Jq2−(pJ ·q)2) ], (12) the production of the D+D¯ state is shown in Fig. 1d).
|Mc|2 = 4gJ2D3∗Dmg2D2NΛc (q2 1m2 )2 Tthheerefdoirsesoictiaistioann iOnZvoIlvaellsowthede pJr/oΨce→ssD. +TD¯hevteorttaelxmaansds
×(pN ·pΛc −mNmJΛc)[(pJ ·pD¯−∗)2D−m2Jm2D∗], (13) othfethineiptiraoldJu/cΨed-mpaesrotinclaesndisnlaurcgleeornthaanndtthheutsottahlesmearsesaoc-f
tions are endothermic. Furthermore, the processes with
witFhinpa≡llyp,ΛFc i+g.p2Na.) shows the J/Ψ+N Λ +D¯ cross sec- D+N and D¯+N scattering are different and both con-
c
→ tribute to J/Ψ dissociation on the nucleon.
tionasafunctionoftheinvariantcollisionenergy√scal-
culated with D (solid line) and D∗ meson (dashed line) The amplitude for the reaction J/Ψ+N→N+D+D¯
can be parametrized as [22,34]
exchanges and with form factors at the interaction ver-
tices. The J/Ψ+N Λ +D¯∗ cross section is shown in
c
→
3
(ǫJ pD) Some of the partial D+N cross sections were cal-
=2g · F(t) , (14)
Md JDD t m2 MDN culated Ref. [21] by an effective Lagrangian approach,
− D where the identity between the K¯+N and D+N inter-
wherep andm aretheD-mesonfour-momentumand
D D action was imposed by the SU(4) relations. Indeed, on
mass, respectively, ǫ denotes the polarization vector of
J the basis of SU(4) symmetry the couplings for the ver-
J/Ψ-meson,tissquaredfour-momentumtransferedfrom tices containing K¯ or K mesons were taken to be equal
initial J/Ψ to final D-meson, while gJDD and F(t) are to those obtained by replacing K¯ or K with D or D¯, re-
thecouplingconstantandtheformfactorattheJ/ΨDD¯
vertex, respectively. Furthermore, is the ampli- spectively. For instance, gπDD∗=gπKK∗, gDNΛc=gKNΛ,
DN etc. In the more general case, the large variety of pos-
tude for D+N or D¯+N scattering,Mwhich is related to sible diagrams describing the K¯+N and K+N interac-
the physical cross section as tions can be identified to those for the D+N and D¯+N
|MDN|2 =16πs1σDN(s1), (15) iRnetfe.ra[3ct5i]onarse, rseuspppeoctritveedlyb.yTthhues,bothsoenesetxicmhaatnegsegfiovremnailn-
where s1 is the squared invariant mass of the D+N or ism. The comparison between the DN scattering cross
D¯+N subsystem. section calculated by an effective Lagrangian and evalu-
The double differential cross section for the reaction ated by the quark diagrammatic approaches is given in
J/Ψ+N N+D+D¯ can be written as Ref. [36]. Furthermore, for the J/Ψ+N N+D+D¯ cal-
→ culations we used coupling constants a→nd form factors
d2σ g2 F2(t) similar to those used for the J/Ψ+N Λ +D¯ reaction.
dtdJsN1 = 96πJ2DqDJ2s qD√s1 (t−mD)2 → c
[(m +m )2 t][(m m )2 t]
J D J D
× −m2 − − σDN(s1), (16)
J
where q is given by
D
q2 = [(mN +mD)2−s1][(mN −mD)2−s1], (17)
D 4s1
and m and m denote the J/Ψ-meson and nucleon
J N
masses, respectively.
As was proposed in Ref. [34], by replacing the elastic
scattering cross section, σ , in Eq. (16) with the to-
DN
tal cross section, it is possible to accountsimultaneously
for all available final states that can be produced at the
relevant vertex. Since we are interested in inclusive J/Ψ
dissociation on the nucleon, in the following we use the
total D+N and D¯+N interaction cross sections. How-
ever, we denote this inclusive dissociation generically as
the J/Ψ+N N+D+D¯ reaction.
