Centrality, Rapidity and Transverse-Momentum Dependence of Cold Nuclear Matter Effects on J/\psi Production in dAu , CuCu and AuAu Collisions at \sqrt{s_{NN}}=200 GeV

Centrality, Rapidity and Transverse-Momentum Dependence of Cold Nuclear Matter Effects on  Production in Au , CuCu and AuAu Collisions at GeV

Abstract

We have carried out a wide study of Cold Nuclear Matter (CNM) effects on  production in Au , CuCu and AuAu collisions at GeV. We have studied the effects of three different gluon-shadowing parametrisations, using the usual simplified kinematics for which the momentum of the gluon recoiling against the  is neglected as well as an exact kinematics for a process, namely as expected from LO pQCD. We have shown that the rapidity distribution of the nuclear modification factor , and particularly its anti-shadowing peak, is systematically shifted toward larger rapidities in the kinematics, irrespective of which shadowing parametrisation is used. In turn, we have noted differences in the effective final-state nuclear absorption needed to fit the PHENIX Au data. Taking advantage of our implementation of a kinematics, we have also computed the transverse momentum dependence of the nuclear modification factor, which cannot be predicted with the usual simplified kinematics. All the corresponding observables have been computed for CuCu and AuAu collisions and compared to the PHENIX and STAR data. Finally, we have extracted the effective nuclear absorption from the recent measurements of in Au collisions by the PHENIX collaboration.

pacs:
13.85.Ni,14.40.Pq,21.65.Jk,25.75.Dw

I Introduction

The  particle is considered to be one of the most interesting probes of the transition from hadronic matter to a deconfined state of QCD matter, the so-called Quark-Gluon Plasma (QGP). In the presence of a QGP, binding of ̄ pairs into  mesons is predicted to be hindered due to color screening, leading to a  suppression in heavy ion collisions Matsui86 ().

The results of the  production in AuAu collisions at =200 GeV Adare:2006ns () show a significant suppression. Nevertheless, PHENIX data on Au collisions Adare:2007gn () have also shown a non trivial behaviour, pointing out that Cold Nuclear Matter (CNM) effects play an essential role at these energies (for recent reviews see Frawley:2008kk (); Rapp:2008tf (); Kluberg:2009wc () and for perspectives for the LHC see Lansberg:2008zm ()).

All this reveals that the interpretation of the results obtained in nucleus-nucleus collisions relies on a good understanding and a proper subtraction of the CNM effects, known to impact the  production in deuteron-nucleus collisions, where the deconfinement cannot be reached.

In previous studies OurIntrinsicPaper (); OurExtrinsicPaper (), we have shown that the impact of the gluon shadowing on  production does depend on the partonic process producing the and then the . Indeed, the evaluation of the shadowing corrections in which one treats the production as a 22 process () shows visible differences in the nuclear modifications factors when compared to the results obtained for a 21 process.

The former partonic production mechanism seems to be favoured by the recent studies based on Colour-Singlet Model (CSM), including Haberzettl:2007kj () or not Brodsky:2009cf () -channel cut contributions. This also seems to be confirmed by the recent studies of higher-order QCD corrections, which have shown, on one hand, that the problematic dependence of the LO CSM CSM_hadron () is cured when going at and  Campbell:2007ws (); Artoisenet:2007xi (); Gong:2008sn (); Artoisenet:2008fc (); Lansberg:2008gk () and, on the other, that the CSM yield at NLO in  Gong:2009kp (); Ma:2008gq () saturates the experimental values by the Belle collaboration Pakhlov:2009nj (). This does not allow for a significant Colour-Octet (CO) component Zhang:2009ym (), which happens to be precisely the one appearing in the low- description of hadroproduction via a process Cho:1995ce (). To summarise, one is entitled to consider that the former kinematics is the most appropriate to account for the PHENIX data.

