Gluonic Hot Spot Initial Conditions in Heavy-Ion Collisions
GGluonic Hot Spot Initial Conditions in Heavy-Ion Collisions
R. Snyder, M. Byres, S.H. Lim, and J.L. Nagle University of Colorado, Boulder, Colorado 80309, USA Pusan National University, Busan, 46241, South Korea (Dated: August 21, 2020)The initial conditions in heavy-ion collisions are calculated in many different frameworks. Theimportance of nucleon position fluctuations within the nucleus and sub-nucleon structure has beenestablished when modeling initial conditions for input to hydrodynamic calculations. However,there remain outstanding puzzles regarding these initial conditions, including the measurementof the near equivalence of the elliptical v and triangular v flow coefficients in ultra-central 0-1%Pb+Pb collisions at the LHC. Recently a calculation termed magma incorporating gluonic hot spotsvia two-point correlators in the Color Glass Condensate framework, and no nucleons, provided asimultaneous match to these flow coefficients measured by the ATLAS experiment, including inultra-central 0-1% collisions. Our calculations reveal that the magma initial conditions do notdescribe the experimental data when run through full hydrodynamic sonic simulations or when thehot spots from one nucleus resolve hot spots from the other nucleus, as predicted in the Color GlassCondensate framework. We also explore alternative initial condition calculations and discuss theirimplications. I. INTRODUCTION
The physics underlying the first fraction of a fm/ c inheavy-ion collisions is of fundamental interest in its ownright, while also a necessary input in order to extractproperties of the created quark-gluon plasma (QGP) thatevolves from this initial state [1, 2]. There are innu-merable modelings of the said initial state ranging fromclaimed ab initio calculations to phenomenological pa-rameterizations [3–5]. A major advance in the field morethan a decade ago was the incorporation of nucleon posi-tion fluctuations via Monte Carlo Glauber calculations [5]and subsequently the realization that odd flow coeffi-cients would be non-zero [6]. Most recently it has be-come clear that sub-nucleon structure is necessary to un-derstand data in proton-proton and proton-nucleus col-lisions [2, 7], as well as collisions of deformed nuclei suchas Uranium-Uranium [8].Monte Carlo Glauber code including nucleon and con-stituent quarks is now publicly available [9]. Such calcu-lations have been incorporated into multiple frameworks,including the often used trento [10] model. In thisframework, each incoming nucleon or sub-nucleon is mod-eled via a two-dimensional Gaussian distribution and thedeposited energy is proportional to the square root of thelocal projectile density times the local target density (inthe trento p = 0 mode). Another such framework is ip-jazma [11], where the deposited energy can be cho-sen as the product of local projectile density times thelocal target density or the square root, as in the trento model. Both examples are purely phenomenological; infact, the trento model has been used in Bayesian anal-yses in an attempt to constrain the initial state parame-ters [12]. A recent comparison of these scalings for mul-tiplicity distributions is detailed in Ref. [13].In contrast, in the weakly-coupled limit, one can inprinciple calculate the initial conditions in the so-calledColor Glass Condensate (CGC) framework (also referred to as the saturation framework) – for useful reviews seeRefs. [14–16]. Although the calculation is termed ab ini-tio , it is an effective theory in the limit as the coupling α s goes to zero and for high gluon occupation number, andthus its applicability in the heavy-ion collision regimeat RHIC and the LHC is unclear. Regardless, withinthis framework one can assume the projectile and tar-get nuclear color charge densities are described by a lo-cal saturation scale Q s and then the deposited energy isproportional to the product of the projectile and targetcolor charge densities [2, 17]. It is notable that as de-rived in Ref. [2], this simple product is also the result forthe deposited energy in the strongly-coupled limit. The ip-glasma code [18] provides a Monte Carlo frameworkfor the calculation of initial conditions via this CGC ef-fective theory. The calculation starts with Monte CarloGlauber with nucleons or sub-nucleons and then asso-ciates a local saturation scale with a two-dimensionalGaussian distribution for each. Additional color chargefluctuations are included on the scale of the lattice spac-ing within the calculation. Finally, the deposited energyis calculated. The ip-glasma code also time evolves theinitial color distribution using the Yang-Mills equationsof motion, and this moderates the dependence of the ad-ditional color charge fluctuations on the lattice spacing.The ip-glasma initial conditions have been successfulat matching experimental flow data when used as inputto viscous hydrodynamic calculations – see for exampleRefs. [19, 20].The ip-jazma phenomenological model [11] was con-structed to specifically evaluate initial conditions fromthe MSTV calculations for small collisions systems whichare calculated in the so-called dilute-dense limit of theCGC framework [21, 22]. ip-jazma can also calculate ini-tial conditions as the simple product of two-dimensionaltarget and projectile Gaussian distributions, the so-calleddense-dense limit. These calculations provide an almostidentical match to the energy deposit initial conditions a r X i v : . [ nu c l - t h ] A ug from the full ip-glasma framework – see details in Ap-pendix A. This agreement emphasizes