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Excitation of Color Degrees of Freedom of Nuclear Matter and $Jpsi$ Suppression

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Excitation of Color Degrees of Freedom of Nuclear Matter and J/ψ Suppression
J¨ org H¨ ufnera,b , Boris Z. Kopeliovichb,c and Alberto Polleria
a

Institut f¨ ur Theoretische Physik der Universit¨ at, Philosophenweg 19, D-69120 Heidelberg, Germany. b Max Planck Institut f¨ ur Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany. c Joint Institute for Nuclear Research, Dubna, 141980 Moscow Region, Russia. Abstract In high energy nuclear collisions, the conventional Glauber model is commonly used to evaluate the contribution to J/ψ suppression originating from the inelastic interaction with colorless bound nucleons. This requires an e?ective value for the J/ψ -nucleon absorption cross section which is larger than theoretically expected. On the other hand, multiple nucleon-nucleon collisions mediated by color exchange interactions, excite their color degrees of freedom. We investigate the importance of this e?ect and ?nd that these excited states provide a larger cross section for J/ψ absorption. We conclude that the related corrections are important to explain the e?ective value extrapolated from experiment.

arXiv:hep-ph/0010282v1 25 Oct 2000

1

Introduction

The origin of the anomalous behavior of the J/ψ production cross section, as measured in Pb+Pb collisions at the CERN-SPS is still debated and several competing interpretations have so far been proposed. For a review on the subject see [1, 2]. In the early stage of a nucleus-nucleus (AB ) collision, the production mechanism of a charmonium is controlled by four length scales. They are the coherence length lc ≤ 2 EJ/ψ /m2 J/ψ , the formation
2 length lf ? 2 EJ/ψ /(m2 ψ′ ? mJ/ψ ), the mean interparticle distance λ ? 2 fm in a nucleus and the mean

nuclear size RA,B ? 5 ? 6 fm for large nuclei. Their interplay is crucial for the correct theoretical formulation of the problem and understanding the nature of early stage suppression. In the kinematic regime of the NA38/50 experiment at the CERN-SPS, which detects charmonia with plab = 50 GeV, one has lc ≤ 2 fm and lf ? 5 fm. Therefore, since lc is shorter than the mean nucleon separation, 1

one can assume that the colorless cc wave packet is produced in elementary nucleon-nucleon (N N ) interactions. On the other hand, the formation length is of the order of the nuclear size and the charmonium takes a ?nite path in the nucleus before being fully formed. The e?ect of these length scales on hadronic processes in a nuclear environment has been extensively studied [3, 4, 5, 6] and experimentally observed in electroproduction of ρ mesons o? nuclei [7]. Nevertheless, J/ψ suppression is mostly treated within the Glauber model, with a constant e?ective cross section for absorption on bound, non-interacting nucleons. Only recently [5, 8], the e?ect caused by the formation length was taken into account in the present context. Notice that while studying the SPS data it is justi?ed to neglect the contribution of the coherence length, for RHIC data it will not be so, being lf ? lc ? RA,B , and a new regime will open up. For the present discussion on SPS data, the relevant scales for J/ψ absorption are lf and RA,B , governing the suppression at the early stage. After the early stage, governed by N N collisions, an excited medium with high energy density is left behind. It is not yet clear how to describe it, but several points of view have been put forward. It is possible that the properties of the excited medium are similar to those of a plasma of de-con?ned quarks and gluons. This was indeed the original motivation for making J/ψ suppression so interesting [9]. It has been argued that such a plasma is very opaque to charmonium, therefore preventing its ?nal observation as a bound state [10, 11]. In the last stage of the reaction, part of the large number of produced hadrons, the so called co-movers, is possibly also responsible for suppression [12, 13, 14]. The present work intends to study the early stage nuclear absorption of J/ψ . It is common practice to approach the problem assigning a constant value to the J/ψ -nucleon absorption cross section. This quantity is then considered as an e?ective parameter, ?tted in order to reproduce the production data in proton-nucleus (pA) collisions. Our goal is to improve the understanding of the physical phenomena behind early stage absorption, neglecting for the time being other possible e?ects and therefore not necessarily being able to describe of the full set of available data. Following a known interpretation of the dynamics of N N collisions, we ascribe the main contribution to the inelastic cross section to color exchange processes. This assumption is well supported by high-energy hadron collision data, showing a plateau in the rapidity distribution of produced particles and indicating that the real part of the scattering amplitude is small with respect to the imaginary one.

