Possible octupole deformation of 208 Pb and the ultracentral v 2 to v 3 puzzle
Patrick Carzon, Skandaprasad Rao, Matthew Luzum, Matthew Sievert, Jacquelyn Noronha-Hostler
PPossible octupole deformation of
Pb and the ultracentral v to v puzzle P. Carzon, ∗ S. Rao, † M. Luzum, ‡ M. Sievert,
4, 1, § and J. Noronha-Hostler ¶ Illinois Center for Advanced Studies of the Universe & Department of Physics,University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Department of Physics and Astronomy, Rutgers University, Piscataway, NJ USA 08854 Instituto de F´ısica, Universidade de S˜ao Paulo, C.P. 66318, 05315-970 S˜ao Paulo, SP, Brazil Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA (Dated: September 10, 2020)Recent measurements have established the sensitivity of ultracentral heavy-ion collisions to thedeformation parameters of non-spherical nuclei. In the case of
Xe collisions, a quadrupole defor-mation of the nuclear profile led to an enhancement of elliptic flow in the most central collisions. In
Pb collisions a discrepancy exists in similar centralities, where either elliptic flow is over-predictedor triangular flow is under-predicted by hydrodynamic models; this is known as the v -to- v puz-zle in ultracentral collisions. Motivated by low-energy nuclear structure calculations, we considerthe possibility that Pb nuclei could have a pear shape deformation (octupole), which has theeffect of increasing triangular flow in central PbPb collisions. Using the recent data from ALICEand ATLAS, we revisit the v -to- v puzzle in ultracentral collisions, including new constraints fromrecent measurements of the triangular cumulant ratio v { } /v { } and comparing two differenthydrodynamic models. We find that, while an octupole deformation would slightly improve theratio between v and v , it is at the expense of a significantly worse triangular flow cumulant ratio.In fact, the latter observable prefers no octupole deformation, with β (cid:46) . Pb, and istherefore consistent with the expectation for a doubly-magic nucleus even at top collider energies.The v -to- v puzzle remains a challenge for hydrodynamic models. I. INTRODUCTION
The precision of heavy-ion collisions has reached thepoint that differences in the collective flow can be pre-dicted between beam energies at the level of a few percent[1–3]. While minor model differences remain in terms ofboth the initial conditions and transport coefficients, thevast majority of collaborations can now fit (and predict)the “bread and butter” observables such as charged par-ticle spectra and two-particle azimuthal anisotropies [4–17]. Extractions from the above theoretical models havegenerally converged on a minimum in the shear viscosityto entropy density ratio of η/s ≈ . ± . ζ/s ≈ . ± .
15. While a number of questions still re-main about the exact scale that initial conditions probe[18–22] and the limitations of hydrodynamics in smallsystems [23], a general consensus has emerged within thecommunity about the applicability of event-by-event rel-ativistic viscous hydrodynamics in large AA collisions.Therefore, when there are deviations of experimentalflow data from theoretical calculations in large systems,it generally signifies missing physics that goes beyondthe above paradigm. One recent example was the mea-surement of a significant enhancement in v { } in central Xe Xe collisions compared to expectations that as-sumed a spherical nucleus. However, the
Xe nucleus ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] ¶ Email: [email protected] is expected to have a small quadrupole deformation, andthis enhancement is explained by such an effect [14, 24–27]. Previous studies [28–31] of collisions of
U, whichis known to have a substantial quadrupole deformation,also found an enhancement in the elliptic flow in ultra-central events. In both of these cases the nuclei havea quadrupole β (and a small hexadecapole β ), leadingto an enhanced elliptic shape and hence to an increasein v { } in ultracentral collisions. Additional propos-als include studying the effect of alpha clustering in O [32–34], scanning small deformed ions [35], and exploringpolarized nuclear beams [36, 37]. Recent work discussingthe details of measuring deformations using heavy-ioncollisions can be found in Refs. [38, 39]. Typically themost sensitivity to nuclear deformation occurs in cen-tral collisions; this is because at zero impact parameterthe anisotropy occurs solely due to fluctuations, whilein other centralities the impact region is dominated bythe almond overlap shape. Furthermore, other sourcesof fluctuations which scale with the system size will havethe smallest effect in central collisions where the overlapregion is the largest.Similarly, it has been noted that ultracentral Pb Pb collisions exhibit a relationship between v { } and v { } that is not shown by any existing the-oretical model [40–43]. While theoretical calculationswhich treat the Pb nucleus as spherical predict a nat-ural hierarchy of v { } > v { } , the data instead show v { } ≈ v { } . As a result, the models all either un-derpredict the triangular flow v { } or overpredict theelliptic flow v { } in ultracentral Pb Pb collisions.A number of attempts have been made to resolve thispuzzle by varying the initial conditions [40, 42, 44], the a r X i v : . [ nu c l - t h ] S e p transport coefficients [40, 43], and the equation of state[13], but it still remains a generic feature of models. Inaddition to this discrepancy among the two-particle cor-relations v { } and v { } , it was also pointed out thatthe four-particle correlation v { } also contains discrep-ancies between theory and data in very central collisions[45]. In the four-particle sector, the hydrodynamic mod-els tended to underpredict the ratio v { } /v { } [46–48], which corresponds to overestimating the width ofthe event-by-event fluctuations in v .The reason that the hierarchy v { } > v { } emergesfrom hydrodynamic models in ultracentral collisions isas follows. In heavy ion collisions and especially in cen-tral collisions of large systems, there is a strong linearmapping from the initial-state geometry to the final-statecollective flow. This mapping is quantified by the nearlylinear proportionality between the initial-state eccentric-ity vectors ε n and the final-state flow vectors v n v n = κ n ε n . (1)Non-linear