The structure of this paper is as follows. In section II, we describe our approach, namely the partonic process, the shadowing parametrisations and the implementation of the nuclear absorption that we have chosen. In section III, we present and discuss our results for versus rapidity, centrality and transverse momentum. We particularly discuss the impact of the partonic process kinematics. Section IV is devoted to the results in AuAu and CuCu collisions. In section V, we present and discuss our extraction of the effective absorption cross section from the PHENIX Au data and we finally conclude.

Ii Our approach

To describe the  production in nucleus collisions, our Monte-Carlo framework OurIntrinsicPaper (); OurExtrinsicPaper () is based on the probabilistic Glauber model, the nuclear density profiles being defined with the Woods-Saxon parametrisation for any nucleus and the Hulthen wavefunction for the deuteron Hodgson:1971 (). The nucleon-nucleon inelastic cross section at is taken to and the maximum nucleon density to .

ii.1 Partonic process for production

Most studies of CNM effects on  production OtherShadowingRefs () rely on the assumption that the pair is produced by a partonic process where both initial particles are gluons carrying some intrinsic transverse momentum . The sum of the gluon intrinsic transverse momentum is transferred to the pair, thus to the  since the soft hadronisation process does not significantly modify the kinematics. This is supported by the picture of the Colour Evaporation Model (CEM) at LO (see Lansberg:2006dh () and references therein) or of the CO mechanism at  Cho:1995ce (). In such approaches, the transverse momentum of the  is meant to come entirely from the intrinsic transverse momentum of the initial gluons.

As just discussed, recent CSM-based studies Haberzettl:2007kj (); Brodsky:2009cf () have shown agreement with the PHENIX  data Adare:2006kf () and the problematic dependence of the LO CSM has been shown to be cured at Tevatron energies when going at and  Campbell:2007ws (); Artoisenet:2007xi (); Gong:2008sn (); Artoisenet:2008fc (); Lansberg:2008gk (). Furthermore, NLO computations in the CSM Gong:2009kp (); Ma:2008gq () leave now too small a room for a CO component which would support a production kinematics Cho:1995ce ().

Parallel to this, intrinsic transverse momentum is not sufficient to describe the spectrum of quarkonia produced in hadron collisions Lansberg:2006dh (). For above approximately 2-3 GeV, one expects that most of the transverse momentum should have an extrinsic origin, i.e. the ’s would be balanced by the emission of a recoiling particle in the final state. The  would then be produced by gluon fusion in a process with emission of a hard final-state gluon. This emission has a definite influence on the kinematics of the  production. Indeed, for a given  momentum (thus for fixed  and ), the process will proceed on average from gluons with larger Bjorken- than . Therefore, they will be affected by different shadowing corrections. From now on, we will refer to the latter scenario as the intrinsic scheme, and to the former as the extrinsic scheme.

In the intrinsic scheme, we use the fits to the rapidity and  spectra measured by PHENIX Adare:2006kf () in  collisions at as inputs of the Monte-Carlo. The measurement of the  momentum completely fixes the longitudinal momentum fraction carried by the initial partons:

(1)

with the transverse mass , being the  mass.

On the other hand, in the extrinsic scheme, information from the data alone – the and  spectra – is not sufficient to determine and . Indeed, the presence of a final-state particle introduces further degrees of freedom in the kinematics allowing more than one value of for a given set – which are ultimately the measured quantities as opposed to . The quadri-momentum conservation explicitly results in a more complex expression of as a function of :

(2)

Equivalently, a similar expression can be written for as a function of . In this case, models are mandatory to compute the proper weighting of each kinematically allowed . This weight is simply the differential cross section at the partonic level times the gluon Parton Distribution Functions (PDFs), i.e. . In the present implementation of our code, we are able to use the partonic differential cross section computed from any theoretical approach. For now, we use the one from Haberzettl:2007kj () which takes into account the -channel cut contributions Lansberg:2005pc () to the basic CSM and satisfactorily describes the data down to very low   . A study using other matrix elements (LO CSM, NLO CEM,…) is planned for future works.