The total D→+N and D¯+N cross sections were evalu-
ated in our previous study [35] by considering the quark
diagrammaticapproach. It was proposedthat the D+N
and D¯+N cross sections should be equal to the K¯+N
and K+N cross sections, respectively, at the same in-
variant collision energy and neglecting the contribution FIG.3. The cross section for J/Ψ dissociation by the nu-
from the baryonic resonances coupled to the K¯+N sys- cleonwithD¯+Dproductionasafunctionofinvariantcollision
tem. The total D¯+N cross section was taken as a con- energy √s. The results are shown for calculations with and
stant, σD¯N=20 mb, while the total D+N cross section without form factor at theJ/ΨD¯D vertex.
can be parameterized as
σD¯N(s1)=(cid:18)[[((mmΛDc++mmNπ))22−ss11]][[((mmΛDc −mmNπ))−ss11]](cid:19)1/2 onFtihgeurneuc3lesohnowwsitthhtehceropsrsodsueccttiioonnfoofrD¯J+/ΨDdpiasisroscciaatlicoun-
− − − lated using Eq. (16), with and without form a factor at
27
+20, (18) the J/ΨD¯D vertex. The lines in Fig. 3 show the results
× s1 where the processes with D¯+N and D+N interactions
were summed incoherently.
where m and m are the Λ -hyperon and pion masses,
L π c
respectively, given in GeV and the cross section is given
in mb.
4
V. TOTAL J/Ψ+N CROSS SECTION AND λ= s−m2J −m2N. (20)
COMPARISON WITH QCD CALCULATIONS 2m
J
Parameterization (20) provides a substantially smaller
Our results for the total J/Ψ+N cross section are
J/Ψ dissociation cross section compared to our result.
shown in Fig. 4 as the solid line a). Here the total cross
section is given as a sum of the partial J/Ψ+N Λ +D¯
c
crosssectioncalculatedwithD andD∗-mesonex→change,
the J/Ψ+N Λ +D¯∗ cross section calculated with D-
c
→
meson exchange and the inclusive cross section for J/Ψ
dissociation with D¯+D production.
FIG. 5. The total cross section for J/Ψ dissociation by
the nucleon as a function of invariant collision energy √s.
The solid line a) shows our calculations with form fac-
tors, while the dashed line indicates the contribution from
theJ/Ψ+N Λc+D¯ andJ/Ψ+N Λc+D¯∗ reactionchannels
→ →
only. The solid line b) indicates the first order calculation
FIG.4. ThetotalcrosssectionforJ/Ψdissociation bythe based on short distance QCD.
nucleon, as a function of invariant collision energy, √s. The
line a) shows our calculation with a form factors. The line
b) indicates the first order calculations based on a short dis- Furthermore, the J/Ψ+N cross section can be eval-
tanceQCD,whilethelinec)showstheparameterizationfrom uated [38,39] using short distance QCD methods based
Ref. [37]. on the operator product expansion [40]. Within the first
order calculation the cross section is given as [41]
We found that the total J/Ψ+N cross section ap- 1
213π (ξx 1)3/2 g(x)
proaches 5.5 mb at high invariant collision energy. We
σ = − dx, (21)
also indicate a partial enhancement of the J/Ψ+N dis- JN 34αsm2c Z (ξx)5 x
sociation cross section at low energies due to the contri- 1/ξ
bution from the J/Ψ+N Λ +D¯ and J/Ψ+N Λ +D¯∗
reaction channels. → c → c where ξ=λ/ǫ0, mc is the c quark mass, αs is the strong
couplingconstantandg(x)denotesthegluondistribution
The line c) in Fig. 4 shows the parameterization from
function, for which we take the form
Ref. [37], explicitly given as
g(x)=2.5(1 x)4. (22)
λ 6.5 −
σ =2.5 1 , (19)
JN
(cid:18) − λ0(cid:19) Calculations with a more realistic gluon distribution
function [42] only change the J/Ψ+N cross section
where the cross section is given in mb, λ0=mN+ǫ0 with slightly at invariant collision energies √s<20 GeV, as
ǫ0 being the Rydberg energy and
5
compared to that obtained with function (22). The dif- thecrosssectionforJ/Ψphotoproductiononthenucleon
ferencesassociatedwithvariousgluonstructurefunctions at t=0 as
canbepredominantlyobservedathigh√s. TheJ/Ψ+N
dσ α
cross section from the first order calculations performed γN→JN = (1+α2 ) σ2 F2(0)
dt (cid:12) 16γ2 JN JN
using short distance QCD is shown by the line b) in (cid:12)t=0 J
Fig. 4. (cid:12)(cid:12) (s−m2N −m2J)2−4m2Nm2J, (25)
We note that the QCD results are in good agreement × (s m2 )2
with our hadronic model calculations at high invariant − N
collisionenergies,butsubstantiallydeviatefromourpre- where αJN is the ratio of the real to imaginary part of
dictionsnearthethresholdforendothermicreaction. We the J/Ψ+N scattering amplitude at t=0.