2

Via color exchange, the 3-quark systems constituting the colliding nucleons become color octet states. It was already mentioned in [15] that each colliding system is colored, although the possible e?ects of this were neglected for simplicity. Color exchange may also be accompanied by gluon radiation. While the contribution of radiated gluons to J/ψ absorption was treated in [15, 16], here we focus on the ?rst part. Having realized that color-singlet nucleons are found in color octet states after an elementary collision, one can expect that repeated scatterings lead to multiple color exchanges. One therefore has the possibility to excite the color degrees of freedom of the 3-quark system also to the decuplet state. Higher color representations can be obtained when radiated gluons are also present, but here we restrict our study on the color dynamics of the constituent quarks of a nucleon. In the next section we calculate the cross sections for the interaction of charmonium with a colored nucleon. We ?nd an increase of 35 % when the 3-quark system is in octet state, with respect to the case when it is a color singlet, while with the decuplet the increase is of 70 %. We analyze and discuss in detail pA collisions in the third section. Finally, we consider the AB case: the e?ect of multiple scattering is studied and then applied to calculate integrated production cross sections. We ?nd an increased absorption, although the experimental suppression in the Pb+Pb cross section cannot be explained by the present treatment. The concluding section critically examines the results and gives an outlook on possible further improvements. Throughout the paper we indicate with Ψ the e?ective charmonium state, composed by a mixture of J/ψ , ψ ′ and χ.

2

Interaction of Charmonium with Colored Nucleons

The main feature we want to stress is the property of the color exchange interaction for the high energy collision amplitude. Repeated collisions allow a nucleon to be in any color SU(3) representation 3 ? 3 ? 3 = 1 ⊕ 8A ⊕ 8S ⊕ 10. To calculate the value of the inelastic cross section for Ψ on a nucleon which has undergone multiple scattering, we use the two-gluon exchange model of Low and Nussinov [17, 18]. Although oversimpli?ed, the model incorporates important properties of high-energy hadronic cross sections such as (approximate) energy independence and scaling of the cross sections according to the mean square radius of hadrons. One can write at once the absorption cross section, making use of the optical theorem, relating it to the forward elastic amplitude. This is convenient, since the

3

Ψ
k
Nn

-k

=

+

+

Figure 1: Top: Diagram illustrating the meson-nucleon elastic amplitude. The dashed line indicates the unitarity cut. Bottom: Vertex Γ between a gluon and the three-quark system of a nucleon. latter amplitude, in the two gluon exchange approximation, is purely imaginary. One obtains
abs σΨ N =

d2 r

Ψ, Nn | [

i,j

V (b + xi ? yj ) ] |Ψ, Nn .

2

(1)

It is a diagonal matrix element between a product state of charmonium Ψ and a nucleon Nn in its color representation n. The variable b is the impact parameter between the scattering hadrons. The transverse coordinates y1,2,3 refer the quarks in the nucleon, while x1,2 are those of Ψ. In the square brackets one has the transverse part of the one-gluon exchange potential. We de?ne its Fourier transform as V (z ) = d2 k (2π )2
1 g λa 2 s (i)

exp( i k z ) k 2 + ?2

1 g λa , 2 s (j )

(2)

where gs is the quark-gluon QCD coupling and ? is an e?ective gluon mass, introduced to incorporate con?nement. The index in parenthesis labeling the Gell-Mann matrices indicates to which quark the vertex is attached. Light-cone wave functions are used for Ψ and Nn in the expectation value (1). For the nucleon, we assume that the transverse positions of the quarks in the nucleon are frozen during the early stage of the collision, and therefore use the same spatial distribution both for singlet and colored states. Inserting in eq. (1) the expression for the gluon potential given in eq. (2) and performing the impact parameter integration, one arrives at the relation
abs σΨ Nn

=

d2 k

(k2

1 Ψ| 1 ? exp( i k x12 ) |Ψ + ?2 )2 (3)

× Nn | Γa (k) Γ? a (k) |Nn , 4

illustrated in the top part of FIG. 1. The ?rst factor in the integrand is the squared gluon propagator, the second contains the phase shift due to the momentum transfer k and is responsible, by the way, for color transparency, while the last factor is the interesting part of the present calculation. It is the product of two vertices between three quarks and a gluon, averaged with the wave function of the colored nucleon. The vertex, illustrated in the bottom part of FIG. 1, is Γa (k) =
16 3 i j k αs (λa (1) )i′ δj ′ δk ′ exp( i k y1 ) a j k i + δi ′ (λ(2) ) ′ δk ′ exp( i k y2 ) j i j a k + δi ′ δ ′ (λ(3) )k ′ exp( i k y3 ) . j

(4)

In order to average the product of the two vertices, one needs to specify the structure of the nucleon wave function. The color part of the wave function is the relevant one, while the spatial part is chosen to be equal for all color multiplets. We (anti)symmetrize quarks 1 and 2 in order to construct the mixed-symmetry octets, therefore use
? ? ? ? ? ? ? ? ? ? ? ? ?
1 √ ? 6 ijk 1 1 r r √ δs ?ijk δk ?ijs ? 3 2 1 r r? √ δi jks + δj ?iks 6 r δ s δ t + perm. of r, s, t δi j k

φn ijk =

{8A } , {8S } , {10} .