corrections to the hydrodynamic response canbecome significant in more peripheral collisions but arenegligible in the ultracentral region considered here [49–51]. The linear response coefficients κ n thus encodethe entire hydrodynamic evolution and properties of themedium.In ultracentral collisions, where the impact parameter b nearly vanishes, the mean-field geometry of the collisionis perfectly round (at least for spherical nuclei). Then ε = ε = 0 on average, and it is the event-by-eventfluctuations of the geometry which produce both ε and ε , leading to ε { } ≈ ε { } . However, this picture ischanged somewhat by the effects of viscosity. Althoughboth the elliptic and triangular harmonics are compara-ble at level of the initial eccentricities, the relative magni-tudes of the final v , v depend on the size of the responsecoefficients κ , κ , and dissipative effects naturally leadto κ < κ . Thus, initial conditions for round nuclei willgenerically exhibit an ε { } ≈ ε { } relationship, leadingto v { } > v { } in the final state, in disagreement withthe data. To overcome the additional viscous suppres-sion of v in the final state, instead one needs an initialstate with ε < ε .In light of the recent studies on deformed nuclei in cen-tral heavy ion collisions, it’s natural to question if Pbmay also be showing signs of nuclear deformation. Thiswould be somewhat surprising since
Pb is a highlystable nucleus with doubly-magic atomic number; nu-clear structure tables indicate that the octupole defor-mation, β , is zero [52]. However, a recent paper did The values of v { } and v { } depend on the range of transversemomentum, and so one does not always have v { } ≈ v { } .Nevertheless, the dependence is the same in theory and experi-ment, and so the discrepancy remains, regardless of the detailsof the measurement. predict a finite β using a Minimization After Projectionmodel [53]. Additionally, nuclear structure experimentsare performed at significantly lower beam energies – mul-tiple orders of magnitude lower than relativistic heavy ioncollisions — and the structure of the nucleus may evolvefrom low to high energies. Finally, nuclear structure ex-periments only probe the electric charge density and itsassociated geometry, whereas heavy-ion collisions probethe color charge density and its associated geometry. Adifference in the geometry of electric charge from the fullgeometry of nuclear matter, such as a neutron skin [54],could modify the initial geometry relevant for heavy-ioncollisions. It seems plausible then, that the deformationparameters such as β could be different between the elec-tric and color charge densities in Pb. Furthermore, theneutron skin has been a point of interest for both low-energy nuclear collisions [55] and also for neutron stars’mass radius relationship [56–58].In this paper we explore the possibility of an octupoledeformation in
Pb. Since the original papers on the v -to- v puzzle came out, the LHC has also had a sub-sequent run at 5.02 TeV with a higher luminosity, whichhas led to significantly smaller error bars in their exper-imental data. Between updates in the medium effects(e.g. improvements in the equation of state) and thesesmaller error bars, we find that the v to v puzzle inultracentral collisions is significantly smaller than orig-inally thought. Nevertheless, we find that an octupoledeformation does not explain the remaining discrepancy.Further, by using recent v { } /v { } data from ATLAS[59] we can place a limit on the octupule deformation of Pb in our models. The rest of this paper is structuredas follows. In Sec. II we discuss the properties of theinitial state, including a discussion in Sec. II A introduc-ing the deformation parameters β n and the calculation inSec. II B of the second and fourth cumulants of the initial-state eccentricities in two different models. In Sec. III wediscuss the two hydrodynamical models we will be com-paring: v-USPhydro and MUSIC. In Sec. IV we comparethe results of the two hydrodynamic simulations againstdata to assess the impact on the v -to- v puzzle, and wesummarize our conclusions in Sec. V. II. INITIAL STATEA. Octupole deformation in Pb Modern simulations of heavy-ion collisions begin bysampling the position of each nucleon in a nucleus on anevent-by-event basis from a Woods-Saxon distribution,in order to obtain the geometrical structure of the heavyion. Generally the two-parameter Woods-Saxon distri-bution is given in spherical coordinates by ρ ( r, θ ) = ρ (cid:20) (cid:18) r − R ( θ ) a (cid:19)(cid:21) − . (2)The Woods-Saxon profile (2) yields a nearly constantdensity for r < R which transitions into an exponen-tial falloff for r > R . For a deformed nucleus, the radius R = R ( θ ) is not taken to be constant, but becomes afunction of the angle θ , which can be decomposed into acomplete sum of spherical harmonics: R ( θ ) = R (cid:16) β Y ( θ ) + β Y ( θ ) + β Y ( θ ) + · · · (cid:17) . (3)For a spherical nucleus we have β = β = β = 0,with nonzero values induced by a quadrupole, octupole,or hexadecupole deformation, respectively. In principle,the summation (3) of Y (cid:96)m extends over all (cid:96) ∈ [0 , ∞ ),but for our present purposes we consider only the oc-tupole deformation β . Initial condition models will varyin the details of effects beyond the nucleon-level positionsampling, such as color charge fluctuations, constitutentquarks, strings, and minijets; however, since the founda-tion is the Wood-Saxon distribution (2), any deformationhas a relatively generic effect.In Ref. [53], a Minimization After Projection modelwas used to predict the octupole deformation β of Pb,finding β ≈ . β = 0. The authors also cite an expected ±
25% error for the comparison of these deformation pa-rameters between theory and experiment. Thus, fromnuclear structure considerations we expect that the oc-tupole deformation of
Pb should be constrained to β (cid:46) .
05. Here we check a range of β for Pb com-pared to experimental data in order to quantify its effect.In order to generate initial conditions with an octupoledeformation, we have modified the Trento initial con-dition model [60] to include a nonzero β . In Trentowe use the geometric mean p = 0 (so that the entropydensity at proper time τ = 0 . c is proportional to √ T A T B ), fluctuation parameter k = 1 .
6, and nucleonwidth σ = 0 .
