Universal geometrical scaling of the elliptic flow

Transcription

1 EPJ Web of Conference 9, 83 (5) DOI:.5/ ejconf/ 5983 C Owned by the author, ublihed by EDP Science, 5 Univeral geometrical caling of the ellitic flow C. ndré, J. Dia de Deu,. Mocoo, and C. Pajare ;a Carlo. Salgado Deartamento de Fíica de Partícula and IGFE, Univeridade de Santiago de Comotela, 578, Santiago de Comotela, Sain CENTR, Deartamento de Fíica, Intituto Suerior Técnico, v. Rovico Pai, 49-, iboa, Portugal btract. The reence of caling variable in exerimental obervable rovide very valuable indication of the dynamic underlying a given hyical roce. In the lat year, the earch for geometric caling, that i the reence of a caling variable which encode all geometrical information of the colliion a well a other external quantitie a the total energy, ha been very active. Thi i motivated, in art, for being one of the genuine rediction of the Color Gla Condenate formalim for aturation of artonic denitie. Here we extend thee reviou finding to the cae of exerimental data on ellitic flow. We find an excellent caling for all centralitie and energie, from RHIC to HC, with a imle generalization of the caling reviouly found for other obervable and ytem. Interetingly, the cae of the hoton, dicult to reconcile in mot formalim, nicely fit the caling curve. We dicu on the oible interretation of thi finding in term of initial or final tate eect. Introduction The dicovery of a izable ellitic flow in colliion, firt oberved at RHIC [, ] and later at HC [3], turned u a an exerimental major breakthrough. The oberved aniotroic flow can excluively be undertood if the meaured article in the final tate deend not only on the hyical condition realized locally at their roduction oint, but alo on the global geometry of the event. Thi non-local information can olely emerge a a collective effect, requiring trong interaction among the relevant degree of freedom, i.e. quark and gluon. The tudy of higher harmonic ha alo hown very intereting feature, including the ridge tructure een in colliion [4 7], Pb colliion [8, 9] and alo in high multilicity colliion []. The conventional undertanding of the ridge i imly related to flow harmonic in a hydrodynamic cenario, where the decrition of the Pb ridge and, ecially, the high multilicity ridge i a challenge. The quetion i whether an initial tate eect could determine the ridge tructure. Or, in other word, if the ellitic flow i an initial tate eect or, on the contrary, a final tate eect amenable to a hydrodynamic decrition [ ]. long thee line, we tudy the oibility that an initial tate roerty, a geometrical caling due to gluon aturation, wa reerved in a imilar caling in the ellitic flow. Similar quetion related to how hydrodynamic decrition could fit caling law oberved in v have already been raied []. a ajare@fax.uc.e Geometrical caling in multilicity ditribution Our work i baed on a reviou reult, the geometrical caling of multilicity ditribution in, and colliion [3 7], namely, dn ch N d T F(); () where i known a the caling variable and i given by T ; () with the aturation cale, and GeV. F() i a function of the caling variable. In the framework of ercolation of tring thi reult arie from the following arametrization of () N 6 ; (3) being N the number of wounded nucleon. () and the roton aturation momentum are given, reectively, by the equation, and () 3 + ln + C (4) W T! ; (5) rticle available at htt:// or htt://dx.doi.org/.5/ejconf/5983