ascribe this discrepancy to the contribution from the The γJ coupling constant can be directly determined
J/Ψ+N Λ +D¯ and J/Ψ+N Λ +D¯∗ reactions. The from the J/Ψ meson decay into leptons [46]
c c
→ →
dashed line in Fig. 5 shows the separate contribution to
athnedtJot/aΨl+J/NΨ+NΛc+crDo¯s∗ssreecatcitoinonfrso,mwhthileeJt/hΨe+soNlid→lΛince+aD¯) Γ(J/Ψ→l+l−)= π3γαJ22 qm2J −4m2l (cid:20)1+ 2mm2J2l (cid:21). (26)
→
again shows our result for the total J/Ψ dissociation by
a nucleon. The solid line b) in Fig. 5 indicates the QCD
result given by Eq. (21). It is clear that the main dif-
ference between our predictionand the QCD calculation
comes from the J/Ψ+N Λ +D¯ and J/Ψ+N Λ +D¯∗
c c
→ →
reactionchannels,whichcontributessubstantiallyatlow
√s.
VI. J/Ψ PHOTOPRODUCTION ON THE
NUCLEON
Within the vector dominance model, the J/Ψ photo-
production invariant amplitude, , on a nucleon can
γJ
M
be related to the J/Ψ+N scattering amplitude, ,
JN
M
as:
√πα
(s,t)= F(t) (s,t), (23)
γJ JN
M γ M
J
wheres is the squared,invariantcollisionenergy,t is the
squared four-momentum transfer, α is the fine-structure
constant and γ is the constant for J/Ψ coupling to the
J
photon. In Eq. (23) F(t) stands for the form factor at
the γ J/Ψ vertex, which accounts for the cc¯ fluctua-
−
tion of the photon [43]. Within the naive VDM only the
FIG. 6. Differential cross section for the reaction
hadronic, but not the quark-antiquark, fluctuations of
γ+N J/Ψ+N at t=0 as a function of the invariant col-
the photonthroughtheγ mixingwiththe vectormesons →
lision energy, √s. Experimental data are from Ref. [54–56].
areconsidered[44,45]and the formfactor F(t) inEq.23
The solid line shows our calculations with the total J/Ψ+N
is therefore neglected. As was calculated in Ref. [43] by
dissociation cross section evaluated using a boson exchange
both the hadronic and quark representations of the cc¯
model, while the dashed line indicates the results obtained
fluctuationof the photon, the formfactorF=0.3att=0.
for the J/Ψ+N cross section given by short distance QCD.
The invariant amplitudes are normalized so that the
J/Ψ+N differential cross section is written as
Theratioα isunknownandasafirstapproximation
dσ 2 JN
= |M| , (24) weputitequaltozero. Thisapproximationissupported
dt 16π[(s−m2N −m2J)2−4m2Nm2J] from two sides.
At high energies Regge theory dictates that the am-
andsimilarlyforthe J/Ψphotoproductioncrosssection.
plitude for hadron scattering on a nucleon is dominated
TakingtherelationofEq.(23)att=0andapplyingthe
by pomeron exchange and is therefore purely imaginary.
optical theorem for the imaginary part of the J/Ψ+N
Thisexpectationisstronglysupportedbytheexperimen-
scatteringamplitude,onecanexpressthetotalcrosssec- tal data [47] available for p, p¯π, K and K¯ scattering by
tion for J/Ψ dissociation by a nucleon σ in terms of
JN a nucleon.