51 fm. This setup was obtained from aBayesian analysis of experimental data [11] and has beensuccessful in a number of predictions across systems size[14]. Additionally, we compare this to an alternative ini-tial condition satisfying a linear scaling T R ∝ T A T B dueto the recent discussion in Refs. [20, 61–63], which wehave also modified Trento to include. With linear scal-ing we must also adjust the multiplicity fluctuations andthere we considered k = 20. However, all hydrodynamicsimulations are performed using p = 0 Trento. B. Eccentricities
While the impact parameter in heavy ion collisionsis not directly measurable, proxy measures such as thenumber of produced particles (which is anticorrelatedwith impact parameter) can be used instead. Whenbinned into centrality percentiles, the produced multi-plicity is greatest in the events with complete overlap T A T B β = β = T A T B β = β = = = Centrality (%) ϵ n { } FIG. 1. (Color online) The RMS eccentricities ε n { } = (cid:112) (cid:104) ε n (cid:105) for the elliptic n = 2 and triangular n = 3 harmonics as afunction of collision centrality for √ s NN = 5 .
02 TeV Pb Pbcollisions. Different curves reflect the choices of initial entropydeposition ( √ T A T B vs T A T B ) and octupole deformation β . of the ions and small impact parameter. For the mostcentral collisions (say 0 −
1% centrality), the impact pa-rameter is essentially zero; finer binning into ultra-centralcollisions is additionally sensitive to nuclear deformationand to fluctuations in particle production [35].Since the number of produced particles in the finalstate is controlled by the entropy of the system at freeze-out, centrality binning by entropy is a good approxima-tion to centrality determined by multiplicity. Moreover,due to the low viscosity of the quark-gluon plasma, thetotal entropy is nearly conserved, allowing us to approx-imately determine an event’s centrality directly from theinitial state. Then, in a given event, the eccentricitiescalculated in the center of mass frame are defined as ε n = − (cid:82) r n e inφ s ( r, φ ) rdrdφ (cid:82) r n s ( r, φ ) rdrdφ (4)with s ( r, φ ) the entropy density in polar coordinates. Inthe center of mass frame, ε = 0 identically , and ε , ε quantify the elliptic and triangular shape of the impactregion, respectively. Because of the (anti)correlation be-tween impact parameter and centrality, there is a strongdependence of the ellipticity on centrality: ε is maximalin mid-central collisions where the elliptical shape of theoverlap region is most prominent.Compared to the ellipticity ε , the triangularity ε iszero at a mean-field level, being driven entirely by theevent-by-event fluctuations away from a smooth Woods-Saxon profile (2). As a result, the dependence of ε ver-sus centrality is therefore flatter than for ε , growing only Not to be confused with the dipole asymmetry, which is oftennotated as ε as the transverse size of the collision area decreases, asshown in Fig. 1. In fact, one can vary β quite signifi-cantly without any visible effect in mid-central to periph-eral collisions. Only in central collisions (i.e. 0 − β .In Fig. 2 we study the central collisions in more detailby comparing the ratio ε { } ε { } of the triangularity to el-lipticity. This initial-state quantity is not equal to v { } v { } in the final state because the response coefficients κ n inEq. (4) do not cancel; still a change in eccentricity ra-tio will be reflected directly in the final-state flow ratio.We show results for Pb Pb collisions at both 2.76 TeVand 5.02 TeV for a range of β octupole values. We alsocompare the eccentricities obtained from the Bayesian-preferred geometric mean s ≈ √ T A T B of Trento versusthose of a linear scaling s ≈ T A T B , inspired by CGCexpectations for the mean energy density at early times[20, 61–63]. We find that √ T A T B is slightly more likelyto provide eccentricities compatible with experimentaldata. This is because in the most central collisions thatwe consider here (0 − √ T A T B leads to ε { } > ε { } by 10 − T A T B scaling which leads to a smaller enhancement of only afew percent. As discussed previously, initial conditionswith ε > ε in central collisions are needed in order toobtain the observed v ≈ v after viscous damping of thehigher harmonics. Finally, we note that there does notappear to be a strong beam energy dependence to thisratio, consistent with other studies [51].While the ratio of v { } and v { } has received themost attention, there are also important discrepancies inthe ratio of v { } /v { } predicted by models. In Ref. [45]it was pointed out that v { } v { } ≈ ε { } ε { } (5)due to the nearly-linear response in central collisions. Toensure that this ratio holds, we ran 11,000 events in the0 −
1% centrality class and compared the initial-state ra-tio of eccentricities versus the final-state hydrodynamicflow. We found that the difference between v { } v { } and ε { } ε { } was less than 1%, giving confidence that we cancompare the eccentricity ratio for the initial conditionmodels directly. This assumption is important since thisobservable requires extremely high statistics, on the or-der of 15 million events to cover a wide centrality range,so that running full hydrodynamic simulations would beprohibitive.In Fig. 3 we plot the cumulant ratio ε { } ε { } as β isvaried and find that the larger β , the more this ratiois enhanced. The previous comparisons for this observ-able [45] were for Pb Pb collisions at 2.76 TeV wherethe estimated experimental errors were quite large. Im-portantly, these uncertainties are certainly overestimatedbecause the covariance was not reported by the experi-mental collaborations, so that the uncertainties in the ratio overcount the correlated errors. Comparing the β variations to previous experimental results at 2.76 TeV,we find that no range of β can be eliminated from ε { } ε { } and, in fact, β = 0 .
12 appears to be preferred comparedto ALICE results [23].However, due to the higher luminosity at 5.02 TeVand because the ATLAS experiment did report the com-plete uncertainty for the ratio v { } v { } , this ratio can beused to put constraints on the β parameter. Since inthis ratio the linear response coefficient κ cancels, itis mostly sensitive to the choice of initial state modelsrather than medium properties. In fact, for √ T A T B scal-ing only β = 0 and β = 0 . β ≤ . T A T B scaling con-sistently overpredicts ATLAS data and appears to be sig-nificantly less sensitive to the β deformation parametercompared to √ T A T B . While √ T A T B scaling is not well-motivated from a theoretical perspective, it appears tobetter capture the features of the experimental data.For completeness, in Fig. 4 we also plot the cumulantratio ε { } ε { } for β of 0 and 0.12 for both scalings. From0-20% Centrality there is good agreement to ALICE data[23] from the β = 0 deformations with the higher defor-mation β = 0 .