that the key in-gredients for the initial geometry are nucleons or sub-nucleons, Gaussian profiles, and taking the local productof these Gaussians. Features in the ip-glasma modelfrom color domains or “spiky” local fluctuations are sub-dominant, and thus not confirmed by agreement withflow data. We highlight that the ip-glasma model alsocalculates the early pre-hydrodynamic time evolution,often up to τ = 0 . /c , and this is not modeledin ip-jazma or trento . The evaluation of this pre-hydrodynamic time evolution and its apples-to-applescomparison with free streaming or strongly-coupled dy-namics is a topic for another paper.A new approach was recently put forward also withinthe CGC framework, termed magma [23]. ∗ A recentanalysis comparing various models including magma isgiven in Ref. [24]. In the following sections we (a) detailthe magma calculation and reproduce their results, (b)show results from magma initial conditions run throughfull hydrodynamic sonic simulations, (c) show how the magma results change if hot spots from one nucleus in-teract with hot spots from the other nucleus – which isnot the default in the magma framework, and finally (d)detail results from alternative initial condition calcula-tions.
II.
MAGMA
CALCULATION
In the magma framework, each nucleus is modeled as atwo-dimensional profile of color charge density calculatedwithin the CGC framework. The density is built from lo-calized color charges. What is notable is that these colorcharges are distributed without any modeling of nucle-ons, and are only bounded by the total size of the nucleus,characterized by the Woods-Saxon parameters for a Pbnucleus ( R = 6 .
62 fm and a = 0 .
55 fm). The numberof localized color charges used in the magma calculationis approximately 100 per nuclei, and thus about half thenumber of nucleons. The authors highlight that this isdistinct from the ip-glasma calculation, where they firstdistribute the 208 nucleons from the Pb nucleus, and thencalculate the saturation momentum depending on the nu-cleon positions. It is unclear what physics justificationallows for neglecting nucleons for heavy-ion collisions atthese energies.Another distinguishing feature is that what is calcu-lated in magma is the energy deposit from localizedcolor charges from the projectile Pb nucleus striking asmooth target nucleus, and then linearly summing the ∗ In the process of finalizing this manuscript, the magma authors,Ref. [23], pointed out a potential problem with the CGC correla-tor used in the model. This issue is under investigation by thoseauthors.
FIG. 1. A schematic display of the initial energy depositioncalculation. The hot spots in nucleus A are multiplied bythe smooth distribution of nucleus B, labeled as B WS andsummed with the inverse, hot spots in nucleus B multipliedby the smooth distribution of nucleus A. The resulting energyspatial distribution is shown on the top right plot. In contrast,the bottom right plot shows the A × B calculation using themodified magma code. energy deposit from localized color charges from the tar-get Pb nucleus striking a smooth projectile Pb nucleus.This is nicely visualized in Figure 1 from the magma pa-per [23] – we have regenerated a version of this represen-tation here as Figure 1, with the resulting energy depositshown in the upper right panel. Each interaction createsa sharply peaked energy deposit that decreases as thedistance squared. The calculation reproduced the one-point and two-point functions of the energy density fieldcalculation in the CGC effective theory [25]. We thuslabel the magma calculation as “ A × B W S + B × A W S ”,which is in striking contrast from the ip-glasma “ A × B ”calculation for the local energy density with two nuclei A and B , with the resulting energy deposit shown in thelower right panel of Figure 1.Using the publicly available Python code from the magma authors, we have reproduced their main result,as shown in Figure 2. The top panel shows the Pb+Pbenergy deposit distribution in arbitrary units. The dis-tribution is dividing into percentiles and then the geo-metric eccentricities are calculated within the individualcentrality selections. This procedure is not identical tothe method of centrality selection in experiment, thoughwe expect this to have negligible impact on our conclu-sions. The second- and fourth- cumulants are calculatedfor the n = 2 and n = 3 flow harmonics as follows. ε { } = (cid:113) (cid:104) ε (cid:105) (1) ε { } = (2 (cid:10) ε (cid:11) − (cid:10) ε (cid:11) ) / (2) ε { } = (cid:113) (cid:104) ε (cid:105) (3)In order to compare with experimental data, one takesadvantage of the fact that hydrodynamics gives an ap-proximately linear relationship between the final flow co-efficient and the initial spatial anisotropy (e.g. v { } = κ × ε { } , v { } = κ × ε { } , and v { } = κ × ε { } ).We note that this ignores potential contributions fromnon-linear response [26], a point we will discuss in thenext section. The κ , κ values depend in detail on theQGP properties such as the shear viscosity to entropydensity ratio ( η /S) and the treatment of hadronic re-scattering after hydrodynamic expansion. However, onecan assume that these values to vary modestly with col-lision centrality, and thus they are fitted to experimentaldata after which one can examine the centrality depen-dence. We highlight that the κ values are numericallydetermined by matching the experimental data at cen-trality = 20, and this is done in a consistent manner inthe later comparisons in the paper. A single value of κ determines the scaling for both the v { } and v { } .Values for κ = 0 .