6
Atlowenergiesthe realpartofthe J/Ψ+N scattering most recent fit [59] to the J/Ψ photoproduction data
amplitude at t=0 can be estimated from the theoreti- from H1 [60] and ZEUS [61,62] experiments.
cal calculations of the J/Ψ-meson mass shift in nuclear The prediction from Regge theory is shown by
matter. As was predicted by the calculations by the op- the dotted line in Fig. 7 together with experimental
erator product expansion [48,49], QCD van der Waals data [54–56,63,64] on J/Ψ photoproduction cross sec-
potential [50] and QCD sum rules [51–53], the mass of tionatt=0asafunctionofthe invariantcollisionenergy
the J/Ψ should only be changed a tiny amount in nu- √s. The solid line in Fig. 7 shows our calculations using
clear matter. However, a partial deviation of our results the boson exchangemodel with form factors. We notice,
from the J/Ψ photoproduction data might be actually that in Refs. [57,59] the Regge model fit to the data on
addressedto the contributionfroma nonvanishingratio γ+N J/Ψ+N cross section at t=0 was not explicitly
→
α . More detailed discussion concerning an evaluation included in an evaluation of the coefficients of Eq. 27,
JN
of the real part of the J/Ψ+N scattering amplitude will which might partially explain some systematic deviation
be given in the following. oftheReggecalculationsfromthephotoproductiondata.
The J/Ψ photoproduction cross section at t=0 is
shown in Fig. 6 as a function of the invariant colli-
sion energy. The experimental data were taken from
Refs. [54–56]. The solid line shows the results calculated
with the total J/Ψ+N cross section given by the boson
exchange model. We found that within the experimen-
tal uncertainties we can reproduce the J/Ψ photopro-
duction data at t=0 at low energies quite well. Let us
recall that this is possible because of the contribution
from the J/Ψ+N Λ +D¯ and J/Ψ+N Λ +D¯∗ reac-
c c
→ →
tion channels.
The dashed line in Fig. 6 indicates the calculations
using Eq. (21), which reasonably describe the data on
photoproductioncrosssectionat√s>10GeV.Wenotice
areasonableagreementbetweenourcalculationsandthe
resultsfromshortdistanceQCDatinvariantcollisionen-
ergies √s>12 GeV. We also recall that in Ref. [39] the
discrepancy between the QCD results and the photopro-
duction data at low √s was attributed to a large ratio
α . Our result does not support this suggestion.
JN
VII. COMPARISON WITH REGGE THEORY
It is of some interest to compare our results with the
predictionsfromReggetheory[57]givenathighenergies.
Furthermore,forillustrativepurposewedemonstratethe FIG.7. Thecrosssectionforthereaction γ+N J/Ψ+N
→
comparison in terms of the cross section at t=0 for the att=0asafunctionofinvariantcollision energy,√s. Experi-
γ+N J/Ψ+N reaction, since the Regge model param- mentaldataarefromRef.[54–56,63,64]. Thesolidlineshows
eters→were originally fitted to the photoproduction data. ourcalculationswiththetotalJ/Ψ+N dissociationcrosssec-
Taking into account the contribution from soft and tion evaluated by boson exchange model. The dotted line
hardpomeronexchangesalone,theexclusiveJ/Ψphoto- shows the prediction (27) from Regge theory with parame-
ters for the soft and hard pomeron from Ref. [59], while the
productioncross sectionat t=0 canbe explicitly written
dashedlineistheresultgivenbyEq.27,butrenormalizedby
as [58]
thefactor 1.3
dσ
=23.15s0.16+0.034s0.88+1.49s0.52, (27)
dt(cid:12)
(cid:12)t=0 In order to illustrate the compatibility of the Regge
(cid:12)
(cid:12) model prediction with our calculations we indicate by
where the cross section is given in nb and the squared
the dashed line in Fig. 7 the result from Eq. (27), renor-
invariant collision energy, s, is in GeV2. The first term
malizedbya factor1.3. We note areasonableagreement
of Eq. (27) comes from the soft pomeron contribution,
between our calculations and the Regge theory within
the second one is due to the hard pomeron, while the
the short range of energies from √s 10 up to 30 GeV.
last term stems from their interference. The parameters ≃
However, considering the result given in this section
for the both pomeron trajectories were taken from the
oneshouldkeepinmindthefollowingcriticalarguments.
7
First, we performedthe calculations using a bosonex- First, we address the partial discrepancy between our
change model, which has some restrictions in its appli- calculationsofthe J/Ψ photoproductioncrosssectionat
cation at high energies. For instance, the form factors t=0 and data at low energies due to the nonzero ratio
introduced at the interaction vertices suppress the con- α appearing in Eq. (25) and evaluate it from the ex-
JN
tribution at large 4-momentum transfer t, which allows perimental measurements as
one to avoid the divergence of the total cross section at
dσexp dσth −1
large collision energies where the range of the available α2 = 1, (29)
t becomes extremely large [22]. A more accurate way to JN dt (cid:12) ×(cid:20) dt (cid:12) (cid:21) −
(cid:12)t=0 (cid:12)t=0
resolve such a divergence at high energies is to use the (cid:12) (cid:12)
wheretheindicesexp(cid:12)andthdenot(cid:12)etheexperimentaland
Reggeized boson exchange model [65–67].