12 showing some tension with data in verycentral collisions. Above 30% Centrality all models startto deviate significantly from data, reflecting the onset ofnonlinear response in mid-central collisions.
III. HYDRODYNAMICAL MODEL
We study the hydrodynamic response to these initialconditions by performing full simulations, and to en-sure the results are not unique to a particular hydro-dynamic model, we perform calculations within two dif-ferent frameworks for comparison.The first model uses the hydrodynamic code v-USPhydro [65, 66] (Parameter 1), which solves the equa-tions of motion using Smoothed Particle Hydrodynamics.There the shear viscosity to entropy density ratio is takento be a constant η/s = 0 . ζ/s = 0, and the freeze-out temperature is T F O = 150MeV. The equation of state used is WB21/PDG16+ [13],which has been shown before to fit experimental data wellacross system sizes and beam energies [13, 14] with otherpredictions in ArAr and OO systems still remaining to beconfirmed [50]. One crucial emphasis of this model is onthe detailed treatment of feed-down decays: the hydro-dynamic phase is followed by direct decays [67, 68] usingall known resonances from the PDG16+ [69]. The philos-ophy behind this method is to use the most state-of-the-art list of particles, which most closely matches LatticeQCD results [69] but this sacrifices hadronic transportafterwards because many of these resonances decay into3 or 4 bodies, which have not yet been incorporated intoSMASH [70] or URQMD. T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 2.76 TeV T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 5.02 TeV T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 2.76 TeV T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 5.02 TeV
FIG. 2. (Color online) Ratio of ε { } /ε { } for PbPb collisions at 2 .
76 TeV (left) and 5 .
02 TeV (right) for varying ocutpoledeformations β . We also compare two models for the initial-state entropy deposition: s ≈ √ T A T B (top row) and s ≈ T A T B (bottom row). The second hydrodynamic model uses MUSIC [71, 72](Parameter 2), a grid-based code. Here all fluid parame-ters are taken as the Maximum a Posteriori (MAP) valuesfrom the Bayesian analysis of momentum integrated ob-servables reported in Table 5.9 of Ref. [11]. Both η/s ( T )and ζ/s ( T ) depend on temperature, as shown in Fig. 5.The equation of state used is s95p-v1.2 [73]. When thesystem reaches a local temperature of 151 MeV, the fluidis converted to hadrons and resonances, which evolve inthe hadron cascade code UrQMD [74, 75]. These param-eters were tuned for a Trento initial condition of energydensity at τ = 0 followed by a period of free streaming τ fs = 1 .
16 fm/ c , whereas here we use Trento to initializethe entropy density at a finite time τ = 0 . c withno initial transverse flow. Nevertheless, the fit to data atmost centralities is expected to be reasonable [22, 76].In both cases, v-USPhydro and MUSIC pass analyt-ical solutions (the Gubser test) [77] so that any differ-ences in the results should lie not in numerics, but inthe different physical ingredients such as the shear andbulk viscosities, the equation of state, the list of hadrons and resonances, and hadronic rescatterings. Note thatthe hydrodynamic equations of motion are different forv-USPhydro and MUSIC. v-USPhydro uses phenomeno-logical Israel-Stewart and MUSIC used DNMR with thesecond order transport coefficients from [78]. IV. RESULTS
We ran our full hydrodynamic machinery for 5,000-10,000 events in the centrality range of 0 −
10% for PbPb5.02 TeV collisions at the LHC. Because these simulationstake a significant amount of time to run, we only con-sider the geometric mean scenario T R = √ T A T B ( p = 0),which we found in the previous section is more likely tofit experimental data.While CMS produced ultracentral data in LHC Run1, ALICE has significantly more data for Run 2 [23],so here we make the comparisons for the LHC Run 2using 0 . < p T < v { } / v { } for Trento+v-USPhydro+decays T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 2.76 TeV T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 5.02 TeV T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 2.76 TeV T A T B β = β = β = β = Centrality (%) ϵ { } / ϵ { } Lead - Lead Collisions at 5.02 TeV
FIG. 3. (Color online) Ratio of ε { } /ε { } for Pb Pb collisions at 2 .
76 TeV (left) and 5 .
02 TeV (right) for varying octupoledeformations β . We also compare two models for the initial-state entropy deposition: s ≈ √ T A T B (top row) and s ≈ T A T B (bottom row). We compare against data at 2.76 TeV from ALICE [47] and ATLAS [48] and at 5.02 TeV from ALICE [23, 64]. (top panel) and Trento+MUSIC+UrQMD (bottom)along with the ALICE data. We see that, inboth cases a finite β deformation significantly im-proves the fit to experimental data. For Trento+v-USPhydro+decays β = 0 .
075 is the best fit, whereasfor Trento+MUSIC+UrQMD β = 0 .