{1} ,

(5)

1 6

Because of the SU (3) structure, the di?erent color multiplets have di?erent number of indices. For simplicity, we label with n the di?erent multiplets. The factor given by the angular brackets in eq. (3) can now be evaluated. It involves the contraction of all the indices, leading to Nn |Γa Γ? a |Nn = N | 1 ? cn exp( i k y ij )|N . (6)

i<j

The e?ect of having di?erent color states is all contained in the coe?cients cn , listed in the second column of TABLE 1. It turns out that matrix elements with n = 8A , 8S are identical. The values of cn found, for n = 8, 10, lead to larger absorption cross sections as compared to the singlet case. To
2q 2 see this it is useful to de?ne the two-quark form factor FN (k ) = N | exp( i k x ij )|N for the nucleon.

Since, within our approximation, the latter does not depend on the colour multiplet, we drop the label n. Then we can rewrite eq. (3) as
abs σΨ Nn = Ψ| σn (r ) |Ψ ,

(7)

5

where the function in the brackets is σn (r ) =
16 3

α2 s

d2 k

(k2

1 [1 ? exp( i k r )] + ?2 )2 (8)

2q 2 × [1 ? cn FN (k )] ,

exhibiting a similar structure to that of the usual dipole-nucleon cross section, but with the important di?erence carried by the coe?cients cn . Notice that with cn = 1 and ? = 0 the integral in eq. (8) is infrared divergent. Therefore, the cut-o? ?, of the order of the inverse con?nement radius, is important. With ? = 0 it holds that σn (r ) → 0 for r → 0, so that one still preserves color transparency, even with a colored nucleon. This is indeed a property of the singlet meson. In our work we are not primarily
abs but rather on its dependence on the color state n with respect interested in the absolute values of σΨ Nn

to the singlet value. Yet we estimated the absolute values choosing αs = 0.6 and ? = 140 MeV. Within
2q 2 em (3k 2 ), the electromagnetic (k ) = FN a harmonic oscillator model of the nucleon, one ?nds that FN

form factor. We assume that this property is general and take the dipole form
em FN (Q2 )

=

?2 λ ? 2 + Q2 λ

2 Q2 →0

? 1?

1 2 2 Q Rp , 6

(9)

? 2 = 12/R2 , with Rp = 0.8 fm. For where the last approximate equality allows to ?x the parameter λ p ? 2 = 3λ2 , so that λ = 2/Rp . With the above de?nitions, the generalized convenience we rescale λ dipole-nucleon cross section can be computed analytically in terms of modi?ed Bessel functions K0 and K1 , with the result σn (r ) = 16πα2 s 3λ2
n (ξ ) 1 ? D2 n H (?r ) ? D2 (ξ ) H (λr ) ξ2

n ? D3 (ξ ) [log(ξ ) + K0 (?r ) ? K0 (λr )] } ,

(10)

n (ξ ) = c /(1 ? ξ 2 )2 , D n (ξ ) = 4c /(1 ? ξ 2 )3 , ξ = ?/λ and H (r ) = 1 ? zK (r ). The results being D2 n 1 n 3

of the calculation, illustrated in FIG. 2, shows the modi?ed dipole-nucleon cross section for the three Table 1: Results of the calculation of the meson-colored nucleon cross sections. Multiplet {n} Singlet {1} Octet {8} Decuplet {10} cn 1 ?1/4 1/2 σψNn [mb] 5.8 7.8 9.9 δn 1.0 1.35 1.7

6

50 1.2 0.9 40

?(r)

0.6 0.3 0 0 0.4 0.8 1.2
r [fm]

[mb]

30

σn (r)

20

10

{1} {8} {10}
0 0.2 0.4 0.6 0.8 1 1.2

0

r [fm]

Figure 2: Dipole cross section for di?erent color states as a function of the quark separation r . Box: The quantity ?(r ) which governs the dependence on color. color states of the nucleon. It largely increases when the dipole scatters o? a colored object. One can get more insight in the e?ect of the di?erent color states rewriting eq. (8) in the form σn (r ) = σ (r ) [ 1 + (1 ? cn ) ?(r ) ] , (11)

where σ (r ) is the usual dipole cross section on a color singlet nucleon and the function ?(r ) is found to vary appreciably with r . Its functional dependence is given in the small box in Fig. 2. To calculate the inelastic cross section according to eq. (7), we must average σn with the meson wave function. Therefore, the dependence of ? on r shows that the e?ect of having colored nucleons is stronger for larger size mesons. Recent calculations [19], based on realistic phenomenology for the dipole cross section and charmonium wave function, gave values for cross sections of various charmonia on √ a nucleon. At center of mass energy s = 10 GeV, it was found that σJ/ψN = 3.6 mb, σψ′ N = 12.2 mb and σχN = 9.1 mb. With a composition of Ψ of 52-60% due to J/ψ , 8-10% due to ψ ′ and 32-40%
abs with a due to χ, a weighted value of 5.8 mb was obtained. In the present case we evaluate σΨ Nn

Gaussian wave function having root mean square transverse separation r 2

1/2

= 0.25 fm, in order to

account for the di?erent states (J/ψ , ψ ′ , χ) implicitly included in Ψ and therefore adjust the singlet cross section to the quoted 5.8 mb. The values found are given in the third column of TABLE 1. In order to minimize the model dependence, we factored out from the colored cross sections the amount 7

abs = σ abs δ . The values for δ are given in the corresponding to the singlet case, therefore writing σψN n ψN n n

last column of TABLE 1. They constitute the important result of this calculation , showing that Ψ scatters with an color octet nucleon with a 35 % larger cross section with respect to a singlet nucleon, while with the decuplet the increase is even of 70 %. The large values found now require a careful analysis to establish the validity of this newly proposed absorption mechanism.