32 and κ = 0 .
31 are obtained, in goodagreement with the numbers quoted in Ref. [23].Figure 2 (above) also shows the experimental data for v { } , v { } , and v { } as measured by the ATLASexperiment [27]. The ATLAS measurements are fromcharged particles in 0 . < p T < | η | < . magma results andthe experimental data is excellent, noting particularlythe splitting between v { } and v { } . Also remarkableis the agreement with both v { } and v { } up to themost central 0-1% Pb+Pb collisions. Figure 3 shows theratio of κ × ε { } / κ × ε { } as a function of collisioncentrality for both data and the magma calculation. Theratio in data (and from magma ) approaches unity whichencapsulates the ultra-central flow puzzle.In the limit of impact parameter b = 0 Pb+Pb colli-sions, the average geometry is circularly symmetric andall spatial anisotropies are zero, i.e. ε n = 0. How-ever, with random fluctuations, for example from nu-cleon position fluctuations, one obtains non-zero eccen-tricities but where all moments are approximately equal,i.e. ε ≈ ε ≈ ε ≈ ... ≈ ε n [28]. However, even inthis case, in general the translation of initial geometryinto flow is less efficient for higher moments and thus onewould expect κ is less than κ – in contradistinction tothe κ , values obtained in the magma fit. For a recentdiscussion of this puzzle, see Ref. [29]. We explore thistranslation of geometry to flow quantitatively in the nextsection. III. FULL HYDRODYNAMIC CALCULATIONS
The linear factors κ , can be determined either phe-nomenologically by matching calculated ε n to measured v n (as is done in the magma result shown above) orcan be calculated directly with viscous hydrodynamicsor parton kinetic theory as examples. It is striking in the magma calculation that κ ≈ κ ; i.e. the elliptic andtriangular flow have the same linear response coefficient.In general, even in the case of small viscous damping,e.g. shear viscosity to entropy density η/s = 1 / π , theresponse coefficient is expected to be smaller for higher · Energy Total [arbitrary units]110
10 0 5 10 15 20 25 30Centrality00.020.040.060.080.10.120.140.16 { m } n v B] · WS +A WS B · MAGMA PbPb [A = 0.32) k ( · }{2 e = }{2 v = 0.32) k ( · {4} e = }{4 v = 0.31) k ( · {2} e = }{2 v }{2 v, }{4 v, }{2 ATLAS data v
FIG. 2. Initial condition calculation with magma . (Top)The total energy deposit distribution over one million events.Solid (dashed) vertical lines correspond to 10% (top 1%) per-centiles. (Bottom) The ε { } , ε { } , and ε { } values asa function of centrality selection, scaled up by the respec-tive κ , values. In comparison, ATLAS experiment data areshown for v { } , v { } , and v { } . moments, i.e. larger n values. This feature has also beenseen in parton transport calculations [6]. We can testthis specifically for the 0-1% Pb+Pb collision initial con-ditions from magma .To this end, we have run 1000 such magma initial con-ditions for events all falling into the 0-1% centrality se-lection through a full hydrodynamic simulation includ-ing hadronic cascade afterburner B3D using the publiclyavailable sonic code [30]. Figure 4 shows time snap-shots of the two-dimensional temperature profile froma single magma initial condition through hydrodynamicevolution. The sonic running conditions for the hydro- Pb+Pb Centrality { } e k / { } e k {2} e k / {2} e k Ratio B) · WS + A WS B · MAGMA (A B) · MAGMA (A B) · IP-Jazma (A B) · WS + A WS B · IP-Jazma (A ) B · A IP-Jazma ( {2} / v{2} ATLAS data v
