Collective nuclear vibrations and initial state shape fluctuations in central Pb+Pb collisions: resolving the v 2 to v 3 puzzle
aa r X i v : . [ nu c l - t h ] A ug Collective nuclear vibrations and initial state shape fluctuations in central Pb+Pbcollisions: resolving the v to v puzzle B.G. Zakharov
L.D. Landau Institute for Theoretical Physics, GSP-1, 117940, Kosygina Str. 2, 117334 Moscow, Russia (Dated: August 18, 2020)We have studied, for the first time, the influence of the collective quantum effects in the nuclearwave functions on the azimuthal anisotropy coefficients ǫ , in the central Pb+Pb collisions at theLHC energies. With the help of the energy weighted sum rule we demonstrate that the classicaltreatment with the Woods-Saxon nuclear density overestimates the mean square quadrupole momentof the Pb nucleus by a factor of ∼ .
2. The Monte-Carlo Glauber simulation of the central Pb+Pbcollisions accounting for the restriction on the quadrupole moment leads to ǫ /ǫ ≈ . v -to- v puzzle. PACS numbers: . The results of experiments on the heavy ion collisions at RHIC and LHC give a lot of evidences for formation ofthe quark-gluon plasma (QGP) in the initial stage of nuclear collisions. The hydrodynamic simulations of the hadronproduction show that the QGP undergoes early thermalization (at the proper time τ ∼ . − η/s = 1 / π ) [1–3]. The most effective constraints on the QGP viscosity come from the hydrodynamicanalysis of the azimuthal dependence of the hadron spectra which is characterized by the Fourier coefficients v n dNdφ = N π ( ∞ X n =1 v n cos [ n ( φ − Ψ n )] ) , (1)where N is hadron multiplicity in a certain p T and rapidity bin, Ψ n are the event reaction plane angles. Forsmooth initial conditions at midrapidity ( y = 0) in the Fourier series (1) only the terms with n = 2 k survive.And the azimuthal anisotropy appears only for noncentral collisions due to the almond shape of the overlap regionof the colliding nuclei in the transverse plane. The event plane (for each n ) in this case coincides with the truereaction plane and Ψ n = 0. However, in the presence of fluctuations of the initial QGP entropy distributions, all theflow coefficients v n become nonzero and the event plane angles Ψ n fluctuate around the true reaction plane. Thefluctuations of the initial fireball entropy distribution is a combined effect of the fluctuations of the nucleon positionsin the colliding nuclei and fluctuations of the entropy production for a given geometry of the nuclear positions. Themost popular method for evaluation of the initial entropy distribution for event-by-event simulation of AA -collisionsis the Monte-Carlo (MC) wounded nucleon Glauber model [4, 5]. The even-by-event hydrodynamic modeling with theMC Glauber (MCG) model initial conditions has been quite successful in description of a vast body of experimentaldata on the flow coefficients in AA -collisions obtained at RHIC and LHC. However, in the last years it was foundthat the hydrodynamical models fail to describe simultaneously v and v flow coefficients in the ultra-central ( c → . For central collisions, at b = 0, the anisotropy of the initial fireball geometryoriginates completely from the fluctuations. The hydrodynamic calculations show [7, 8] that for small centralities ineach event the v n for n ≤ v n ≈ k n ǫ n , (2)where ǫ n are the Fourier coefficients characterizing the anisotropy of the initial fireball entropy distribution, ρ s ( ρ ), inthe transverse plane defined as [9, 10] ǫ n = (cid:12)(cid:12)R d ρ ρ n e inφ ρ s ( ρ ) (cid:12)(cid:12)R d ρ ρ n ρ s ( ρ ) . (3)Here it is assumed that the transverse vector ρ is calculated in the tranverse c.m. frame, i.e., R d ρρ ρ s ( ρ ) = 0. Thehydrodynamic calculations give k /k >
1, and this ratio grows with increase of the QGP viscosity. On the other Experimentally, the centrality, c , of an event is defined in terms of the charged particle multiplicity. To a very good accuracy (exceptfor the most peripheral collisions) c in terms of the impact parameter b reads c = πb /σ AAin [6]. hand, the MCG calculations show that at b = 0 ǫ and ǫ are close to each other (and are ∼ . v /v >
1. But experimentally it was observed that v is close to v in the ultra-central2 .
76 and 5 .