3

Analysis of proton-nucleus data: is the color mechanism already at work ?

Since the absorption pattern observed in pA collisions is usually considered to be the baseline for the AB case, it is necessary to examine it in some detail in the light of the possibility of a color mechanism for charmonium absorption. Consider the illustration given in FIG 3. There one can see the longitudinal and transverse geometry of the collision between a proton and the row of target nucleons with which it collides. On the left hand side one sees the projectile P which collides with the target nucleon T1 , producing a would be Ψ meson. Successively the projectile interacts with the target nucleon T2 which, then, collides with the Ψ meson. According to our initial argument, each nucleon-nucleon collision leaves the nucleons in a color octet state. As illustrated on the right hand picture, since Ψ is produced by P, it is overlapping with it. If then the charmonium is struck by T2 , P and T2 must also overlap, indicating that they have previously interacted. Therefore, in a pA collision all the nucleons with which Ψ has an interaction are color octets. This means that the e?ective absorption cross section normally used in Glauber model calculations in order to reproduce pA data, should be the cross section with a color octet nucleon. On the other hand, another simple geometrical argument shows that this might not be the full truth. In fact, in order to claim that Ψ interacts with a colored nucleon, the charmonium state itself is required to be well separated from the parent projectile 3-quark system, preceeding Ψ in propagation through the nucleus. This takes a ?nite time, since the charmonium is produced inside the projectile and has a smaller velocity. In the rest frame of the nucleus, the di?erence in velocities between Ψ and the parent
2 2 2 3-quark system is ?v ? m2 Ψ /2EΨ ? mp /2Ep , being v = (0)

1 ? m2 /E 2 ? 1 ? m2 /2E 2 . Therefore, one

requires a ?nite length L = Rp /?v , with Rp = Rp /γp , in order to have the charmonium separated from the projectile. In the following we focus on the kinematics of the NA38/50 experiments at the 8

t

Ψ
T2

T1

Ψ
z P T1 T2 P

Longitudinal

Transverse

Figure 3: Left: Longitudinal space-time representation of charmonium production in a pA collision, in the c.m. frame of a N N collision. Right: Illustration of the overlap between projectile (P), parent target (T1 ) and nucleon with which Ψ scatters inelastically (T2 ). CERN-SPS, for charmonia with laboratory momentum pΨ = 50 GeV and consider projectile protons with at least Ep = 200 GeV. Then, since γp = Ep /mp , taking Rp = 0.8 fm we obtain L = 2 fm. The physical meaning of L is the following: After its creation, the charmonium has to travel a length L (in the target rest frame) before one can safely assume that the next interaction occurs with a colored nucleon. Since the mean length which Ψ travels inside a nucleus is 2/3 RA , the retardation length L reduces the e?ect of colored nucleons considerably. It destroys it in light nuclei with A ≤ 50, and it reduces it to about 50% in heavy nuclei. Now, before we analyze the data for charmonium production in pA collisions, we discuss the importance of charmonium formation time, introduced in a quantum-mechanical way in [3, 5]. Within a simpli?ed version of a two-channel model [8], the ΨN absorption cross section evolves in the eigentime τ of Ψ as
abs in (0) in σΨ ? σΨ N (τ ) = σΨN + (σ N ) cos(τ /τf ) , (0)

(12)

in is the inelastic ΨN where τf = (mψ′ ? mJψ )?1 = 0.3 fm is the charmonium formation time, σΨ N abs (τ = 0) is the cross section of the so called “pre-meson” on a nucleon. cross section and σ (0) = σΨ N in = 5.8 mb [19]. Unfortunately σ (0) has not yet As mentioned in the last section, calculations give σΨ N

been calculated. We therefore use the value 2.7 mb from [8]. Making use of eq. (12), one can then write the expression for Ψ production in pA collisions, normalized to one nucleon, as
pA NN = σΨ σΨ

1 A

d2 b dz

(13)

9

× 1?

+∞ z

dz



abs z ′ ?z σΨ N ( γΨ )

ρA (z, b)/A

A?1

.