Figure 1: (Color online) comparison of the gluon shadowing parametrisations EKS98 Eskola:1998df (), EPS08 Eskola:2008ca (), nDS deFlorian:2003qf () at LO and EPS09 Eskola:2009uj () at LO in a Lead nucleus at GeV. Adapted from Eskola:2009uj ().
Figure 2: (Color online)  nuclear modification factor in Au collisions, , at versus in the intrinsic (top row) and extrinsic (lower row) schemes using the following gluon shadowing parametrisations: EKS98 Eskola:1998df () (left), EPS08 Eskola:2008ca () (middle), nDSg deFlorian:2003qf () (right). The vertical lines indicate qualitatively the antishadowing peak and its shifts towards larger in the extrinsic scheme.

ii.2 Shadowing

To get the  yield in  and  collisions, a shadowing-correction factor has to be applied to the  yield obtained from the simple superposition of the equivalent number of  collisions. This shadowing factor can be expressed in terms of the ratios of the nuclear Parton Distribution Functions (nPDF) in a nucleon of a nucleus  to the PDF in the free nucleon.

These parametrisations provide the nuclear ratios of the PDFs:

(3)

The numerical parametrisation of is given for all parton flavours. Here, we restrain our study to gluons since, at high energy,  is essentially produced through gluon fusion Lansberg:2006dh ().

We shall consider three different shadowing parametrisations for comparison: EKS98 Eskola:1998df (), EPS08 Eskola:2008ca () and nDSg deFlorian:2003qf () at LO. Recently, a new parametrisation, EPS09 Eskola:2009uj (), has been made available. It offers the possibility of properly taking into account the errors arising from the fit procedure. Yet, in the case of gluon nPDF, as illustrated by Fig. 1, the region spanned by this parametrisation is approximately bounded by both the nDS and EPS08 ones. However, we shall not consider the nDS parametrisation which shows such small shadowing corrections that no significant yield corrections are expected. We shall prefer to use nDSg as done in other works since it provides a lower bound of EPS09 in the antishadowing region. Furthermore, the central curve of EPS09 is very close to EKS98. In this context, we have found it reasonable to limit our studies to EKS98, EPS08 and nDSg and to postpone to a further study the analysis of the error correlations and their impacts of production studies2.
Note that the spatial dependence of the shadowing has been included in our approach, assuming an inhomogeneous shadowing proportional to the local density Klein:2003dj (); Vogt:2004dh ().

(a)  EKS98
(b)  EPS08
(c)  nDSg
Figure 3: (Color online)  nuclear modification factor in Au collisions, , at versus the number of collisions for three rapidity ranges and for four values of the nuclear absorption (from top to bottom: 0, 2, 4 and 6 mb) using a) EKS98, b) EPS08 and c) nDSg in the extrinsic scheme.

ii.3 The nuclear absorption

The second CNM effect that we have taken into account concerns the nuclear absorption. In the framework of the probabilistic Glauber model, this effect refers to the probability for the pre-resonant pair to survive the propagation through the nuclear medium and is usually parametrised by an effective absorption cross section . Our results will be first shown for different values of using the three aforementioned shadowing parametrisations. Afterwards, we will extract the values that provides the best fit to the PHENIX data. We note here that this effective cross section may also account for initial state effects, such as parton energy loss in the nuclear target.

Iii Results for collisions

The  suppression is usually characterised by the nuclear modification factor , i.e., the ratio obtained by normalising the  yield in ion collisions to the  yield in proton collisions at the same energy times the average number of binary inelastic nucleon-nucleon collisions :

(4)

Any nuclear effect affecting production leads to a deviation of from unity.

(a)  EKS98
(b)  EPS08
(c)  nDSg
Figure 4: (Color online)  nuclear modification factor in Au collisions, , at versus the transverse momentum for three rapidity ranges and for four values of the nuclear absorption (from top to bottom: 0, 2, 4 and 6 mb) using a) EKS98, b) EPS08 and c) nDSg in the extrinsic scheme.

iii.1 vs rapidity: distribution shift

In Fig. 2, we show vs according to both the extrinsic and intrinsic schemes. The results are displayed for four values of for each of the aforementioned shadowing parametrisations and are compared with the PHENIX data Adare:2007gn (). The best fit result as performed in section V is also shown.