02 TeV Pb+Pb collisions [11, 12]. Since the hydrodynamic prediction for k /k seems to be very reliable,this situation looks very puzzling (it is called in the literature v -to- v puzzle). This leads to a serious tension for thehydrodynamic paradigm of heavy ion collisions.There were several attempts to resolve the v -to- v puzzle by modifying: the initial conditions [13, 14], the viscositycoefficients [15], and the QGP equation of state of [16]. However, these attempts have not been successful. Thecommon feature of all previous analyses devoted to the v -to- v puzzle is the use of the Woods-Saxon (WS) nucleardistribution for sampling the nucleon positions in the MC simulations of Pb+Pb collisions. In fact, this is an universalchoice in the physics of high-energy heavy ion collisions. However, the MC sampling of nucleon positions with theWS distribution completely ignores the collective nature of the long range fluctuations of the nucleon positions. Itis well known that the long range 3D fluctuations of the nuclear density have a collective nature and are closelyrelated to the giant nuclear resonances [17, 18] (for more recent reviews see [19, 20]). The major vibration modeof the spherical Pb nucleus corresponds to excitation of the isoscalar giant quadrupole resonance [17, 18]. Thesecollective quantum effects are completely lost if one samples the nuclear configurations with the WS distribution. Itis clear that an inappropriate description of the 3D long range fluctuation of the nucleon positions in the collidingnuclei will translate into incorrect long range fluctuations of the 2D initial fireball entropy density, which are crucialfor ǫ , in the central AA -collisions, when they are driven by fluctuations.In the present paper we demonstrate that the WS distribution overestimates considerably the mean square nuclearquadrupole moment of the Pb nucleus as compared to that obtained in the quantum treatment of the quadrupolevibrations. We calculate the azimuthal anisotropy coefficients ǫ , in Pb+Pb collisions in the MCG model by samplingthe nuclear configurations for ordinary WS distribution and a modified one which reproduces the quantum meansquare nuclear quadrupole moment of the Pb nucleus. Our results show that for the quantum version the ratio ǫ /ǫ becomes substantially smaller than that for ordinary WS distribution. The magnitude of the obtained ǫ /ǫ issmall enough to resolve the v -to- v puzzle.Note that the ordinary MC simulation is also inadequate for the isovector dipole mode, which plays an importantrole in fluctuations of electromagnetic fields in AA -collisions at the RHIC and LHC energies [21] (the classicaltreatment overestimates the mean square dipole moments for Au and
Pb by a factor of ∼ ǫ /ǫ in AA -collisions turns out tobe very small. For the first time, the problem with description of the mean square quadrupole nucleus moments inthe ordinary MC simulations with the WS nuclear distribution and its importance for the event-by-event analyses of AA -collisions was noted in [22]. . We assume that Pb nucleus is spherical, and the nuclear density is given by the ordinary WS nuclear density ρ A ( r ) = ρ r − R A ) /a ] (4)with parameters R A = (1 . A / − . /A / ) = 6 .
49 fm, and a = 0 .
54 fm [5]. Let us first consider classical calculationof the nuclear mean square multipole moment. We define the isoscalar L -multipole operator as (see, e.g. [17, 18, 20])in terms of the spherical harmonics F L = A X i =1 r Li Y Lm (ˆ ρ i ) (5)with ˆ ρ i = ρ / | ρ | . Assuming that the many-body nuclear density factorizes into a product of the single nucleon WSdensities, one can easily obtain (we ignore a very small effect of the c.m. correlations) h F + L F L i W S = A (2 L + 1) h r L i π . (6)Of course, this formula becomes invalid in the presences of the nucleon correlations. Usually, in the MC simulationsof AA -collisions, the effect of the nuclear correlations is included in the approximation of a hard-core repulsion. Theshort range N N -expulsion somewhat suppresses the mean square quadrupole moment. But this suppression is notvery strong. More important effect on the multipole moments may come from the long range correlations due toquantum collective nuclear excitations.The quantum calculation of the mean square quadrupole moment of the
Pb nucleus can be performed with thehelp of the energy weighted sum rule (EWSR) (for a review, see [23]) for strength function S ( ω ) of the isoscalarquadrupole operator. For the nuclear ground state the strength function of an operator F reads S ( ω ) = X n |h n | F | i| δ ( ω − ω n ) , (7)where ω n = E n − E and E n are the energies of the nucleus states. The ground state expectation value of the operator F + F can be written as h | F + F | i = m , (8)where m is the zeroth order moment of the strength function. For an arbitrary k the moment m k is defined as m k = Z ∞ dωω k S ( ω ) . (9)The ratio m /m characterizes the typical energy of the states excited by the action of the operator F on the groundstate, which is usually called the centroid energy E c . Then, in terms of E c we can write h | F + F | i = m E c . (10)For the case of interest F = F L the moment m can be evaluated accurately using the EWSR, that gives for L ≥ m = AL (2 L + 1) h r L − i πm N , (11)where m N is the nucleon mass. Then from (6) and (10), we obtain for the ratio of the classical to the quantum meansquare moments r = h | F + L F L | i c h | F + L F L | i q = 2 m N E c h r L i L (2 L + 1) h r L − i . (12)In the case of the isoscalar L = 2 operator the EWSR is exhausted by the isoscalar giant quadrupole resonance(ISGQR) with ω q ≈ .