theoretical results for the photoproduction cross section
Second, the Regge theory has limitations for applica-
at t=0, respectively.
tions atlow energies [47]. Here we adopt the parameters
Themodulusoftheratioα evaluatedfromthedata
JN
for soft and hard pomeron trajectories that were origi-
is shown in Fig. 8 as a function of the J/Ψ momentum,
nallyfixed[59]usingthelargesetofdataat√s 40GeV
≥ pJ, in the nucleon rest frame. It is clear that the sign of
and the very reasonable agreement between the Regge
the ratio cannot be fixed by Eq. (29), but some insight
modelcalculationswith the data,evenatlowerenergies,
mightbegainedfromtheReggetheorycalculation,which
might in some sense be viewed as surprising.
is shown in Fig. 8 by the dashed line. Here again we use
On the other hand, the D and D∗ meson exchanges
theparametersofthepomerontrajectoriesfromRef.[59].
used in our calculations cannot be related to pomeron
Now one can conclude that the f(0) is positive at low
exchange, rather they can be related to the Regge tra- ℜ
J/Ψmomenta,whichagreeswiththepredictions[48–53]
jectories,whose contributionto J/Ψ photoproductionat
onthe reductionof the J/Ψ mesonmass in nuclearmat-
high energies was assumed [57–59]to be negligible. This
ter. However,thisspeculationistrueonlyiftherealpart
doesnotcontradictthe resultsshowninFig.7,sinceour
oftheJ/Ψ-nucleonscatteringamplitudeatt=0doesnot
calculations are substantially below the data at high en-
change its sign at some moderate momentum.
ergies.
VIII. EVALUATION OF THE REAL PART OF
J/Ψ+N SCATTERING AMPLITUDE
One of the crucial ways to test the coherence of our
calculations consists of an evaluation of the real part of
the J/Ψ+N scattering amplitude at t=0 anda compari-
sontothe variousmodelpredictions[51–53]for f(0)at
ℜ
zero J/Ψ momentum. The latter is of course related to
theJ/Ψ-mesonmassshiftortherealpartofitspotential
in nuclear matter.
Within the low density theorem the J/Ψ meson mass
shift∆m innuclearmatteratbaryondensityρ canbe
J B
relatedtotherealpartofthescatteringamplitude f(0)
ℜ
as [52,68,69]
m +m
N J
∆m = 2π ρ f(0). (28)
J B
− m m ℜ
N J
TherecentQCDsumruleanalysis[51–53]withtheop-
erator product expansion predicts attractive mass shifts
of about –4 –10 MeV for J/Ψ meson in nuclear mat-
÷
ter,whichcorrespondstosmall f(0)about0.1 0.2fm.
ℜ ÷
Thisresultisveryclosetothepredictionsfromtheoper-
ator product expansion[48,49]and the calculations with FIG.8. Themodulusoftheratioofrealtoimaginarypart
of the J/Ψ+N scattering amplitude at t=0 extracted from
QCD van der Waals potential [50].
experimental data on cross section for γ+N J/Ψ+N reac-
Therearetwowaystolinktheseresultsfortheeffective →
J/Ψ mass in nuclear matter or the J/Ψ-nucleon scatter- tion as a function of J/Ψ momentum pJ in the nucleon rest
frame. Experimental data used in the evaluation are from
inglengthwithourcalculations. Bothofthemethodsap-
Ref. [54–56]. The solid line shows the results from the dis-
pliedbelowcontainanumberofuncertaintiesandshould
persion relation. The dotted line shows the prediction from
be carefully considered.
Regge theory,which fixed thepositive sign of theratio.