The comparison between the three plots on the upper row and their corresponding plots on the lower one shows a striking – but expectable – feature : the rapidity distribution in the extrinsic scheme is systematically shifted toward larger compared to the intrinsic case. This is particularly visible when one focuses on the anti-shadowing peak, which we have indicated qualitatively with vertical lines.

Such a shift is, in fact, not surprising at all. It simply reflects the larger value of the gluon momentum fraction in the nucleus, , needed to produce a when the momentum of the final state gluon is explicitly taken into account in the computations.

As mentioned above, recent theoretical studies of production in vacuum, i.e. in collisions, support at low and mid a partonic production mechanism as given by LO pQCD, namely , as opposed to a process. In this context, we claim that this rapidity shift – evident for any shadowing parametrisation – is a feature of production in Au that should be systematically accounted for. Along the same lines of arguments, we shall focus in the following discussions on the results obtained in the extrinsic scheme ( case), except for the extraction of using the and results.

(a)  EKS98
(b)  EPS08
(c)  nDSg
Figure 5: (Color online)  central to peripheral nuclear modification factor in Au collisions, , at versus rapidity for three centrality ranges and for four values of the nuclear absorption (from top to bottom: 0, 2, 4 and 6 mb) using : a) EKS98, b) EPS08, c) nDSg.

iii.2 vs centrality

In Figs. 3 we present our results for versus centrality, expressed as the number of collisions. We have taken into account the three shadowing parametrisations and four values of . One observes the effect of the impact parameter dependence of the shadowing –increasing for inner production points– which induces a progressive increase (resp. decrease) vs in the backward (resp. forward) due to the anti-shadowing (resp. shadowing) effect. Indeed, for collisions with larger , the  creation points are on average closer to the gold nucleus center where the shadowing is expected to be stronger. For the same reason, the absorption suppresses the yield more strongly for larger .

The overall effect (see Figs. 3) matches the trend of the PHENIX data Adare:2007gn (), showing a decrease vs stronger in the forward region than in the backward.

We also note that STAR collaboration has recently released a preliminary measurement of in the region using the most central collisions (Perkins:2009tp (): . Higher statistics are however needed to draw conclusions from those data.

iii.3 vs transverse momentum

It is important to note that, in order to predict the transverse momentum dependence of the shadowing corrections, one needs to resort to a model which contains an explicit dependence on . Studies were earlier carried on using the CEM at NLO in Bedjidian:2004gd (). However, due to the complexity inherent to the NLO code used, it was not possible to implement the impact parameter dependence of the shadowing, needed to reproduce, for instance, the centrality dependence Klein:2003dj (); Vogt:2004dh () as just discussed. Thanks to the versatility of our Glauber code, we can carry on such computation including such an impact parameter dependence as well as involved production mechanisms containing a non-trivial dependence on .

In Figs. 4, we show our results on versus the transverse momentum. We emphasise that the growth of is not related to any Cronin effect, since it is not taken into account here. It simply comes from the increase of for increasing as given by Eq. (2). Hence, in the mid and forward rapidity region, goes through the anti-shadowing region and one observes an enhancement in . In the backward region, where sits in anti-shadowing region, one only sees a decrease. A similar behaviour is obviously expected in vs as will be discussed in section IV.3.







iii.4

New results for the  from the 2008 Au run with approximately thirty times larger integrated luminosity than the 2003 Au results are emerging, with the first preliminary result in terms of  daSilva:2009yy (),