We now want to compare this expression with the conventional one obtained in the Glauber model,
abs (τ ) → σ ef f . containing only an e?ective absorption cross section, therefore making the replacement σΨ N ΨN

In this way one can perform the z -integration and obtain the well known formula
pA σΨ NN = σΨ

d2 b

ef f 1 ? 1 ? σΨ N TA (b) / A ef f σΨ N A

A

,

(14)

being TA (b) the nuclear thickness function. For A = 208 and pΨ = 50 GeV, using a realistic pro?le for Pb [20] and forcing the two expressions
ef f in eqs. (13) and (14) to be equal, we obtain σΨ N = 3.8 mb. This number has to be interpreted as

the value of the e?ective absorption cross section of Ψ on a color singlet nucleon, including formation time e?ects. To obtain the corresponding value for the case when Ψ scatters on a color octet nucleon, according to the result of the previous section we have to enhance the singlet result by 35%, obtaining
ef f 5.1 mb. The values of σΨ N = 3.8 mb and 5.1 mb are indicated as full lines in FIG. 4. Because of a

possible retardation of the color mechanism, we expect the experimental values to lie between these two lines. We have then re-analyzed the existing data taken by 3 ?xed target experiments: NA3 at CERN [21], which measured the ratio of Ψ production cross sections between Pt and H2 for 200 GeV projectile protons; NA38/51 [22] at CERN, which measured the cross section for di?erent targets (H2 , D2 , C, Al, Cu, W) struck by 450 GeV protons; E866 at FERMILAB [23], which measured the ratio of cross sections between W and Be for 800 GeV protons. In order remove the e?ects on absorption due to di?erent Ψ momenta, we impose the same kinematic conditions, considering Ψ’s of 50 GeV momentum in the laboratory frame, corresponding to xF = 0.15, 0.01, ?0.04 respectively, for the three experiments. We then assume that the absorption mechanism is parameterized by the e?ective
ef f cross section σΨ N , appearing in the Glauber model expression given in eq. (14). The ratio of cross

sections is de?ned as
pA pB RA/B = σΨ / σΨ .

(15)

N N and σ ef f , the ratio depends Therefore, while the pA cross section depends on the two parameters σΨ ΨN

only on the latter. We made three separate ?ts using the MINUIT package from CERNLIB, including 10

quoted statistical and systematic errors for the various cross sections and ratios. We extract the e?ective absorption cross section by minimizing χ2 . The result of these ?ts is presented in the second
2 , as σ = column of TABLE 2. The average is obtained, using the weights wi = 1/δσi i wi σi / i wi ,

while the error is δσ = (

?1/2 . i wi )

The values extracted from experiment are represented as open

points in FIG. 4. The mean value of 3.5 ± 0.6 mb seems to favor the singlet mechanism, for which a cross section of 3.8 mb is theoretically expected. On the other hand, the assumption that Ψ is produced on a bound nucleon with the same cross section as on a free one, used in eq. (14), can be an oversimpli?cation of the physics under study. We know that in a nuclear environment parton distributions are substantially modi?ed [24, 25, 26]. In particular, the gluon distribution in nuclei exhibits anti-shadowing e?ects at xBj ? 0.1, i.e. the gluon distribution gA (x2 , Q2 ) in a nucleus is larger than the one in a nucleon g(x2 , Q2 ). We expect
g (x2 , Q2 ) = gA (x2 , Q2 )/g(x2 , Q2 ). that charmonium production on a nucleus is enhanced by a factor RA g We estimate these e?ects by rescaling the Ψ cross section with the A-dependent correction RA =

0.032 log(A) + 1.006, extracted from [26]. For a heavy nucleus this amounts to an enhancement of
ef f the production cross section by a factor of nearly 1.2, therefore requiring a larger σΨ N . We ?t the

data once more, obtaining the results listed in the third column of TABLE 2. One sees that the new average value of 5.7 ± 0.6 mb now seems to favor the octet mechanism, for which we estimated a value of 5.1 mb. The results of the ?ts, together with the theoretical expectations, are all plotted in FIG. 4, where the full points are the result of accounting for anti-shadowing. One observes that the situation is, in all, quite uncertain. The above analysis is based on the assumption that there is no dependence of nuclear e?ects

Table 2: Results of the ?ts to pA data for the various experiments considered. See text for details. Experiments pψ = 50 GeV NA3 NA38/51 E866 Averages Theory
abs [mb] σψN Glauber ?t Gluon corr. 4.5 ± 1.7 7.3 ± 1.9 7.1 ± 1.6 10 ± 1.8 2.7 ± 0.7 4.7 ± 0.7 3.5 ± 0.6 5.7 ± 0.6 3.8 {1} 5.1 {8}

11

12 11 10 9 8 7 6 5 4 3 2 1 0 0

pψ (lab) = 50 GeV

abs ψN

[mb]

{8} {1}

σ

NA3

NA38/51

E866

100 200 300 400 500 600 700 800 900 1000 Ep (lab)
[GeV]