2 EPJ Web of Conference Figure. (Colour online.) Multilicity deendence on. data from [3 37] (circle), CuCu (triangle) and uu (tar) from [38], PbPb (tar) from [39]. Curve obtained from eq. (7): (N, ) for (grey line); (N 5, 63) for CuCu (blue line); and (N 75, ) for uu/pbpb (red line). Color online. Figure. (Colour online.) Multilicity deendence on centrality (the number of articiating nucleon Nart N where N i the number of articiant er nucleu). CuCu (triangle) and uu (tar) data from [38], and PbPb (circle) from [39]. Curve: (.4, 6.4, GeV) for CuCu (blue), ( 9.6, 6.4, 3, GeV) for uu (green), and (.76, 3., 3.9, 5.5 TeV) for PbPb (red). Color online. with GeV, W :7. 3, 45 GeV and The function () in (4) ha to do with energy conervation in the multiarticle roduction roce. In gluon aturation model, a in the glama icture of the color gla condenate or in tring ercolation, color flux tube (tring) are formed, which ubequently give rie to article via fragmentation. Even at moderate high energie (RHIC energie) the number of color tring i very large for central heavy ion colliion. The fragmentation of tring require a minimum of energy, around :5 GeV, to create at leat a coule of hadron. However, the total available energy i which, at low and intermadiate energie, i not enough to hare with uch a large number of tring. ymtotically, the function () goe to 3 and for central colliion behave, a uual, like 3. Thi arametrization of () ha been reviouly ued in the framework of ercolation of tring to decribe the multilicity ditribution of and colliion at all centralitie and raiditie and at SPS, RHIC and HC energie [8, 9]. The cale indicate when the energy-momentum conervation eect become mall and the behaviour of the eective number of colliion tart to change from N to N 43. The validity of thi arametrization wa teted in reviou tudie [3] and the reult are hown in Fig. and in the form of fit to multilicity ditribution a a function of energy and centrality, reectively. In Fig. 3, the overall geometrical caling for all rojectile, target and energie i hown. reaonable caling i oberved in the range : < <. Detailed tudie of the caling have been reviouly done [3 6] howing dierent ratio of dierent tye of colliion, ecially for heavy ion colliion which are the main concern of our work. Figure 3. (Colour online.) Charged article multilicity er articiant in the eudoraidity range : < <:4 for all the heavy ion colliion conidered [4 43] veru for :7. 3 Geometrical caling of the ellitic flow In order to relate the geometrical caling of the tranvere momentum ditribution, equation (), with the caling of the ellitic flow, we define an azimuthal angle aturation momentum,., on average, the gluon denity i larger for maller R (maller ), we aume R. In addition, hould be roortional to the invere of the mean free ath, mf, normalized to the length ize of the cattering,, i.e. inverely roortional to the Knuden number k n mf. Therefore, we can write: mf R k n R Notice that h i i not, indeed, R : (6) 83-.

3 XIV International Symoium on Multiarticle Dynamic (ISMD 4) h i hr i R k n mf h i <<. mf k n : (7) i very mall for heavy nuclei colliion, Now, the caling variable, T r be relaced by T r + r T B@ + T, hould C : (8) the integrated tranvere momentum ditribution decreae trongly with T, and h i i much maller, than the contribution of the additional term to thee ditribution, concerning the geometrical caling, i negligible. However, thi term i reonible of the caling of the ellitic flow. In fact, 4 R d co dn d T d 4 R d co F( ) v : dn F() d T (9) Exanding F( ) in ower of R R and retaining the firt non-vanihing term, we have, v Z R R 4 d co R F() where we aroximate R. Denoting by we have or 4 df F() d df ; () d (); () v () () vk n (); (3) which i the caling law we were looking for, where and Z R R d co R ; R R in( ) ; in (4) Z b arcin( in ); R hr R i dr (5) being R the radiu of the nucleu and the length aociated to the ize of the colliion area at a given imact arameter and energy. Indeed, the roduct i the invere of the Knuden number, k n, i.e., the mean free ath normalized to the length meaured a the number of cattering center. i a meaure of the eccentricity of the colliion. It doe not deend on the ditribution of cattering center (arton or nucleon) in the tranvere lane and it i determined only by the almond hae of the colliion at a given imact arameter. The caling law obtained for v i baed excluively on two ingredient: the geometrical caling of tranvere momentum ditribution for < (intial tate eect) and the aumtion for the azimuthal aturation momentum (6) which encode all the angle deendence. The main quetion i whether the aumtion (6) can be conidered a a natural conequence of the tructure of the initial tate or, on the contrary, it i a final tate eect. On the one hand, the oibility of domain of color flux tube or cluter of tring having dierent azimuthal angle, ha been ointed out in everal aroache [5 7, ]. Thi would be the origin of the ridge tructure. In thi inital tate aroach the equation (6) i a natural aumtion. On the other hand, the mean free ath or the Knuden number of equation (6) can be regarded a a meaure of the ath needed to way out the colliion and, conequently, a a meaure of the energy lot by the arton roduced in the fragmentation of a color flux tube (or in a cluter of tring) interacting with the color field of other color flux tube []. 4 Comarion with exerimental reult In Fig. 4 (a) we lot the meaured value of v( T ) for u- u colliion for dierent centralitie at RHIC [44] and for PbPb colliion at HC [45] divided by the roduct comuted for each centrality and energy. We take the uual value of b and N for each centrality to comute and uing the equation (4), (5) and (3) reectively. i a meaure of the number of longitudinal cattering, which in the Glauber model i roortional to N 3. Neverthele, we ue ( + N 3 ), which i ued by mot of the tring model a dual arton model [46, 47], quark gluon tring model [48], Venu [49] or EPOS [5]. The olid black line correond to a fit to thee data, given by v ab ; (6) where a :64 :76 and b :44 :5. The Fig. 4 how that thi caling i atified. In order to ee the quality of thi caling we how in Fig. 4 (b) the ratio of Pb-Pb - % at.76 TeV, Pb-Pb 4-5 % at.76 TeV, u-u -3% at GeV and u- u 3-4% at GeV over Pb-Pb 3-4% at.76 TeV a a function of. ll the ratio lie in the range :8 :5 for the whole conidered, howing that the caling i quite good (mot of the exerimental error data are of the order of %). Changing the eccenticity,, by the uual eccentricity, hy x ihy + x i, or by the articiant eccentricity, the caling i not atified for both Monte-Carlo Glauber and Color Gla ditribution. uming that the v caling can be extended to colliion, we comute the ellitic flow a a function of the tranvere momentum, v( T ), for <. In Fig. 5 we 83-.3