(5)
(a)  EKS98
(b)  EPS98
(c)  nDSg
(a)  EKS98
(b)  EPS98
(c)  nDSg
Figure 6: (Color online) centrality dependence of (solid lines) and (dashed lines) for five values of the nuclear absorption (from top to bottom: , 2, 4, 6 and 8 mb) using a) EKS98, b) EPS08, c) nDSg, compared with the corresponding PHENIX data Adare:2006ns ()
Figure 7: (Color online) centrality dependence of (solid lines) and (dashed lines) for five values of the nuclear absorption (from top to bottom: 0, 2, 4, 6 and 8 mb) using a) EKS98, b) EPS08, c) nDSg, compared with the corresponding PHENIX data Adare:2006ns ().
Figure 6: (Color online) centrality dependence of (solid lines) and (dashed lines) for five values of the nuclear absorption (from top to bottom: , 2, 4, 6 and 8 mb) using a) EKS98, b) EPS08, c) nDSg, compared with the corresponding PHENIX data Adare:2006ns ()
Figure 8: (Color online) comparison of the gluon shadowing parametrisations EKS98 Eskola:1998df () and nDSg deFlorian:2003qf () in lead and copper nuclei at GeV3.

Those recent data show a significant dependence on the rapidity, with a visible suppression for the most forward points in the three centrality ranges (, and ). In the negative rapidity region, which is expected to be dominated by large contributions4, the data (see Fig. 5) show approximately no nuclear effects within the uncertainties, but a visible suppression is observed at mid and forward rapidities. We also note that the suppression is stronger in most central collisions. As regards our results, Fig. 5 shows the rapidity dependence of the modification factor for the three centrality ranges using the EKS98, EPS08 and nDSg shadowing parametrisations. Different break-up cross sections have also been considered, from 0 to 6 mb. As announced, we have chosen to focus on the extrinsic scheme. Like for , this induces differences on the shadowing impact compared to an intrinsic scheme used in frawley-INT (). One observes on Fig. 5 that the trend of the data is reasonably described with a in the range of 2-4 mb, while the most forward points seem to decrease more than our evaluation. However, the uncertainties affecting the present preliminary measurements precludes drawing firmer conclusions. We shall analysis this in more detail in section V.

Iv Results for and collisions

iv.1 Centrality dependence5

In Fig. 7 and Fig. 7, we present the centrality dependence of the nuclear modification factor and in the forward and mid rapidity regions. This has been computed for the three shadowing parametrisations and for five .

As we have already observed in OurExtrinsicPaper (), is systematically smaller in the forward region that in the mid rapidity region (see also the next section). The difference increases for more central collisions since we have used an impact parameter dependent shadowing. While this difference (approximatively 20% for rather central collisions) matches well the one of the data when , one would have to choose a large if one wanted to reproduce the normalisation of the AuAu data, disregarding any effects of Hot Nuclear Matter (HNM). However, for such large , surviving from inner production points would be so rare that the difference between shadowing effects at mid and forward rapidities would nearly vanish. Note that for a in the range of 2-4 mb, a difference remains.

While most of the above discussion is similar for CuCu collisions using the EKS and EPS shadowing, one observes a peculiar feature for nDSg. Indeed, one does not observe any effect. This comes for the very weak shadowing encoded in nDSg for Cu. This is illustrated on Fig. 4, where one sees that nDSg shadowing in Cu nucleus ends up to be very small. Indeed, while there is a moderate difference between the EKS shadowing in Cu and in Au, there is a significant difference for nDSg.

(a)  EKS98
(b)  EPS08
(c)  nDSg
(a)  EKS98
(b)  EPS08
(c)  nDSg
Figure 9: vs for AuAu collisions for five (from top to bottom: 0, 2, 4, 6, 8 mb) using a) EKS98, b) EPS08, c) nDSg in 4 centrality bins.
Figure 10: vs for CuCu collisions for five (from top to bottom: 0, 2, 4, 6, 8 mb) using a) EKS98, b) EPS08, c) nDSg in 4 centrality bins.
Figure 9: vs for AuAu collisions for five (from top to bottom: 0, 2, 4, 6, 8 mb) using a) EKS98, b) EPS08, c) nDSg in 4 centrality bins.

iv.2 Rapidity dependence

We now discuss the rapidity dependence of the nuclear modification factor in the case of CuCu and AuAu collisions. It is displayed on Fig. 10 and Fig. 10 for the three shadowing parametrisations and for five . As in OurExtrinsicPaper (), slightly peaks at reducing the need for recombination effects recombinationRefs () which are particularly concentrated in the mid rapidity region and which could elegantly explain the differences of between the forward and mid rapidity regions.