89 MeV and Γ q ≈ E c ≈ . . Using this centroid energy, we obtainfrom (12) for the quadrupole mode r ≈ . . This says that the simple probabilistic treatment of the Pb nucleuswith the factorized WS many-body nuclear density considerably overestimates the 3D-quadrupole fluctuations. Onecan expect that this can lead to incorrect predictions for the 2D-fluctuations of the QGP fireball in the MC simulationof AA -collisions as well. One of the ways to cure this problem is to use in the MC sampling of the nucleon positionsthe nuclear configurations that have the distribution function in the square quadrupole moment (we denote it Q ) ofthe form P sq ( Q ) = rP ( rQ ) , (13)where P is the native distribution function of the squared quadrupole moment for the WS nuclear distribution (i.e.it is calculated without imposing any filter on the nucleon positions). The MC sampling of the nucleon positionswith the squeezed distribution P sq automatically guarantees that the colliding nuclei will have correct mean squarequadrupole moments. . We consider the initial condition for the QGP fireball in Pb+Pb collisions in the central rapidity region ( y = 0).For evaluating the distribution of the entropy density in the transverse plane we use the MCG approach developedin [25, 26]. The MCG scheme of [25, 26] allows to perform calculations describing the nucleon as a one-body stateand accounting for the meson-baryon component of the physical nucleon. This model describes very well the data The strength function is proportional to the imaginary part of the quadrupole polarisability(susceptibility) α q (see Eq. (20) of [22])which satifies the relation α q ( − ω ∗ ) = α ∗ q ( ω ). For this reason there must be used a double Breit-Wigner parametrization with the polesat ± ω q − i Γ q / There was an error in the code used in [22]. This led to underestimating the EWSR mean square quadrupole moment by a factor of ∼ .
95. For this reason the quadrupole contribution in Figs. 2, 3 (dashed lines) of [22] should be multiplied by this factor.
Pb+Pb 2.76 TeV Pb+Pb 5.02 TeVMC with P ( Q ) MC with P sq ( Q ) MC with P ( Q ) MC with P sq ( Q ) ǫ { } ǫ { } ǫ { } /ǫ { } ǫ , { } and the ratio ǫ { } /ǫ { } for central 2 .
76 and 5 .
02 TeV Pb+Pb collisions obtainedwithin the MCG model of [26] with and without (numbers in brackets) the meson-baryon component in the nucleon. For eachenergy the left column shows the results for the sample of nucleon configurations without restrictions on the squared quadrupolemoment Q (i.e. for the native distribution P ( Q ) for the WS nuclear density), and the right one for the sample correspondingto the squeezed distribution P sq ( Q ) (see main text for details). on the centrality dependence of the midrapidity charged particle density in 0 . .