FIG. 3. Ratio of v { } /v { } as a function of Pb+Pb collisioncentrality as measured by the ATLAS experiment. Multipletheoretical calculations are also shown.FIG. 4. Time evolution event display for a Pb+Pb 0-1%most central collision using magma initial conditions runthrough sonic viscous hydrodynamics. The time snapshotsshow the temperature in the transverse (x,y) plane and areat t = 0 . , . , . , . dynamic stage include shear viscosity to entropy den-sity η /S = 1 / π and bulk viscosity to entropy den-sity ζ/s = 0. The hydrodynamic initial time is set to τ = 0 . T f = 170 MeV.In Figures 5 and 6, we plot for Pb+Pb centralities 0-1%and 25-26%, respectively, the flow coefficients v (upper)and v (lower) for three different p T selections versus the magma initial geometry ε and ε for 1000 individualevents. One sees a reasonable linear relationship in allcases as indicated via the calculated Pearson coefficients shown in the legend. Each panel is fit to a line with theintercept forced at zero and the slope corresponding tothe κ n value. It is noticeable that for the 25-26% central-ity events, where the events extend out to larger valuesof ε there is a clear non-linearity contribution - whichis reasonably described by a quadratic fit. Another ob-servation is that the Pearson coefficients are significantlylower for the v in the 25-26% centrality compared withthe 0-1% centrality, i.e. there is a lot more event-to-eventspread around the central linear relation. Lastly, the κ n values increase with increasing p T . Since the ε n valuesfor each event do not depend on particle p T , this increaseis simply a reflection of the larger v n as a function of p T .We highlight that in general any linear approximationof flow coefficients with eccentricities is not known to bevalid for p T -differential anisotropies and thus integrat-ing over a finite p T range introduces a sensitivity on theinfrared-cut used [31].Figure 7 shows for Pb+Pb collisions of 0-1% (left) and25-26% (right) centralities the κ (upper) and κ (mid-dle) coefficients and their ratio (lower) as a function ofcharged hadron p T . The κ and κ p T -integrated valuesover the range 0.5 - 3.0 GeV are shown as solid horizon-tal lines. The lower p T selection is made to match theATLAS measurement range and the upper p T selectionis nearing the limit where the hydrodynanic calculationshas significant systematic uncertainties. The values forPb+Pb 0-1% centrality are κ = 0 .
38 and κ = 0 . κ = 0 .
28 and κ = 0 .
24. Thus, the assumption used in the magma comparison in Figures 2 and 3 of κ n independent of cen-trality is significantly in error. It is also notable thatthe ratio of κ /κ varies between these two centrality se-lections, 0.73 (0-1%) and 0.85 (25-26%). Both of thesevalues are substantially lower than the 0.31/0.32 = 0.97obtained from the magma fit shown in Figure 2.We have mapped out the κ n values over the full central-ity range 0-30% corresponding to charged hadrons with p T = 0 . − . κ ( κ ) values. Inthe inset, we show the event-by-event distribution of κ values for the specific Pb+Pb 5-6% centrality. There is anon-Gaussian high-side tail which is dominated by eventswith very small values of ε . We have also fit these dis-tributions to a Gaussian and shown the Gaussian meanvalues in Figure 8 as closed points. There is a clear andsubstantial centrality dependence for both κ and κ val-ues. The method for comparison of measured v { } with κ × ε { } for example, used in the magma analysis inFigure 2, would be technically more comparable to ex-tracting κ from the sonic hydrodynamic calculation asthe RMS of v n divided by the RMS of ε n . These valuesare also shown in Figure 8 and are only very modestlydifferent from the Gaussian mean values.We have also run calculations with nearly ideal hydro-dynamics η/s = 0 .
02 = 1 / × / π and find for Pb+Pb0-1% centrality value of κ = 0 .
53 and κ = 0 .