As we have noted in the previous section, shadowing effects exhibit naturally such a rapidity dependence. This happens for the three shadowing parametrisations we have used for AuAu, confirming that this is a feature, rather than an accident. This effect is, however, reduced when an absorption cross section is accounted for. It is widely accepted that HNM effects are responsible for an extra suppression. If this suppression – by the creation of a QGP for instance – is not correlated to the path of the on its way out of the nuclei, it may not damp down the difference between and , as does a larger .

iv.3 Transverse-momentum dependence

(a)  EKS98
(b)  EPS08
(c)  nDSg
Figure 11: (Color online)  nuclear modification factor in CuCu collisions at versus for the mid rapidity region for five values of the nuclear absorption (from top to bottom: 0, 2, 4, 6, 8 mb) using : a) EKS98, b) EPS08, c) nDSg. Those are compared with the two STAR points Abelev:2009qaa () and the PHENIX preliminary results Leitch:2009xt ().

We now move on to the discussion of the transverse momentum dependence in the mid rapidity region, analysed both by the PHENIX Leitch:2009xt () and STAR Abelev:2009qaa () collaborations. As announced during the discussion of the dAu results, versus increases with (see Fig. 11). In fact, the growth partially matches the trend of the PHENIX and STAR data. We should, however, mention here that there is no consensus for now on whether one should expect a nuclear modification factor larger than one for around 8 GeV as seems to indicate the published STAR results Abelev:2009qaa ().

In the case of a confirmation of a non-suppression of at large , one could say that it does not behave as the other hadrons, which are significantly suppressed in central Cu+Cu collisions and for increasing (see adare:2008cx () for and Garishvili:2009ei () for “heavy-flavour” muons) In fact, the seems to adopt a behaviour closer to the one of prompt photons Reygers:2008pq () than to the one of other (heavy-flavoured) hadrons. We also note a non-suppression of meson in central Cu+Cu collisions Naglis:2009uu ().

If one goes further, one may want to extract information about the production mechanism at work here. Indeed, although the energy loss of a colored object in CNM is limited to be constant, rather than scaling with energy, by the Landau-Pomeranchuk-Migdal effect Brodsky:1992nq (), its magnitude per unit of length will be significantly larger for a CO than for a CS state propagating in the nuclear matter. This is especially relevant since, in the mid rapidity region, the pair hadronises outside the nucleus. This would naturally lead us to the conclusion that it is rather a colourless state than a coloured one which propagates in the nuclear matter.

V Extraction of break-up cross section by fits on Au data

By comparing our results in the intrinsic and extrinsic approaches, we have learnt that one of the consequences of this kinematical change implies a shift of the rapidity distribution. The latter is shifted as a whole to larger values of rapidity in the extrinsic case. As usual a break-up cross section, , has to be accounted for to describe the normalisation of . In practice, it is fit from the data and then used in the description of nucleus-nucleus collisions. In view of the differences of the shadowing impact induced by one or the other kinematics, it is natural to wonder what the corresponding variations of fit to the data are.

v.1 Fitting data

For this purpose, we have used PHENIX measurements of  Adare:2007gn () in order to obtain the best fit of for each of the shadowing parametrisation considered in both the intrinsic and extrinsic schemes. Based on the method used by PHENIX in Adare:2007gn () and Adare:2008cg (), we have computed the in the different cases, including both statistical and systematic errors.