8
Now, f(0) is given as a product of the ratio α predictions [48–53] at p =0. Notice, that the shadowed
JN J
ℜ
extractedfromthephotoproductiondata[54–56]andthe areais extendedto higher momenta only fororientation.
imaginary part of the J/Ψ+N scattering amplitude at Within the uncertainties of the method we could notde-
t=0, which is related to the total J/Ψ+N cross section tectanydiscrepancywiththeabsolutevalueofthe f(0)
ℜ
by the optical theorem (30) as given in Refs. [48–53]. However, we emphasize again,
that this conclusion should be accepted very carefully.
p
f(0)= J σ , (30) AdifferentwaytoevaluatetherealpartoftheJ/Ψ+N
JN
ℑ 4π scattering amplitude at t=0 involves the dispersion rela-
tion that is given as [70]
Let us recall that in our calculations the total J/Ψ+N
cross section is given only for OZI allowed processes,i.e. 2(ω2 ω2)
atenergies√s≥mΛc+mD. Thisthresholdcorrespondsto ℜf(ω)=Ref(ω0)+ π− 0
theJ/Ψmesonmomentumof 1.88GeV/c. Thus,within
≃ ∞
thepresentmethodwecannotprovidetherealpartofthe ω′ f(ω′)
P ℑ dω′, (31)
scatteringamplitudeatpJ=0andmakeadirectcompar- × Z (ω′2 ω2)(ω′2 ω2)
ison with the absolute value of the f(0) predicted in ωmin − 0 −
ℜ
Refs. [48–53]. However, the estimate of f(0) at min-
ℜ where ω stands for the J/Ψ total energy, ω is the
imal J/Ψ momenta allowed by OZI reactions might be min
threshold energy and f(ω) denotes the imaginary part
done.
ℑ
of the J/Ψ-nucleon scattering amplitude at t=0, as
a function of energy ω, which can be evaluated from
Eq. (30). Furthermore, Ref(ω0) is the subtraction con-
stant taken at J/Ψ energy ω0.
Being more fundamental, this method involves sub-
stantialuncertainties for the followingreasons. First, we
should specify the subtraction constant. By taking it
at p =0 from Eq. (28) we could not further verify the
J
consistencyof our calculationswith the predictions from
Refs. [48–53]. Second, to evaluate the principal value of
the integral of Eq. 31 one should know the imaginary
part of the J/Ψ+N scattering amplitude at t=0 or the
total cross section up to infinite energy.
Since ourmodel couldnotprovidethe highenergybe-
havior of σ one should make an extrapolationto high
JN
energiesbyusingeithershortdistanceQCDorReggethe-
ory. As was shown in Ref. [39] the total J/Ψ+N cross
section at high energies is dependent on the gluon dis-
tribution function, g(x), appearing in Eq. (21). Further-
more, both distributions given by Eq. (22) and taken
from Ref. [42] were unable to reproduce [39] the ex-
perimental data on the γ+N J/Ψ+N cross section at
→
√s>70 GeV. Thus we could not extrapolate our results
to high energiesby the shortdistance QCDcalculations.
As shown in Fig. 7, the Regge theory reproduces the
J/Ψphotoproductiondataatt=0quitewell,whenrenor-
FIG.9. Therealpart f(0) oftheJ/Ψ+N scatteringam-
ℜ malizedbyafactorof1.3. Thismeetsourcriteriasinceit
plitudeextractedfromexperimentaldata[54–56]onthecross
also fits our calculations at √s 20 GeV. Thus we adopt
section for γ+N J/Ψ+N reaction at t=0 as a function of ≃
→ theReggemodelfitfortheextrapolationofourresultsto
J/Ψ momentum, pJ, in the nucleon rest frame. The solid
highenergiesinordertoevaluatethedispersionrelation.
line shows the f(0) evaluated using the dispersion relation.
ℜ It is important to notice, that, in principle, there is no
Theshadowedareaindicatestherangeofthepredictionsfrom
Refs.[48–53]givenatzeromomentumoftheJ/Ψmeson. The needtoaddressJ/Ψphotoproductioninordertoextrap-
extension of the shadowed area to higher momenta is shown olate our results to infinite energy using Regge theory.
only for orientation and has no physicalmeaning. Onecanadopttheenergydependenceofthedissociation
cross section given by soft and hard pomeron exchanges
andmakethe absolutenormalizationbyourcalculations
The f(0) extracted from the experimental data on √s>20 GeV. This is reasonable, because the absolute
ℜ
J/Ψ meson photoproduction cross section on a nucleon normalization from the Regge model can be considered
att=0isshowninFig.9asafunctionofJ/Ψmomentum to some extent as a free parameter,not fixed by the the-
in the nucleon rest frame. The shadowed area shows the ory.