12 provides a bet-ter fit. We also note that the absolute magnitudes of v { } v { } are smaller in v-USPhydro than in MUSIC, whichcan be attributed to the smaller shear viscosity η/s usedin v-USPhydro. This is because, unlike the ratio v { } v { } of different cumulants of the same harmonic, the ratio v { } v { } has significant a viscosity dependence; v { } ismore strongly damped by viscosity than v { } .Comparisons of the ratio v { } / v { } shown in Fig.6 are one important element of the v -to- v puzzle, butit is also essential to compare the absolute magnitudesof both quantities as well. These comparisons are shownin Fig. 7 for the spherical case β = 0. Generally, bothTrento+MUSIC and Trento+v-USPhydro fit v { } wellfor centralities larger than 10% centrality, but in the cen- tral collisions shown here, they both overpredict the ex-perimental data. Meanwhile, Trento+MUSIC managesto fit v { } quite well from 0 −
10% centrality whileTrento+v-USPhydro overpredicts the data. Thus, basedon these two representative examples of hydrodynamicmodels, it appears that the crux of the v -to- v puzzlemay lie in the overprediction of v { } rather than an un-derprediction of v { } . From this perspective, it seemsunlikely that boosting the triangularity ε by adding anoctupole deformation β will improve the agreement withdata.Indeed, this is what’s seen in Fig. 8 when we considerthe impact of the β deformation. While a larger β gen-erally leads to a larger v in most cases, there appear tobe some non-monotonic fluctuations, and the β defor-mation can non-trivially affect v as well.As was noted previously in Sec. II B and plotted inFig. 3, increasing the octupole deformation β does notimprove the agreement of the calculated ε { } / ε { } with data. Instead, the increasing disagreement withthe data allows us to set an upper bound before this T A T B β = β = T A T B β = β = Centrality (%) ϵ { } / ϵ { } FIG. 4. (Color online) Ratio of ε { } /ε { } for PbPb colli-sions at 5 .
02 TeV for varying octupole deformations, β , and √ T A T B and T A T B scaling. We compare against data at 2.76TeV from ATLAS [48]. / s MUSICv-USPhydro0.10 0.15 0.20 0.25 0.30T(GeV)0.000.020.04 / s FIG. 5. Comparison of the bulk and shear viscosities enteringthe two hydrodynamic models using MUSIC and v-USPhydro. ratio becomes incompatible with the data. Thus anyimprovement caused by β in the ratio v { } / v { } inFig. 6 must be weighed against the deterioration of theagreement of v { } / v { } . We illustrate this in Fig. 9by plotting the two observables simultaneously on inde-pendent axes. There are a number of caveats to thiscomparison which should be immediately pointed out.The “experimental” point here represents a combina-tion of the ATLAS data for v { } / v { } [48] and theALICE data for v { } / v { } [23, 64]. On the theoryside, the points represent combinations of the initial-state ε { } / ε { } (without running hydrodynamics) with the spherical Pb ( β = ) deform Pb ( β = ) deform Pb ( β = ) ● ● ● ● ● ● ● ● ● ●● ALICE 5.02TeVtrento + v - USPhydro + decays % v { } / v { } spherical Pb ( β = ) Pb ( β = ) Pb ( β = ) trento + MUSIC + UrQMD ● ● ● ● ● ● ● ● ● ●●
ALICE 5.02TeV
Centrality % v { } / v { } FIG. 6. (Color online) Ratio of v { } /v { } in PbPb 5.02TeVcollisions compared to experimental data from ALICE [23,64]. full final-state evaluation of v { } / v { } for differentvalues of the deformation β and for the two hydrody-namic models. Despite these caveats, the plot clearlyillustrates the inability of an octupole deformation β toresolve the v -to- v puzzle: the improved agreement in v { } / v { } is more than offset by the worsened agree-ment in v { } / v { } . V. CONCLUSIONS
By this point, a number of possibilities to resolve the v -to- v puzzle have been explored and exhausted. Vari-ations in the initial conditions (assuming spherical Pbnuclei) [42], the transport coefficients [43], and the equa-tion of state [13] have all been considered, yet the prob-lem still remains. No initial condition models assumingspherical Pb ions managed to capture the triangular flowfluctuations in ultra central PbPb collisions despite beingable to reproduce elliptic flow fluctuations [45].In this paper, we considered the possibility of a nonzerooctupole deformation β in Pb as a means of im-proving the mismatch in the ratio v { } / v { } in data.Those results from full hydrodynamics (Fig. 6) and trento + MUSICtrento + v - USPhydro β = Centrality (%) v { } trento + MUSICtrento + v - USPhydroALICE
Centrality (%) v { } FIG. 7. (Color online) Comparison of v { } and v { } in PbPb 5.02TeV collisions from v-USPhydro and MUSIC compared toexperimental data from ALICE [23, 64]. β = β = β = = = Centrality (%) v n { } β = β = β = - USPhydron = = Centrality (%) v n { } FIG. 8. (Color online) Absolute values of v { } and v { } in PbPb 5.02TeV collisions from v-USPhydro and MUSICcompared to experimental data from ALICE [23, 64] varying β . the cumulant ratio ε { } / ε { } from the initial state(Fig. 3) are the main results of this work. We findthat, while a finite β does improve the agreement in v { } / v { } with data, it simultaneously worsens theagreement in v { } / v { } . Moreover, we have also per- Parameter 1Parameter 2Experimental β = β = β = ν { }/ ν { } ϵ { } / ϵ { } FIG. 9. Influence of the octupole deformation β on mul-tiple observables. Parameter 1 is the ratio of ε { } / ε { } from Trento initial conditions with ε { } / ε { } the param-eter tune from vUSPhydro. Parameter 2 is the ratio of ε { } / ε { } from Trento initial conditions with ε { } / ε { } with the parameter tune from MUSIC. Clearly adding anonzero β does not improve the overall agreement with data.Details of this plot are discussed in the text. formed direct comparisons to the absolute magnitudes of v { } and v { } , rather than just their ratio (Fig. 8).We find that, in both hydrodynamic models, the biggestdiscrepancy