36. Theseare significantly higher than the values quoted above for
FIG. 5. Results from sonic hydrodynamic calculations for magma initial condition 0-1% central Pb+Pb collisions. Shownare results in the upper (lower) panels from individual events for v ( v ) versus the initial geometric ε ( ε ). The linear fitsrepresent the κ response coefficients. Also shown are the Pearson coefficients indicating the degree of linear correlation.FIG. 6. Results from sonic hydrodynamic calculations for magma initial condition 25-26% central Pb+Pb collisions. Shownare results in the upper (lower) panels from individual events for v ( v ) versus the initial geometric ε ( ε ). The linear fitsrepresent the κ response coefficients. Also shown are the Pearson coefficients indicating the degree of linear correlation. FIG. 7. Results from sonic hydrodynamic calculations for magma initial condition 0-1% (left) and 25-26% (right) Pb+Pbcollisions. Shown are results in the upper (middle) panels are the κ ( κ ) values as a function of charged hadron p T . Thehorizontal line represented the p T -integrated value over the range 0.5 - 3.0 GeV. The lower panel shows the ratio of κ /κ asa function of p T . The dashed line is set at one for reference. Pb+Pb Centrality % - i n t e g r a t e d . - . G e V ] T [ p n k Pb+Pb (MAGMA I.C. + SONIC), mean k , Gaussian fit mean k ) e )/RMS( , RMS(v k , mean k , Gaussian fit mean k ) e )/RMS( , RMS(v k k dist. Pb+Pb 5-6% k FIG. 8. Results from sonic hydrodynamic calculations for magma initial condition 0-30% Pb+Pb collisions quantifyingthe κ κ values both via the mean values and the Gaus-sian fit mean values. Also shown are values of κ n determinedas the RMS of v n divided by the RMS of ε n . An example fitis shown in the inset.FIG. 9. Results from sonic hydrodynamic calculations for magma initial conditions 0-1% for the ratio κ /κ for twodifferent values of η /S. η/ S = 1 / π as expected since there is less viscous damp-ing and hence stronger flow. Shown in Figure 9 is acomparison of the κ /κ ratio with two different valuesof η/ S. There are modest difference that again highlightthat medium properties do not completely cancel out inthese ratios.The testing of magma initial conditions with full hy-drodynamics reveals that in fact the magma initial con-ditions do not match experimental data to resolve theultra-central puzzle. Any resolution of the ultra-centralpuzzle from an initial geometry picture must be coupledwith full transport calculations for confirmation. { m } n v B] · MAGMA PbPb [A = 0.28) k ( · }{2 e = }{2 v = 0.28) k ( · {4} e = {4} v = 0.15) k ( · {2} e = {2} v {2} , v{4} , v{2} ATLAS data v
FIG. 10. Initial condition calculation with magma but modi-fied to run where the energy deposit is proportional to A × B .The ε { } , ε { } , and ε { } values as a function of centralityselection are scaled up by the respective κ , values. In com-parison, ATLAS experiment data are shown for v { } , v { } ,and v { } . IV. ALTERNATIVE
MAGMA
MODELING
Next we test whether the magma results are highlightdependent on the non-standard A × B W S + A W S × B cal-culation of energy deposit. To this end, we have modifiedthe magma code to calculate the energy deposit as A × B ,more in line with the weakly-coupled ip-glasma calcu-lation. Figure 10 shows the comparison of eccentricityand flow cumulants from the modified- magma calcula-tion. The splitting between the v { } and v { } is nolonger captured by the calculation. Also the agreementwith both v { } and v { } is not maintained. These re-sults plotted as the ratio of v { } / v { } are also shownin Figure 3 and the modified- magma calculation onlyreaches 0.5 in the most central events. Matching the ex-perimental v n data, the new value for κ = 0 .
15 is nowmuch smaller than κ = 0 .