By using the data on  versus rapidity, we have obtained the values of given in Table 1 for each of the shadowing parametrisations and for both production schemes. The resulting curves are shown on Fig. 2 for dAu, on Fig. 13 and Fig. 13 for AuAu and CuCu.

    
EKS98 Int. 0.9
EPS08 Int. 1.1
nDSg Int. 1.3
EKS98 Ext. 1.1
EPS08 Ext. 0.5
nDSg Ext. 1.2
Table 1: extracted from fit of (All cross section in units of mb)

In general, fitting the rapidity with a constant leads to more than acceptable . The largest ones are obtained for the nDSg parametrisation; this confirms the impression that parametrisation with “significant” shadowing and/or anti-shadowing (EKS98, EPS08) are preferred by the data (see also Fig. 2). The only systematic one can really see is that the 3 values extracted using the extrinsic scheme are larger than the corresponding ones obtained with the intrinsic scheme, as expected due to the increase of in the extrinsic case compared to the intrinsic one. For the time being, the data are not precise enough to draw further conclusions from such fits.

(a)  EKS98
(b)  EPS08
(c)  nDSg
(a)  EKS98
(b)  EPS08
(c)  nDSg
Figure 12: (Color online) vs for AuAu collisions for the best as obtained from the fit to the dAu data using a) EKS98, b) EPS08, c) nDSg in 4 centrality bins.
Figure 13: (Color online) vs for CuCu collisions for the best as obtained from the fit to the dAu data using a) EKS98, b) EPS08, c) nDSg in 4 centrality bins.
Figure 12: (Color online) vs for AuAu collisions for the best as obtained from the fit to the dAu data using a) EKS98, b) EPS08, c) nDSg in 4 centrality bins.

v.2 Fitting data

We have also fit the new data with a constant in each rapidity regions. Our results are given in Table 9 along with the one of frawley-INT () based on a kinematics and using the EKS98 shadowing. We need to make clear at this point that the intrinsic scheme used in frawley-INT () slightly differs from the one we have used for instance to fit the dAu data as shown in the previous section. The difference appears at the level of the running of the scale of the shadowed gluon distribution and the invariant mass of produced system. However, this is not expected to modify the following discussion (see OurIntrinsicPaper ()).

                    all
EKS98 Int. frawley-INT () N/A
EKS98 Ext.
EPS08 Ext.
nDSg Ext.
Table 2: and their errors6 extracted from fit of the data averaged on three rapidity region as well as on the entire range. (All cross section in units of mb).
(a)  EKS98
(b)  EPS08
(c)  nDSg
Figure 14: (Color online) versus obtained by fitting the data for Au, in the extrinsic scheme (in red closed circles) compared to the intrinsic scheme frawley-INT () (in blue open circles) using a) EKS98, b) EPS08 and c) nDSg.

As can be seen in Table 9, it appears that the strong suppression at forward rapidity and the lack of suppression at backward rapidity cannot be described using a fixed breakup cross section with EKS98 in the intrinsic scheme frawley-INT (). One also seems to observe an increase of with rapidity in our analysis (extrinsic) for EKS98 and nDSg, but a constant behaviour cannot be ruled out. The increase is in any case softer when the final-state gluon momentum is taken into account. It seems that forward rapidity is maybe the most interesting region for such investigations. Interestingly, in the case of EPS08, the value we have extracted for the forward region is equal to the one for the backward region.

As we have argued earlier, EPS08 can be used as a good indicator of the strongest shadowing reachable within the uncertainty of EPS09. From this viewpoint, the recent update of the intrinsic analysis, where frawley-BNL () no increase of is observed for the strongest shadowing of EPS09, confirms our findings.

To get a more precise view on the situation, it is useful to plot the effective absorptive cross section as function of the rapidity, without averaging on the three rapidity regions. The result can be seen on Figs. 14 with our result in the extrinsic scheme (red closed circles) and the one of frawley-INT () (blue open circles)7 in the intrinsic scheme. We have obtained an increasing absorptive cross section with rapidity, but the increase is much less pronounced than in the intrinsic case. In fact, in the EPS08 case, the increase does not appear statistically significant.