underlying the v -to- v puzzle lies with anoverprediction of v { } rather than the underpredictionof v { } .Clearly, an octupole deformation of Pb is not a vi-able resolution of the v -to- v puzzle in ultracentral col-lisions within these models. While different in the de-tails of their implementation and emphasizing differentphysics, both v-USPhydro and MUSIC are examples ofreasonable hydrodynamic models that can describe bulkflow observables in heavy ion collisions. The two hydro-dynamic models differ in various details, but in particularthe larger viscosity in MUSIC (see Fig. 5) leads to notice-able differences. That larger viscosity creates additionalviscous suppression of higher harmonics like v , bringingMUSIC into better quantitative agreement with v { } than v-USPhydro, which overpredicts the data.This difference arising from the viscosity illustrates animportant correlation between the parameters like η/s entering the hydrodynamics models, and parameters like β entering the initial state. By adjusting β , one candial the ratio v { } / v { } but not control the absolutemagnitudes of either quantity. Those magnitudes are af-fected directly by the choice of viscosity, in correlationwith the prior choice of β . Thus it seems reasonablethat a simultaneous fit of the unknown properties of theinitial state (such as the deformation parameters β , β )and the hydrodynamic transport parameters could helpto resolve the v -to- v puzzle.For the hydrodynamic models, clearly the choice ofviscosity has a significant effect on the v -to- v puzzle,possibly indicating that a more sophisticated treatmentof out-of-equilibrium effects could be important. Addinga temperature-dependent η/s to the v-USPhydro modelcould also be an important improvement.Following our analysis, we stress that studying the v -to- v puzzle only through the ratio v { } v { } can be highlymisleading. Independent observables, like the multipar-ticle correlations v { } v { } , are vital to constrain the originof these discrepancies. Additionally, it is critical to sep-arately examine the absolute magnitudes of v { } and v { } as the baseline for the ratio v { } v { } . For now, the v -to- v puzzle in ultracentral Pb Pb collisions remainsa challenge for hydrodynamic models. ACKNOWLEDGEMENTS
J.N.H., M.S., and P.C. acknowledge support from theUS-DOE Nuclear Science Grant No. DE-SC0019175, theAlfred P. Sloan Foundation, and the Illinois CampusCluster, a computing resource that is operated by theIllinois Campus Cluster Program (ICCP) in conjunctionwith the National Center for Supercomputing Applica-tions (NCSA) and which is supported by funds from theUniversity of Illinois at Urbana-Champaign. M.L. ac-knowledges support by FAPESP projects 2016/24029-6,2017/05685-2 and 2018/24720-6 and by project INCT-FNA Proc. No. 464898/2014-5.
Appendix A: Other Observables1. Symmetric Cumulant εNSC (3 , As part of the v -to- v puzzle, it is natural to askwhether the correlation between these quantities, thesymmetric cumulant N SC (3 , εN SC (3 ,
2) among the eccentrici- (cid:1) = (cid:1) = (cid:1) = (cid:1) = - % (cid:2) N S C ( , ) ( ) Centrality (%)
FIG. 10. (Color online) Normalized symmetric cumulants εNSC (3 ,
2) of the initial-state eccentricities in central col-lisions for PbPb 5.02TeV for varying octupole deformations, β . ties ε and ε , which was shown in Refs. [12, 79, 80] tobe closely related to the final-state N SC (3 , N SC (3 ,
2) in Fig. 10for a range of octupole deformations β . Even with 3million events we find that all β variations are indistin-guishable within the statistical error bars. Thus, we findthat N SC (3 ,
2) is not an observable that is sensitive todeformations.
2. Linear Response Pearson Coefficients Q n It is clear from the discussion of Fig. 8 that v-USPhydro and MUSIC produce different final flow har-monics. This is due to differences in their “best fit”medium parameters — primarily viscosity, although theyalso differ slightly in other respects, such as the equationof state as well. In ultracentral collisions, both modelsare well described by linear response theory, with themodel differences yielding different constants of propor-tionality. Aside from this, the two models also may re-ceive small contributions from nonlinear response, andthese contributions can be different. The deviations fromlinear response are encapsulated in the linear (Pearson)correlation coefficients Q n between initial eccentricity ε n and the final flow harmonic v n .We find in Fig. 11 that the MUSIC simulation hasa very strong linear relation (4), event-by-event for v ,but a weaker linear correlation for v . Conversely, thev-USPhydro show a similar linear correlation for bothharmonics. Without a dedicated study, it is difficult tofurther disentangle how the various features of the modelsresult in the overall mapping from initial state eccentric-ities to final state flow harmonics. REFERENCES MUSIC β = β = - USPhydro β = - USPhydro β = n Q n FIG. 11. (Color online) Pearson coefficients for v and v fromMUSIC and v-USPhydro depending on the deformation.[1] J. Noronha-Hostler, M. Luzum, and J.-Y. Ollitrault,Phys. Rev. C93 , 034912 (2016), arXiv:1511.06289 [nucl-th].[2] H. Niemi, K. J. Eskola, R. Paatelainen, and K. Tuomi-nen, Phys. Rev.
C93 , 014912 (2016), arXiv:1511.04296[hep-ph].[3] J. Adam et al. (ALICE), Phys. Rev. Lett. , 222302(2016), arXiv:1512.06104 [nucl-ex].[4] H. Song, S. A. Bass, U. Heinz, T. Hirano, and C. Shen,Phys. Rev. Lett. , 192301 (2011), [Erratum: Phys.Rev. Lett.109,139904(2012)], arXiv:1011.2783 [nucl-th].[5] P. Bozek and I. Wyskiel-Piekarska, Phys. Rev.
C85 ,064915 (2012), arXiv:1203.6513 [nucl-th].[6] F. G. Gardim, F. Grassi, M. Luzum, and J.-Y. Ollitrault,Phys. Rev. Lett. , 202302 (2012), arXiv:1203.2882[nucl-th].[7] P. Bozek and W. Broniowski, Phys. Rev.
C88 , 014903(2013), arXiv:1304.3044 [nucl-th].[8] H. Niemi, K. J. Eskola, and R. Paatelainen, Phys. Rev.