4 EPJ Web of Conference Figure 6. (Color online.) v divided by the roduct for direct hoton at -% and -4% u-u colliion at GeV [54] and direct hoton at -4% Pb-Pb colliion at.76 GeV [55] lotted a a function of. The olid black line i a fit to data according to (6). Figure 4. (Color online.) (a) v divided by the roduct for -%, -3%, 3-4% and 4-5% u-u colliion at GeV [44], for -%, -3%, 3-4% and 4-5% Pb-Pb colliion at.76 GeV [45] in term of. The olid black line i a fit to data according to (6). (b) Ratio of Pb-Pb -%, Pb-Pb 4-5% at.76 TeV [45], u-u -3% and u-u 3-4% at GeV [44] over Pb-Pb 3-4% at.76 TeV [45] veru. how our rediction for 4 TeV and for imact arameter value of b :5 fm and b :7 fm.the v( T ) obtained i much maller than the comuted one uing hot ot inide the roton [5] and only lightly maller than the one found conidering uual imact arameter ditribution [5, 53]. For b :7 the multilicity would not be very dierent from the minimum bia which in many model i redicted to be around 7: at central raidity [8]. We have not included in our analyi the v data on Pb colliion due to the uncertaintie in the value of N at a given imact arameter. Moreover, in thi tage of our reearch, we have not tudied the caling for ecified article. far a geometrical caling i atified for, and [5], we exect that there will alo be a v caling for identified article, uing m T m intead of T. In thi way, we could comute v( T ) of any article whoe momentum ditribution verifie the caling. In addition, ince direct hoton roduction atifie geometrical caling [7], it ellitic flow may be of the ame ize and T hae of the ret of article. In order to check thi oint, in Fig. 6 we lot the ICE reliminary data [55] and the PHENIX data [54] at dierent centralitie. PHENIX collaboration quote two dierent oint at the ame T and centrality obtained by dierent analyi method (BBC and RXN detector). In any cae, we oberve that the data are cloe to the caling curve. 5 Concluion Figure 5. (Color online.) v rediction for colliion at 4 TeV for imact arameter value of b :5 fm (olid black curve) and b :7 fm (dahed red curve) a a function of T. We have hown that the exerimental data on the elitic flow of charged article for u-u and Pb-Pb colliion for dierent centralitie at RHIC and HC energie atify a caling law. The hoton data, deite their large uncertaintie, alo atify thi caling. Other than the eccentricity, thi caling law involve the number of cattering and a function which deend only on T. The number 83-.4