Vi Conclusion

Taking advantage of the probabilistic Glauber Monte-Carlo framework, JIN, discussed in OurIntrinsicPaper (); OurExtrinsicPaper (), we have (i) considered three different gluon shadowing parametrisations – EKS98, EPS08 and nDSg – taking into account a dependence on the impact parameter and the momentum of the gluon recoiling against the , (ii) shown that the rapidity dependence of is shifted towards larger rapidities irrespective of the shadowing parametrisation, and particularly the anti-shadowing peak, (iii) shown that the anti-shadowing peak is reflected in a rise of the nuclear modification factor for increasing , (iv) compared our results with the experimental measurements of the nuclear modification factors and from Au collisions presently available at RHIC and extracted the favoured values of the absorption cross section in the nuclear matter, and finally (v) shown that the effective absorption cross section increase at forward rapidity, obtained from the recent analysis of PHENIX data frawley-INT () in which the final-state gluon momentum is neglected (intrinsic case), is less marked when it is taken into account; that is in the extrinsic case.

Acknowledgments

We would like to thank S.J. Brodsky, A. Linden-Levy, C. Lourenço, N. Matagne, J. Nagle, T. Ullrich, R. Vogt, H. Wöhri for stimulating and useful discussions. This work is supported in part by Xunta de Galicia (2008/012) and Ministerio de Educacion y Ciencia of Spain (FPA2008-03961-E/IN2P3), the Belgian American Educational Foundation, the Francqui Foundation and the U.S. Department of Energy under contract number DE-AC02-76SF00515.

Footnotes

  1. Present address at Ecole polytechnique.
  2. Note that, as shown in Fig. 1, the EPS08 parametrisation shows a strong gluon shadowing at very small due to the inclusion of forward-rapidity BRAHMS data in the fit. An explanation for such an effect has been based on the idea that, in this kinematic region that corresponds to the beam fragmentation region at large Feynman , one can reach the smallest values of the momentum fraction variable in nuclei. It makes it possible to access the strongest coherence effects such as those associated with shadowing or alternatively the Color Glass Condensate (CGC). Nevertheless, as it has been argued in Kopeliovich:2008 (), although at forward rapidities one accesses the smallest in the nuclear target, one simultaneously gets into the region of large of the projectile nucleon where energy conservation becomes an issue. Indeed, as stated in Kopeliovich:2005 (), at large (or ) one expects a suppression from Sudakov form factors, giving the probability that no particle be produced as , as demanded by energy conservation. In a pA collision, the multiple interactions of the nucleon remnants with the nucleus makes this less likely to occur and the suppression is expected to be stronger. In fact, factorisaton breaks down and the effective parton distributions in the projectile nucleon then become correlated with the nucleus target, see Eq. (15) of Kopeliovich:2005 (). This effect should not be confused with gluon shadowing or other manifestations of coherence. Because of this, the strength of EPS08 gluon shadowing may be overestimated, since it includes data at large which were analysed assuming that the suppression was attributed only to a reduction of the gluon distribution in the nucleus. However, we would like to emphasize that, while the gluon shadowing in EPS08 and EKS98 differ by a factor 3 at GeV, the difference is already less than 20 at which is the relevant scale for our analysis.
  3. The plot has been generated by the nPDF generator http://lappweb.in2p3.fr/lapth/npdfgenerator
  4. One recalls here that in the extrinsic scheme there is no one-to-one mapping between the rapidity and the momentum fraction of the initial gluon. Yet they are correlated.
  5. As announced, we focus on the extrinsic case in all the following discussions.
  6. The errors quoted for the first line are extracted differently than ours. The errors shown here should only be compared for a given analysis.
  7. We recall here that the error bars on Figs. 14 are extracted with two different procedures in both case. They should only be compared within one approach.

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