C93 , 024907 (2016), arXiv:1505.02677 [hep-ph].[9] S. Ryu, J. F. Paquet, C. Shen, G. S. Denicol, B. Schenke,S. Jeon, and C. Gale, Phys. Rev. Lett. , 132301(2015), arXiv:1502.01675 [nucl-th].[10] S. McDonald, C. Shen, F. Fillion-Gourdeau, S. Jeon,and C. Gale, Phys. Rev.
C95 , 064913 (2017),arXiv:1609.02958 [hep-ph].[11] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu,and U. Heinz, Phys. Rev.
C94 , 024907 (2016),arXiv:1605.03954 [nucl-th].[12] F. G. Gardim, F. Grassi, M. Luzum, andJ. Noronha-Hostler, Phys. Rev.
C95 , 034901 (2017),arXiv:1608.02982 [nucl-th].[13] P. Alba, V. Mantovani Sarti, J. Noronha, J. Noronha-Hostler, P. Parotto, I. Portillo Vazquez, and C. Ratti,Phys. Rev.
C98 , 034909 (2018), arXiv:1711.05207 [nucl-th].[14] G. Giacalone, J. Noronha-Hostler, M. Luzum, andJ.-Y. Ollitrault, Phys. Rev.
C97 , 034904 (2018),arXiv:1711.08499 [nucl-th].[15] K. J. Eskola, H. Niemi, R. Paatelainen, and K. Tuomi- nen, Phys. Rev.
C97 , 034911 (2018), arXiv:1711.09803[hep-ph].[16] R. D. Weller and P. Romatschke, Phys. Lett.
B774 , 351(2017), arXiv:1701.07145 [nucl-th].[17] B. Schenke, C. Shen, and P. Tribedy, Phys. Rev.
C99 ,044908 (2019), arXiv:1901.04378 [nucl-th].[18] F. G. Gardim, F. Grassi, P. Ishida, M. Luzum, P. S. Ma-galhes, and J. Noronha-Hostler, Phys. Rev.
C97 , 064919(2018), arXiv:1712.03912 [nucl-th].[19] J. Noronha-Hostler, J. Noronha, and M. Gyulassy, Phys.Rev.
C93 , 024909 (2016), arXiv:1508.02455 [nucl-th].[20] J. L. Nagle and W. A. Zajc, Phys. Rev.
C99 , 054908(2019), arXiv:1808.01276 [nucl-th].[21] M. Hippert, J. G. P. Barbon, D. Dobrigkeit Chinel-lato, M. Luzum, J. Noronha, T. Nunes da Silva, W. M.Serenone, and J. Takahashi, (2020), arXiv:2006.13358[nucl-th].[22] T. Nunes da Silva, D. Chinellato, M. Hippert,W. Serenone, J. Takahashi, G. S. Denicol, M. Luzum,and J. Noronha, (2020), arXiv:2006.02324 [nucl-th].[23] S. Acharya et al. (ALICE), Phys. Rev. Lett. , 142301(2019), arXiv:1903.01790 [nucl-ex].[24] S. Acharya et al. (ALICE), Phys. Lett.
B784 , 82 (2018),arXiv:1805.01832 [nucl-ex].[25] S. Acharya et al. (ALICE), Phys. Lett.
B788 , 166 (2019),arXiv:1805.04399 [nucl-ex].[26] C. Collaboration (CMS), (2018).[27] T. A. collaboration (ATLAS), CERN (CERN, Geneva,2018).[28] L. Adamczyk et al. (STAR), Phys. Rev. Lett. ,222301 (2015), arXiv:1505.07812 [nucl-ex].[29] A. Goldschmidt, Z. Qiu, C. Shen, and U. Heinz, Phys.Rev.
C92 , 044903 (2015), arXiv:1507.03910 [nucl-th].[30] P. Dasgupta, R. Chatterjee, and D. K. Srivastava, Phys.Rev.
C95 , 064907 (2017), arXiv:1609.01949 [nucl-th].[31] G. Giacalone, Phys. Rev.
C99 , 024910 (2019),arXiv:1811.03959 [nucl-th].[32] S. H. Lim, Q. Hu, R. Belmont, K. K. Hill, J. L. Nagle,and D. V. Perepelitsa, Phys. Rev.
C100 , 024908 (2019),arXiv:1902.11290 [nucl-th]. [33] Z. Citron et al. , in Report on the Physics at theHL-LHC,and Perspectives for the HE-LHC , edited byA. Dainese, M. Mangano, A. B. Meyer, A. Nisati,G. Salam, and M. A. Vesterinen (2019) pp. 1159–1410,arXiv:1812.06772 [hep-ph].[34] M. Rybczyski and W. Broniowski, Phys. Rev.
C100 ,064912 (2019), arXiv:1910.09489 [hep-ph].[35] J. Noronha-Hostler, N. Paladino, S. Rao, M. D. Sievert,and D. E. Wertepny, (2019), arXiv:1905.13323 [hep-ph].[36] P. Bozek and W. Broniowski, Phys. Rev. Lett. ,202301 (2018), arXiv:1808.09840 [nucl-th].[37] W. Broniowski and P. Boek, (2019), arXiv:1906.09045[nucl-th].[38] G. Giacalone, (2019), arXiv:1910.04673 [nucl-th].[39] G. Giacalone, (2020), arXiv:2004.14463 [nucl-th].[40] M. Luzum and J.-Y. Ollitrault,
Proceedings, 23rd Inter-national Conference on Ultrarelativistic Nucleus-NucleusCollisions : Quark Matter 2012 (QM 2012): Washing-ton, DC, USA, August 13-18, 2012 , Nucl. Phys.
A904-905 , 377c (2013), arXiv:1210.6010 [nucl-th].[41] C. Collaboration (CMS), (2012).[42] C. Shen, Z. Qiu, and U. Heinz, Phys. Rev.
C92 , 014901(2015), arXiv:1502.04636 [nucl-th].[43] J.-B. Rose, J.-F. Paquet, G. S. Denicol, M. Luzum,B. Schenke, S. Jeon, and C. Gale,
Proceedings, 24thInternational Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark Matter 2014): Darmstadt,Germany, May 19-24, 2014 , Nucl. Phys.