Figure 4: Results of the ?ts to pA data for the various experiments considered. Open dots refer to the Glauber model ?t while full dots account for gluon anti-shadowing corrections. The horizontal lines indicate the theoretical expectations for the singlet and octet absorption mechanisms, respectively. on the proton energy, is the charmonium energy is the same in all experiments. This is justi?ed in many phenomena, e.g. the coherence time e?ect [4]. However, there might still be some dependence on the proton energy. According to [27], energy loss of the projectile gluon in the nuclear medium substantially reduces the production rate of charmonia at large xF and is still rather important at small xF . In order to compensate for energy loss, the gluon distribution must be evaluated at a higher value of x1 , therefore suppressing Ψ production. This will in turn reduce the ?nal state absorption on nucleons in a ?t to the data. To account for this e?ect, already observed in Drell-Yan data [28], further study, which goes beyond the scope of this paper, is required. To summarize, the data seem to suggest that Ψ is absorbed by colored nucleons in pA collisions. The picture is though not yet conclusive due to the large experimental errors. Nevertheless, we follow our theoretical prediction and examine the potential role of the color mechanism in the more complex case of AB collisions.

4

Color dynamics in nucleus-nucleus collisions

Preparing the ground for the forthcoming discussion, it is useful to recall the conventional expression within the Glauber model for the Ψ production cross section at impact parameter b. Generalizing

12

eq. (13), to the case of AB collisions, one simply obtains
AB d2 σΨ

d2 b

(b) =

NN d2 bA dzA d2 bB dzB δ2 (bA + bB ? b) σΨ A?1 B ?1

abs × ρA (zA , bA ) 1 ? σΨ N TA (zA , bA )/A abs × ρB (zB , bB ) 1 ? σΨ N TB (zB , bB )/B

× SF.S.I.(zA , zB , bA , bB ) ,

(16)

which describes the collision of two rows at impact parameters bA and bB , overlapping due to the δ-function. ρA and ρB are the nuclear densities while
zA

TA (zA , bA ) =

?∞

′ ′ , bA ) dzA ρA (zA +∞ zB

and

TB (zB , bB ) =

′ ′ , bB ) dzB ρB (zB

are the thickness functions. The last factor SF.S.I.(zA , zB , bA , bB ) represent the probability of survival due to any additional “anomalous” suppression factor which goes beyond conventional nuclear e?ects, that is ?nal state interactions with the produced matter (QGP, hadron gas, etc.). Here we are not primarily interested in these contributions, therefore set SF.S.I. = 1. On the other hand, we focus on the reinterpretation of the terms describing nuclear e?ects discussing, in the next two subsections, color exchange in the context of multiple scattering, then applying the obtained results to the calculation of Ψ production in AB collisions.

1

Multiple color exchange in a nucleus

Having completed the computation of the Ψ cross sections with colored nucleons, we need to look at how these color objects develop in a collision between nuclei. Consider a nucleon of one of the nuclei colliding with a row of nucleons of the other nucleus. It undergoes multiple scattering, with repeated color exchanges. Keeping the projectile quark coordinates y1,2,3 frozen, according to the high-energy approximation, it is possible to calculate the probability Pn ({yj }, z, b) for a nucleon in color state n = 1,8,10 after having traveled through the row up to z at impact parameter b. This can be done in a very elegant way, by solving an evolution equation for the color density matrix of the 3-quark system. The problem has been considered some time ago with the general aim of studying the color 13

exchange interaction in the nuclear environment. A detailed calculation [29] shows that 1 + 20 F + 2 G 1 16 ? 40 F + 8 G × ? 27 ? 10 + 20 F ? 10 G
9 16 i<j

Pn = where

? ? ?

{1} {8} , {10}

(17)

F ({yj }, z, b) = exp [ ? and G ({yj }, z, b) =
i<j

σ (yij ) T (z, b) ]

(18)

exp [ ? 9 σ (yij ) T (z, b) ] . 8

(19)

The coe?cients F and G are expressed in terms of the standard dipole cross section σ (yij ), appearing in eq. (11), which one can construct by pairing two of the three quarks in a nucleon. Their arguments are the di?erences of quark coordinates yij = yj ? yj . The thickness function T (z, b) can be taken as that of nucleus A or B , depending on which nucleus one is considering. We can average the probabilities with spatial nucleon wave functions as Pn (z, b) = N | Pn ({yj }, z, b) |N , (20)

therefore integrating over internal quark coordinates. Due to the structure of eq. (17), the averaging procedure expressed by eq. (20) is transferred to an average of F and G . In analogy to the de?nition of Pn (z, b), we will de?ne also the functions F (z, b) and G(z, b). This average can in principle be exactly evaluated, although in practice one needs to know the correct form of the dipole cross section and of the spatial distribution of the three quarks in a nucleon. We therfore choose to average the exponents appearing in eqs. (18) and (19), instead of the full exponentials, i.e. evaluating exp(? N |X |N ) instead of N | exp(?X )|N for a given X . This corresponds to neglecting inelastic shadowing corrections
in for which, according to [30, 31], are small. Due to rotational symmetry one has σ (yij ) = 2/3 σN N in = 30 mb. In this way one obtains all i, j . We took the value σN N

where

1 + 20 F (z, b) + 6 G(z, b) 1 × 16 ? 40 F (z, b) + 24 G(z, b) Pn (z, b) = 27 ? ? 10 + 20 F (z, b) ? 30 G(z, b) F (z, b) = exp ? 9 σ in T (z, b) 8 NN 14

? ? ?