28. The original magma cal-culation has a larger contribution from the intrinsic ge-ometry, encapsulated in the smooth nuclear distribution,compared to fluctuations. This modified magma calcula-tion has relatively larger geometry fluctuations and thusthe v has a flatter centrality dependence and the relativescaling to match v and v are very different. Thus, the magma results are very sensitive to this non-standardcalculation of energy deposit, and do not match experi-mental data using the more standard A × B method. FIG. 11. Initial condition calculation with ip-jazma runwhere the energy deposit is proportional to A × B , as is truein the ip-glasma model. The ε { } , ε { } , and ε { } valuesas a function of centrality selection, scaled up by the respec-tive κ , values. In comparison, ATLAS experiment data areshown for v { } , v { } , and v { } . V. ALTERNATIVE INITIAL CONDITIONS
Within the ip-jazma framework, we can calculate ini-tial conditions in a variety of modes. First, we showresults in Figure 11, where the energy deposit is chosento be proportional to the local energy density in the pro-jectile times the local energy density in the target ( A × B mode). As detailed in Appendix A, this models the ini-tial spatial energy distribution in ip-glasma almost per-fectly. The agreement with experimental data is reason-able, although there is more splitting between v { } and v { } in the calculation. Also, as shown as a ratio inFigure 3, this geometry does not resolve the ultra-centralpuzzle. Interestingly, the phenomenologically fitted κ isnow 50% larger than κ , more in line with hydrodynamicexpectations. We highlight that this calculation is withnucleons as two-dimensional Gaussians, and ignores sub-nucleons. Sub-nucleon degrees of freedom in this contexthave been explored in Ref. [9] and they do not resolvethe ultra-central puzzle.Next, we show results in Figure 12, where the energydeposit is chosen to be proportional to the square root ofthe local energy density in the projectile times a smoothtarget nucleus summed with the local energy density inthe target times a smooth projectile ( √ A × B as donein the trento calculation with p = 0). The agreementwith experimental data is quite good, though as shownin Figure 3, the results are still below the experimen-tal data for v { } / v { } . It is again notable that the κ is approximately 50% larger than κ , more in linewith hydrodynamic expectations. We note however thatthese κ n values have been extracted by fitting the entire FIG. 12. Initial condition calculation with ip-jazma runwhere the energy deposit is proportional to √ A × B as istrue in the trento model with parameter p = 0. The ε { } , ε { } and ε { } values as a function of centrality selection,scaled up by the respective κ , values. In comparison, AT-LAS experiment data are shown for v { } , v { } and v { } . κ n vary with centrality. Thus, afinal evaluation can only be made with full hydrodynamiccomparison to the data.In both of these cases, it is important to run full hy-drodynamic simulations and with variations on mediumproperties to have a precision test of the centrality depen-dence and whether the ultra-central puzzle is reconciled.Such simulations with sonic are underway. VI. SUMMARY
In summary, we have reproduced the results from the magma initial condition model and its agreement withelliptic and triangular flow coefficients in Pb+Pb colli-sions at the LHC. However, we find that these results arehighly dependent on the energy deposit being propor-tional to hot spots in the projectile hitting a smooth nu-clear target and hot spots in the target hitting a smoothnuclear projectile, i.e. the hot spots do not “see” eachother. In addition the translation factors κ , implied bythe data comparison are in contradistinction from whatwe find with full sonic hydrodynamic simulations. Acritical take away is that any precision test of initial ge-ometry must be carried out with full evolution to flow co-efficients. We have explored other initial condition mod-eling (e.g. ip-glasma , trento ) within the ip-jazma framework and find the large triangular flow coefficientin ultra-central Pb+Pb collisions remains a puzzle requir-ing further investigation. VII. APPENDIX A
As detailed earlier, the ip-glasma is a first princi-ples calculation in the CGC weakly-coupled limit. Fo-cusing only on the initial distribution of energy depositin the transverse plane, we find that there are no non-trivial manifestations of color domains or “spiky” fluctu-ations that have been confirmed by experiment throughcomparisons of flow measurements and initial conditionsrun through hydrodynamics. This is in line with stud-ies on the lack of sensitivity to fine-scale structures inlong-wavelength hydrodynamics [32]. This observationis based on direct comparisons of calculated geometriesbetween two models, ip-jazma and ip-glasma . As abrief reminder, the ip-jazma calculation has no quan-tum fluctuations and only has Gaussian distributions as-sociated with nucleons and energy deposit proportionalto A × B . Using identical Monte Carlo Glauber ini-tial conditions for Au+Au events at b = 0, fed throughthe ip-jazma and ip-glasma calculations yield essen-tially identical ε − distributions – shown in Figure 13.We quantify this comparison for the second and thirdharmonics with the following values from ip-glasma ( ip-jazma ): (cid:104) ε (cid:105) = 0 .
104 (0 . = 0 .
054 (0 . = 0 .
271 (0 . (cid:104) ε (cid:105) = 0 .
089 (0 . = 0 .
047 (0 . = − .