76 TeV Pb+Pb collisions [26]. The theoretical predictions for 5 .
02 TeV Pb+Pb, and 5.44 Xe+Xe collisions [27] arealso in very good agreement with the data. In the present analysis we perform calculations for the versions with andwithout the meson-baryon component. Both the versions lead to very close predictions for the ratio ǫ { } /ǫ { } we areinterested in. Here we briefly sketch the algorithm used in our MCG model for the version without the meson-baryoncomponent (for this case our model is similar to the well known MCG generator GLISSANDO [5]). The interestedreader is referred to [25, 26] for the detailed description of the model and the parameters of the model.We use two-component scheme [28] with two kinds of the entropy sources: corresponding to the wounded nucleons(WN) and to the hard binary collisions (BC). The center of each WN source coincides with the position of the WN.And the center of each BC source is located in the middle between the pair of the colliding nucleons. The suppressionof the probability of hard BC for a given N N -interaction is controlled by the parameter α . The total event entropydensity in the transverse plane is given by ρ s ( ρ ) = N wn X i =1 S wn ( ρ − ρ i ) + N bc X i =1 S bc ( ρ − ρ ′ i ) , (14)where the S wn terms corresponds to the sources for wounded constituents and S bc terms to the binary collisions, N wn and N bc are the number of the WNs and BCs, respectively. The entropy distribution for WN and BC sources arewritten as S wn ( ρ ) = (1 − α )2 s ( ρ ) , S bc ( ρ ) = s ( ρ ) , (15)where for s ( ρ ) we use a Gaussian distribution s ( ρ ) = s exp (cid:0) − ρ /σ (cid:1) /πσ (16)with s the total entropy of the source, and σ width of the source. We perform calculations for σ = 0 . σ .We describe fluctuations of the total entropy for each source by the Gamma distribution. The parameters ofthe Gamma distribution have been adjusted to fit the experimental pp data on the mean charged multiplicity andits variance in the unit pseudorapidity window | η | < . dS/dy = CdN ch /dη , with C ≈ .
67 [29]. In the version with the meson-baryon component of our MCG generator[25, 26] the entropy sources can be produced in BB , M B , and
M M collisions. Both the versions of the model givesimilar predictions for the charged multiplicity. But the optimal values of the parameter α are somewhat smallerfor the version with the meson-baryon component. The fit to the data on centrality dependence of the midrapiditycharged particle density gives α ≈ . .
14) for the versions with(without) the meson-baryon component (see [26, 27]for details).We performed numerical calculations of the rms coefficients h ǫ n i / (they are usually denoted as ǫ n { } ) for n = 2and 3 by MC generation of 5 × central ( b = 0) Pb+Pb collisions at √ s = 2 .
76 and 5 .
02 TeV. The results for boththe versions, with and without the meson-baryon component, are summarized in table I. From table I one can seethat the quantum collective effects for the quadrupole deformations do not affect ǫ { } . But they give a noticeablereduction of ǫ { } . For the quantum version with the meson-baryon component we obtain ǫ { } /ǫ { } ≈ .
8. For theversion without the meson-baryon component we obtain a bit bigger ǫ , . But the change in the ratio ǫ { } /ǫ { } isvery small (it is increased by ∼ . ǫ { } /ǫ { } allows to resolve the v -to- v puzzle in the ultra-central Pb+Pbcollisions. Because the hydrodynamic calculations give k /k ≈ . − . c ∼ < ǫ { } /ǫ { } we obtain v /v ≈ . − . . This is in reasonable agreement with the ALICE measurements [12] for 2.76 and 5.02 TeVPb+Pb collisions that give in the limit c → v /v ≈ . ± . b . Due to fluctuations of the multiplicity (at agiven impact parameter), there is some mismatch/smearing between b and c which experimentally is measured viathe multiplicity. We checked that the effect of this smearing on our predictions is very small. Also, to understand thesensitivity of the results to the form of the squeezed distribution P sq ( Q ) used for filtering the nucleon configurationsin our MCG simulations, we also performed calculations for sampling the nuclear configurations with a sharp cutoff in Q . The cutoff on the squared quadrupole moment has been adjusted to fit the the EWSR mean square quadrupolemoment of the Pb nucleus. This ansatz leads to the value of ǫ { } /ǫ { } which is in perfect agreement with thatfor the ansatz given by (13). This test demonstrates high stability of our predictions for ǫ { } /ǫ { } against thechanges of the P sq ( Q ) distribution. It means that for ǫ { } /ǫ { } the only crucial quantity is the total mean squarequadruple moment of the colliding nuclei.In this preliminary study, we have ignored possible inadequacy of the MCG simulation with the WS density for theoctupole ( L = 3) vibrations of the Pb nucleus. The mean square octupole moment can be defined via the EWSRin the same way as for the quadrupole mode using the ratio of the moments m /m calculated via the experimentaldata on the strength function. However, for the octupole mode the strength function is not exhausted by a singleresonance, but it gets contribution from a broad range of ω . It has peaks at ω ∼ . ω ∼