A931 , 926(2014), arXiv:1408.0024 [nucl-th].[44] F. Gelis, G. Giacalone, P. Guerrero-Rodrguez, C. Mar-quet, and J.-Y. Ollitrault, (2019), arXiv:1907.10948[nucl-th].[45] G. Giacalone, J. Noronha-Hostler, and J.-Y. Ollitrault,Phys. Rev.
C95 , 054910 (2017), arXiv:1702.01730 [nucl-th].[46] S. Chatrchyan et al. (CMS), Phys. Rev.
C89 , 044906(2014), arXiv:1310.8651 [nucl-ex].[47] K. Aamodt et al. (ALICE), Phys. Rev. Lett. , 032301(2011), arXiv:1105.3865 [nucl-ex].[48] G. Aad et al. (ATLAS), Eur. Phys. J.
C74 , 3157 (2014),arXiv:1408.4342 [hep-ex].[49] J. Noronha-Hostler, L. Yan, F. G. Gardim, andJ.-Y. Ollitrault, Phys. Rev.
C93 , 014909 (2016),arXiv:1511.03896 [nucl-th].[50] M. D. Sievert and J. Noronha-Hostler, Phys. Rev.
C100 ,024904 (2019), arXiv:1901.01319 [nucl-th].[51] S. Rao, M. Sievert, and J. Noronha-Hostler, (2019),arXiv:1910.03677 [nucl-th].[52] P. Mller, A. J. Sierk, T. Ichikawa, and H. Sagawa,Atom. Data Nucl. Data Tabl. , 1 (2016),arXiv:1508.06294 [nucl-th].[53] L. M. Robledo and G. F. Bertsch, Phys. Rev.
C84 ,054302 (2011), arXiv:1107.3581 [nucl-th].[54] C. M. Tarbert et al. , Phys. Rev. Lett. , 242502 (2014),arXiv:1311.0168 [nucl-ex].[55] S. Abrahamyan et al. , Phys. Rev. Lett. , 112502(2012), arXiv:1201.2568 [nucl-ex].[56] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. , 5647 (2001), arXiv:astro-ph/0010227 [astro-ph].[57] F. J. Fattoyev and J. Piekarewicz, Phys. Rev. C86 ,015802 (2012), arXiv:1203.4006 [nucl-th].[58] F. J. Fattoyev, J. Piekarewicz, and C. J. Horowitz, Phys.Rev. Lett. , 172702 (2018), arXiv:1711.06615 [nucl-th].[59] M. Aaboud et al. (ATLAS), JHEP , 051 (2020),arXiv:1904.04808 [nucl-ex].[60] J. S. Moreland, J. E. Bernhard, and S. A. Bass, Phys.Rev. C92 , 011901 (2015), arXiv:1412.4708 [nucl-th].[61] T. Lappi, Phys. Lett. B , 11 (2006), arXiv:hep-ph/0606207.[62] P. Romatschke and U. Romatschke,
Relativistic FluidDynamics In and Out of Equilibrium , Cambridge Mono-graphs on Mathematical Physics (Cambridge UniversityPress, 2019) arXiv:1712.05815 [nucl-th].[63] G. Chen, R. J. Fries, J. I. Kapusta, and Y. Li, Phys.Rev. C , 064912 (2015), arXiv:1507.03524 [nucl-th].[64] S. Acharya et al. (ALICE), JHEP , 103 (2018),arXiv:1804.02944 [nucl-ex].[65] J. Noronha-Hostler, G. S. Denicol, J. Noronha, R. P. G.Andrade, and F. Grassi, Phys. Rev. C , 044916 (2013),arXiv:1305.1981 [nucl-th].[66] J. Noronha-Hostler, J. Noronha, and F. Grassi, Phys.Rev. C , 034907 (2014), arXiv:1406.3333 [nucl-th].[67] J. Sollfrank, P. Koch, and U. W. Heinz, Z. Phys. C ,593 (1991).[68] U. A. Wiedemann and U. W. Heinz, Phys. Rev. C ,3265 (1997), arXiv:nucl-th/9611031.[69] P. Alba et al. , Phys. Rev. D96 , 034517 (2017),arXiv:1702.01113 [hep-lat].[70] A. Ono et al. , Phys. Rev. C , 044617 (2019),arXiv:1904.02888 [nucl-th].[71] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. C ,014903 (2010), arXiv:1004.1408 [hep-ph].[72] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. Lett. ,042301 (2011), arXiv:1009.3244 [hep-ph].[73] P. Huovinen and P. Petreczky, Nucl. Phys. A , 26(2010), arXiv:0912.2541 [hep-ph].[74] S. Bass et al. , Prog. Part. Nucl. Phys. , 255 (1998),arXiv:nucl-th/9803035.[75] M. Bleicher et al. , J. Phys. G , 1859 (1999), arXiv:hep-ph/9909407.[76] T. Nunes da Silva, D. Dobrigkeit Chinellato, R. Der-radi De Souza, M. Hippert, M. Luzum, J. Noronha, andJ. Takahashi, MDPI Proc. , 5 (2019), arXiv:1811.05048[nucl-th].[77] H. Marrochio, J. Noronha, G. S. Denicol, M. Luzum,S. Jeon, and C. Gale, Phys. Rev. C , 014903 (2015),arXiv:1307.6130 [nucl-th].[78] G. Denicol, S. Jeon, and C. Gale, Phys. Rev. C ,024912 (2014), arXiv:1403.0962 [nucl-th].[79] J. Adam et al. (ALICE), Phys. Rev. Lett. , 182301(2016), arXiv:1604.07663 [nucl-ex].[80] G. Giacalone, L. Yan, J. Noronha-Hostler, and J.-Y. Olli-trault, Phys. Rev. C94