{1} {8} , {10}

(21)

(22)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -8 -6

ρPb(z,b=0)
(1) (8) (10)

Color Probability

-4

-2

0 2 z [fm]

4

6

8

Figure 5: Evolution of the color probabilities P1 , P8 , P10 , along z and with b = 0. Also shown is the pro?le ρPb (z, b = 0) of a Pb nucleus. and
3 in σN N T (z, b) . G(z, b) = exp ? 4

(23)

One notices several properties of the probabilities. First of all, the limit T → 0 implies F, G → 1. This means that P1 → 1 and P8 , P10 → 0. In other words the nucleon, before entering the nucleus, is in singlet state as it should be. Moreover, P1 + P8 + P10 = 1 for any T , therefore probability is always conserved. Finally, if the nucleus is large enough one has T ? 1 and F, G ? 1. This implies that if su?cient scatterings have taken place, the color probabilities reach the statistical limit, given by the ?rst coe?cients of eqs. (21). In other words, the 3-quark system becomes unpolarized in color after many collisions. Out of 27 states, one is the singlet, 16 are in the two octets and 10 are in the decuplet. The z -dependence of the color probabilities is illustrated in FIG. 5, together with the longitudinal pro?le ρPb (z, b) of a Pb nucleus. Soon after the nucleon has penetrated the nuclear pro?le, a process that involves ? 4 fm corresponding to 2 nucleon mean free paths, the statistical limit is essentially reached in which about 16/27 ? 2/3 of the nucleons are in octet state while 10/27 ? 1/3 are in decuplet. The amount of singlet is negligible.
abs , we With the calculated probabilities Pn and the previously evaluated color cross sections σΨ Nn

can now construct the cross section for Ψ when it scatters with a nucleon, taking into account all its

15

color states. We obtain
abs σΨ N (z, b) = n abs σΨ Nn Pn (z, b) .

(24)

As obvious from FIG. 5, the projectile nucleon which collides with several target nucleons quickly reaches a “color equilibrium”. Using the values for δn given in TABLE 1, that is 1.35 for the octet cross section and 1.7 for the decuplet one, the e?ective absorption cross section of Ψ on a nucleon, within a nucleus-nucleus collision, becomes
abs (z, b) σψN

z →∞ b=0

?→ ?

16 10 1 abs + × 1.35 + × 1.7 × σψN 27 27 27
abs 1.47 × σψN = 5.5 mb ,

(25)

abs = 3.8 mb has been used, according to our calculation in the previous section. where the value σψN

We can therefore summarize saying that, although the cross section for Ψ with a singlet nucleon is 3.8 mb, including formation time e?ects, its e?ective value in AB collisions with large nuclei is 5.5 mb. As a ?nal remark, notice that the calculation leading to eqs. (21), (22), (23) and (24) is performed assuming that each projectile nucleon undergoes collisions with singlet nucleons in the target. This neglects the color excitation of the target. A complete treatment of the problem is very involved and is not expected to change the picture drawn. In fact, as seen in FIG. 5, color excitation is a rather fast process and accounting for the neglected features would make it even faster. Moreover, as discussed in the introduction, gluon radiation, here neglected, would speed up color excitation even more.

2

Charmonium Suppression in AB collisions

Having discussed in detail colored cross section and multiple color exchange, we obtained, with eq. (24), the important ingredient needed to calculate in detail the Ψ production cross section in AB collisions. It is ?rst of all necessary to discuss some kinematical features concerning Ψ absorption and follow the longitudinal space-time diagram of FIG. 6, where the kinematics is illustrated. Considering the absorption caused by one of the nucleons of nucleus B , with coordinate z ′B , one notices that, before interacting with the meson at point DB , it has traveled through nucleus A from +∞ down to z ?A . The position z ?A depends on the Ψ velocity in the center of mass frame. This, in turn, depends on xF = 0.15, from which one obtains that vΨ =
2 2 x2 F s/(4mΨ + xF s), where s is the center of mass

16

t

Ψ
DB Q CA DA CB P

z
~ zA zA ’ zA zB ~ zB zB ’

A

B

Figure 6: Longitudinal space-time representation of a row on row collision taking place at impact parameters (bA , bB ) within a AB collision. The Ψ production point is indicated with P, while the absorption points by DA and DB , do to nucleons of A and B respectively. energy of a N N collisions. With an elementary geometrical reasoning, making use of the triangle P QDB for nucleus B , and DA CB P for nucleus A, one arrives at the relations
?1 ′ ′ z ?A = zA ? (zB ? zB ) RΨ , z ?B = zB ? (zA ? zA ) RΨ ,

(26)

where RΨ = (1 ? vΨ )/(1 + vΨ ). These must be used as arguments for the e?ective cross section in eq. (24). Therefore one has two cross sections, one for absorption by nucleus A, taking into account the multiple scattering of its nucleons with nucleus B , and one for nucleus B , depending on A. We can now modify eq. (16) by making the replacements
abs σΨ N TA (zA , bA ) → abs σΨ N TB (zB , bB ) → zA ′ ′ , bA ) , ΣB (? zB , bB ) ρA (zA dzA ′ ′ dzB ΣA (? zA , bA ) ρB (zB , bB ) .