023 ( − . ip-glasma code has a Glaubertwo-nucleon exclusion radius d = 0 . ip-glasma and ip-jazma weredifferent, one might mistakenly attribute that differencewith the models rather than the inputs for the MonteCarlo Glauber.Another comparison of relevance is the two-pointenergy-energy correlator. Shown in the left panels ofFigure 14 are energy deposit displays from ip-glasma (upper panel) and ip-jazma (lower panel) using anidentical Monte Carlo Glauber Au+Au event at b =0. Right panels of Figure 14 show the correlator( (cid:104) ε × ε (cid:105) / ( (cid:104) ε (cid:105) (cid:104) ε (cid:105) )), integrated over many events, asa function of distance scale from the ip-glasma (up-per panel) and ip-jazma (lower panel) calculations. Onesees a large distance scale (approximately 6 fm) structurefrom the size of the nucleus and a narrower (approxi-mately 1 fm) structure from nucleons both in ip-glasma and ip-jazma . Only the narrowest scale structure at < .
05 fm in the ip-glasma calculation is absent inthe ip-jazma calculation – see the inset for a zoomed inview. This structure in ip-glasma scales linearly withthe lattice spacing used in the calculation and is put inby hand. Again, it is notable that there is no visibleevidence of color domains in the energy deposit struc-ture. We note that this “spiky” structure gets washedout with the subsequent time evolution in the ip-glasma framework. ip-jazma is a model only for the initial en- ergy density and does not perform a time evolution as ip-glasma does.We have run the same comparison with sub-nucleonstructure (for example with three constituent quarks)and then there appears another structure in both ip-glasma and ip-jazma (approximately 0.2–0.3 fm), in-dicating that smaller structures can be seen in principle.Again, they do not appear to reflect any CGC-specificphysics.Lastly, we show a comparison of initial geometry fromthe ip-jazma calculation in a trento p = 0 like mode.The results from ip-jazma with energy deposit propor-tional to √ T A × T B , i.e. the square root of the localnuclear thickness values, are shown in Figure 15. Sincethe energy distribution from each nucleon is distributedas a two-dimensional Gaussian, if one considers that oneis taking the square root, i.e. √ T A × √ T B , the Gaussian σ should be increased by √
2. However, one is locallysumming the Gaussian contributions from all nucleonsin a nucleus and then taking the square root, so thereis no perfect match between trento p = 0 style and ip-glasma style geometry. ACKNOWLEDGMENTS
We gratefully acknowledge useful discussions with Giu-liano Giacalone as well as for sharing the magma
Pythoncode. We also acknowledge useful discussions and a care-ful reading of the manuscript by Jean-Yves Ollitrault,Paul Romatschke, Anthony Timmins and Bill Zajc. Weacknowledge Bjoern Schenke for the publicly available ip-glasma code and Paul Romatschke for the publiclyavailable sonic code. We highlight that the ip-jazma and trento codes are also publicly available. RS, MB,JLN acknowledges support from the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics underContract No. DE-FG02-00ER41152. SHL acknowledgessupport from Pusan National University Research Grant,2019.0 e Au+Au b=0IP-GlasmaIP-Jazma e e e e FIG. 13. Distributions of spatial eccentricities ε − from ip-glasma and ip-jazma for 1000 Monte Carlo Glauber Au+Auevents of impact parameter b = 0. FIG. 14. An identical Monte Carlo Glauber Au+Au event at b = 0 fed through the ip-glasma (upper panel) and ip-jazma (lower panel) calculations. The energy density distributions are shown in the left panels, with a zoom inset for more detail. Alsoshown are the two-point energy-energy correlator averaged over many events in the ip-glasma (upper panel) and ip-jazma (lower panel) frameworks. e Au+Au b=0IP-Glasma B T · A T (cid:181) e IP-Jazma w/Trento Style (p=0) B T · A T (cid:181) e s *2IP-Jazma IPSAT w=Trento Style (p=0) B T · A T (cid:181) e Au+Au b=0IP-Glasma B T · A T (cid:181) e IP-Jazma w/Trento Style (p=0) B T · A T (cid:181) e s *2IP-Jazma IPSAT w=Trento Style (p=0) B T · A T (cid:181) e Au+Au b=0IP-Glasma B T · A T (cid:181) e IP-Jazma w/Trento Style (p=0) B T · A T (cid:181) e s *2IP-Jazma IPSAT w=Trento Style (p=0) B T · A T (cid:181) e e e e e FIG. 15. 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