5 XIV International Symoium on Multiarticle Dynamic (ISMD 4) of cattering in the only involved quantity in relation with final tate eect. The ret ha to do with the geometry and gluon aturation. The caling law obtained can be related to the known geometrical caling of the tranvere momentum ditribution by encoding the azimuthal deendence i a new aturation momentum which, beide thi angular deendece, i roortional to the number of cattering. Thi aturation momentum can be interreted a the lot of momentum of a arton in it way out of the colliion. It would be intereting to look for a caling law imiliar to (3) for the ret of the harmonic moment. The total ditribution might factorize in two term: one with the roduct of the number of cattering and a caling function on T and the other with the um of the roduct of the dierent eccentricitie with the correonding azimuthal deendence. cknowledgement We thank N. rmeto for very ueful dicuion. Thi work ha been done under the roject FP-776 of MINECO (Sain), the Sanih Conolider CPN roject, FEDER fund and Xunta de Galicia. Reference [] K. drox et al. (PHENIX Collaboration), Nucl. Phy. 757, 84 (5) [] J. dam et al. (STR Collaboration), Nucl. Phy. 757, (5) [3].K. amodt et al. (ICE Collaboration), Phy. Rev. ett. 5, 53 () [4] J. dam et al. (STR Collaboration), Phy. Rev. C73, 6497 (6) [5]. dare et al. (PHENIX Collaboration), Phy. Rev. C78, 49 (8) [6] B. lver et al. (PHOBOS Collaboration), Phy. Rev. C8, 3495 () [7] S. Chatrchyan et al. (CMS Collaboration), Eur. Phy. J. C7, 5 () [8] B. belev et al. (ICE Collaboration), Phy. ett. B79, 9 (3) [9] S. Chatrchyan et al. (CMS Collaboration), Phy. ett. B78, 795 (3) [] V. Khachatryan et al. (CMS Collaboration), JHEP 9, 9 () [] S. Gavin,. Mcerran, G. Mochelli, Phy. Rev. C79, 59 (9) [] K. Duling, R. Venugoalan, Phy. Rev. D87, 544 (3) [3]. Bzdak, B. Schenke, P. Tribedy, R. Venugoalan, Phy. Rev. C87, 6496 (3) [4]. Dumitru et al., Phy. ett. B697, () [5]. Kovner, M. ublinky, Phy. Rev. D83, 347 () [6]. Kovner, M. ublinky, Int. J. of Mod. Phy. E, 33 (3) [7]. Dumitru,. Giannini, arxiv: [he-h] (4) [8] Y.V. Kovchegov, D.E. Werteny, Nucl. Phy. 96, 5 (3) [9] K. Werner, I. Karenko, T. Pierog, Phy. Rev. ett. 6, 4 () [] I. Bautita, J. Dia de Deu, C. Pajare, Eur. Phy. J. C7, 38 () [] M.. Braun, C. Pajare, V. Vechernin, Nuc. Phy. 96, 4 (3) [] G. Torrieri, Phy. Rev. C89, 498 (4) [3]. Mcerran, M. Prazalowicz, cta Phy. Polon. B4, 97 () [4] M. Prazalowicz, Phy. Rev. ett. 6, 4 () [5] M. Prazalowicz, Phy. ett. B77, 46 (3) [6] C. ndré,. Mocoo, C. Pajare, Nucl. Phy. 9, 4 (3) [7] C. Klein-Boeingand,. Mcerran, arxiv:43.74v [nucl-th] (3) [8] I. Bautita, J.G. Milhano, C. Pajare, J. Dia de Deu, Phy. ett. B75, 3 () [9] I. Bautita, C. Pajare, J.G. Milhano, J. Dia de Deu, Phy. Rev. C86, 3499 () [3] I. Bautita, J. Dia de Deu, J. Guilherme Milhano, C. Pajare, Phy. ett. B 75, 3 () [3] U Collaboration, Nucl. Phy. B 335, 6 (99) [3] G. J. lner et al. (U5 Collaboration), Phy. Re. 54, 47 (987) [33] STR Collaboration, Phy. Rev. C 79, 3499 (9) [34] CDF Collaboration, Phy. Rev. D 4, 33 (99) [35] K. amodt et al. (ICE Collaboration), Eur. Phy. J. C 68, 89 () [36] V. Khachatryan et al. (CMS Collaboration), JHEP, 4 () [37] V. Khachatryan et al. (CMS Collaboration), Phy. Rev. ett. 5, () [38] B. lver et al. (PHOBOS Collaboration), Phy. Rev. C 83, 493 () [39] K. amodt et al. (ICE Collaboration), Phy. Rev. ett. 6, 33 () [4] B.B. Back et al. (PHOBOS Collaboration), Phy. Rev. ett. 94, 834 (5) [4] B. lver et al. (PHOBOS Collaboration), Phy. Rev. ett. 96, 3 (6) [4] B. belev et al. (ICE Collaboration), Phy. ett. B7, 5 (3) [43] B. belev et al. (ICE Collaboration), Phy. Rev. ett., 83 (3) [44]. dare et al. (PHENIX Collaboration), Phy. Rev. ett. 98, 63 (7) [45]. amodt et al. (ICE Collaboration), Phy. Rev. ett. 5, 53 () [46]. Caella et al., Phy. Re. 36, 5 (994) [47]. Caella, C. Pajare,.V. Ramallo, Nucl. Phy. B4, 75 (984) 83-.5

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