(27) (28)

?∞ +∞ zB

The position-dependent e?ective cross sections ΣA (? zA , bA ) =
n A abs zA , bA ) , σΨ Nn Pn (? B abs zB , bB ) , σΨ Nn Pn (? n

(29) (30)

ΣB (? zB , bB ) = are obtained from eq. (24).

Now eq. (16) can be used to evaluate total cross sections, with a further integration in impact parameter. We compute values to compare with all available data measured by the NA38/50/51 17

collaboration at the SPS [32]. We use realistic nuclear pro?les [20] and scale the center of mass √ 12 N N ? (1 ? m energy according to the parameterization σΨ J/ψ / s) . The results are compared with the data in FIG. 7. The dashed curve corresponds to a standard Glauber model calculation with the
abs = 3.8 mb, as evaluated in Sect. 3, while the thick curve refers singlet absorption cross section σΨ N1

to the improved calculation which takes into account color excitations, studied in Sect. 4. In order to emphasize the di?erence between the two absorption mechanisms, we neglect here anti-shadowing
abs by 35 %, therefore used σ abs = 5.1 e?ects. For the pA calculation we increased the value of σΨ N1 Ψ N8

mb. This is the extreme situation, when Ψ always encounters color octet nucleons. For the AB case we used eqs. (29) and (30). Since our main intent is to compare the the new absorption mechanism to the conventional Glauber model one, we do not present the calculation of the E⊥ -dependent cross section for Pb+Pb collisions, which we cannot explain. Di?erent from the Glauber model result, our improved one provides a better agreement with the data, except for the Pb+Pb point. In other words, the proposed absorption mechanism due to color dynamics is found to be an important contribution to nuclear e?ects in Ψ production. It must be accounted for, together with other mechanisms of suppression [15, 16, 33] which we here put aside. In the search for new physics one must in fact rely on a solid baseline, employing the best of our knowledge of the dynamics of Ψ absorption at the early stage of a nuclear collision.

5

Conclusions

In this work we studied the dynamics underlying Ψ suppression due to nuclear e?ects, employing the idea that color exchange processes in multiple N N collisions may play a signi?cant role. First of all, we calculated inelastic cross sections for Ψ scattering on colored nucleons and found an increase of about 35 % when the nucleon is in a color octet state as compared with the singlet case, while the enhancement factor for the color decuplet is about 70 %. Making use of available calculations of ΨN cross sections, we included formation time e?ects and
abs = 3.8 mb. Scaling this value obtained the e?ective value for Ψ absorption on singlet nucleons σΨ N1 abs = 5.1 mb and with a 70 % increase σ abs with a 35 % increase we obtained σΨ ΨN10 = 6.4 mb. N8

We then performed a ?t to available data on Ψ production cross sections in pA collisions. The result

18

3.0

2.5

Glauber Colours

[nb] B?+?- σ(J/ψ) / AB

2.0
pp pd

1.5

pC

pAl

1.0

pCu

pW pU

OCu

OU SU PbPb

0.5

0.0

100

101

102 AB

103

104

105

Figure 7: Calculations of total J/ψ production cross sections in AB collisions and comparison with the NA38/50/51 data. The curves, drawn to guide the eye, correspond to Glauber absorption (dashed) and to absorption by colored nucleons (full). of the ?t, including gluon anti-shadowing corrections, is that the e?ective absorption cross section for
ef f Ψ is σΨ N = 5.7 ± 0.6 mb, therefore suggesting that in pA collisions, the produced charmonium interacts

with color octet nucleons. The conclusion is, though, rather weak because the experimental points are quite scattered. Total production cross sections for Ψ were then calculated and compared in FIG. 7 with available data for AB collisions. Compared to the Glauber model result, the agreement is improved and we
abs = 5.5 conclude that e?ects due to color exchange processes are substantial and provide the value σΨ N

mb for the e?ective absorption cross section on nucleons in AB collisions (See eqs. (24) and (25)). In conclusion, we have completed a study of nuclear e?ects in charmonium production in AB collisions, providing a better baseline for future calculations. These amount to an improved treatment of gluon bremsstrahlung, including non-linearities arising because of gluon cascades, to the calculation of e?ects caused by gluon energy loss and to the evaluation of E⊥ -dependent cross sections. This will hopefully provide the starting point from which to address more precisely the nature of anomalous suppression in Pb+Pb collisions.

19

Acknowledgments
This work has been supported in part by the B.M.B.F. under contract number 06 HD 642.

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