9
higMhoerneeorvgeyr,,ωw0e=t1a0k3eGtheVe,suabntdraficxtiiotnbycoRnesgtgaentthℜefor(yω0w)itaht ddσt(cid:12)(cid:12)t=0 = 161π (1+α2JN)σJ2N. (33)
the predictedratioα . Thus wecanindependently use (cid:12)
JN The slope b (cid:12)can be estimated from the differential
an evaluated real part of the J/Ψ+N scattering ampli-
γ+N J/Ψ+N crosssectionandwasfitted inRef.[39]
tude at t=0 for comparison with the predictions [48–53]
→
to available experimental data as b= 1.64+0.83 lns.
given at p =0.
J −
Finally, the elastic J/Ψ+N cross section is given by
Our results for the ratio α evaluated by the disper-
JN
sion relation (31) are shown in Fig. 8 by the solid line.
1
Obviously, the calculations match the Regge model pre- σ = (1+α2 )σ2 , (34)
el 16πb JN JN
dictions since they were used to fixed both the high en-
ergy behavior of the total J/Ψ+N cross section and the and is shown in Fig. 10 as a function of J/Ψ meson mo-
subtraction constant for the real part of the scattering mentum. ThedashedareainFig.10indicatestheelastic
amplitude at t=0. Furthermore, the dispersion calcula- cross section at p =0 given within the scattering length
J
tions give a surprisingly good fit to the ratios αJN ex- approximationin Refs. [48–53] as
tracted from the experimental data [54–56] on the cross
section at t=0 of the γ+N J/Ψ+N reaction. σel =4π f(0)2, (35)
→ |ℜ |
The realpartof theJ/Ψ+N scattering amplitude cal-
thatisvalidasfaras f(0)=0atp =0. Note,thatEq.34
culatedbydispersionrelationisshownbythesolidlinein ℑ J
couldnotprovidetheestimateforelasticcrosssectionat
Fig. 9 as a function of J/Ψ meson momentum in the nu-
p <1.88 GeV, i.e. below the Λ +D¯ reaction threshold,
cleon rest frame. Our calculations describes reasonably J c
wherethedissociationcrosssectionσ =0. Inprinciple,
well the data evaluated from the J/Ψ photoproduction JN
onecaninterpolateσ fromzeroJ/Ψ momentumto our
att=0, thatareshownby the solidcircles inFig.9. It is el
results.
important that our results, normalized by a subtraction
constant at high energies, p =103 GeV, simultaneously
J
match the predictions [48–53] given at p =0.
J
Concludingthissectionletusmakeafewremarks. The
twomethods appliedto the evaluationofthe realpartof
the J/Ψ-nucleon scattering amplitude at t=0 are actu-
ally different, but nevertheless provide almost identical
resultsforthe f(0)overalargerangeofJ/Ψmomenta.
ℜ
Theconsistencyofourapproachcanbeprovedbyagree-
ment between our calculations simultaneously with the
Regge model predictions for f(0) at high energy and
ℜ
the predictions from various models [48–53]for f(0) at
ℜ
p =0. ¿From these results we confirm that below the
J
momentum of p 1.88 GeV, which corresponds to the
J
OZI allowed J/Ψ+≃N Λ +D¯ threshold, the J/Ψ inter-
c
→
actionswithanucleonarepurelyelastic. TheJ/Ψmeson
doesnotundergosignificantdissociationonabsorptionat
momenta p <1.88 GeV.
J
IX. ESTIMATE OF THE ELASTIC J/Ψ+N CROSS
SECTION
Bydefinition,thetotalelasticJ/Ψ+N J/Ψ+N cross
→
section,σ , isrelatedto the elasticdifferentialcrosssec-
el
tion as
FIG.10. Theelastic J/Ψ+N crosssection asafunctionof
dσ the J/Ψ meson momentum in the nucleon rest frame. The
σ = exp(bt), (32)
el Z dt(cid:12) solid lines show ourcalculations. The dashed area illustrates
(cid:12)(cid:12)t=0 predictionsfromRefs.[48–53]givenatpJ=0. Thedashedline
where b is an exponential(cid:12) slope of the differential cross indicates our interpolation for the elastic cross section based
section and the cross section at the optical point, t=0, on the QCD sum rule results alone [51–53].
is related to the total J/Ψ+N cross section and the ra-
tio αJN of the real and imaginary part of the scattering As we already discussed, the uncertainty in the elas-
amplitude as tic cross section at p =0, given in Refs. [48–53], is very
J
large and to make a less unambiguous interpolation let
10