20 MeV (see,e.g. [24, 37, 38]). From the available experimental data [24, 37, 38] one can conclude that for L = 3 the ratio r given by (12) is close to one or a bit smaller. The octupole strength function calculated in [39] within the randomphase approximation for the Skyrme interaction also leads to r ≈
1. However, of course the determination of r fromthe experimental data seems to be preferable. But experimental uncertainties for contribution of the low and highenergy ω -regions to the EWSR for the octupole mode are rather large. This renders difficult an accurate calculationof the ratio (12). We checked that the scenario with r < L = 3) leads an increase of the value of ǫ { } , andthe ratio ǫ { } /ǫ { } will be somewhat smaller than that obtained in the present analysis. We leave a detailed MCsimulation for this scenario with accounting for filtering for both the quadrupole and octupole moments for future work. . In summary, we have studied, for the first time, the influence of the collective quantum effects in the nuclear wavefunctions on the azimuthal anisotropy coefficients ǫ , in the central Pb+Pb collisions at the LHC energies. We havecompared the predictions for the mean square quadrupole moment of Pb obtained in the classical probabilistictreatment with the WS nuclear distribution with that obtained from the quantum analysis using the EWSR andthe experimental data on the isoscalar giant quadrupole resonance. This analysis shows that the classical treatmentoverestimates the mean square quadrupole moment of the
Pb nucleus by a factor of r ≈ .
2. In our MCGsimulations of Pb+Pb collisions, we cure this problem by sampling the nucleus configurations for the distribution inthe squared quadrupole moment squeezed by the factor r . This guarantees that the colliding nuclei have the meansquare quadrupole moment predicted by the quantum EWSR. We have found that the EWSR version of the MCGsimulation leads to a noticeable reduction of the azimuthal asymmetry ǫ , as compared to the ordinary MC samplingwithout restrictions on the quadrupole moments of the colliding nuclei. The values of ǫ for two versions of theMCG simulations are practically the same. For the EWSR version we obtained ǫ { } /ǫ { } ≈ .
8. This leads to v { } /v { } ≈ . − .
12 (if one adopts the hydrodynamic linear response coefficients k , from [13, 14, 30, 31]),which is in rather good agreement with the data from ALICE [12].In the present analysis we have addressed only the case of the spherical Pb nucleus. However, it is clear that forhigh-energy collisions of the non-spherical nuclei, like
Au+
Au and U+ U, the MCG simulations with theordinary WS density may be inadequate as well. This fact may be important for interpretation of the results of theevent shape engineering, which uses the event multiplicity to select the events with a certain initial system geometry The numbers in table I were obtained for the factorized WS distribution without short range NN -correlations. We also performed theMCG simulation with the hard repulsion for the expulsion radius r c = 0 . . ǫ { } /ǫ { } ≈ .
845 and 0 . v /v which is in rather reasonable agreement with the data.However, one should bear in mind that from the point of view of the entropy production in AA -collisions the real situation with thecontribution to the entropy density of the short range NN -pairs may differ from that in the picture with a big expulsion volume (as, e.g.,in [32]). Say, for a successful dibaryon paradigm of the short range NN -interaction (for reviews, see [34, 35]) the expulsion region is notempty, but occupied by a 6 q -cluster. As in the case of hD -scattering [36], the 6 q -states can participate in the color exchanges betweenthe colliding nuclei and contribute to the entropy generation. For this reason, in reality the effect of the short range NN -configurationsmay be of the opposite sign. (e.g., the tip-tip collisions of the prolate U nuclei as in the STAR experiment [40]).The quantum collective effects discussed in the present analysis may be important for analyses of the data onthe flow effects in Au+Au collisions in future experiments at NICA. In the NICA energy region the critical pointeffects may influence the medium evolution, and accurate treatment of the initial state geometry becomes especiallyimportant.
Acknowledgments
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