ΛΛ polarization from thermalized jet energy Willian Matioli Serenone a, ∗ , Jo˜ao Guilherme Prado Barbon a , David Dobrigkeit Chinellato a , Michael Annan Lisa b ,Chun Shen c,d , Jun Takahashi a , Giorgio Torrieri a a Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas, Campinas, Brasil b The Ohio State University, Columbus, Ohio, USA c Department of Physics and Astronomy, Wayne State University, Detroit, MI 48201, USA d RIKEN BNL Research Center, Brookhaven National Laboratory Upton, NY 11973, USA
We examine the formation of vortical “smoke rings” as a result of thermalization of energy lost by a jet. We simulatethe formation and evolution of these rings using hydrodynamics and deﬁne an observable that allows to probe thisphenomenon experimentally. We argue that observation of vorticity associated with jets would be an experimentalconﬁrmation of the thermalization of the energy lost by quenched jets, and also a probe of shear viscosity.
Two of the most studied results in heavy ion physicsat ultra-relativistic energies are jet energy loss [1, 2,3, 4, 5] and ﬂuid behavior [6, 7, 8, 9, 10, 11]. Theﬁrst shows that colored degrees of freedom form “amedium” opaque to fast partons, and the second showsthis medium thermalizes very quickly and subsequentevolution is nearly inviscid. Both results are usually in-terpreted as evidence that the medium created in heavyion collisions is a “strongly coupled liquid”.However, considerable theoretical uncertainty existsregarding the fate of the energy lost by the jet. If theplasma is a very good ﬂuid it is a reasonable hypothe-sis that the jet energy should thermalize and contributeto the ﬂuid ﬂow gradients. However, we do not have aclear experimental signature of this. Partially, this is be-cause the models of parton-medium interaction are in-conclusive , and partially it is because direct signa-tures of ﬂuid behavior, such as “Conical ﬂow”, have notbeen conclusively observed [13, 14].Recently, a new intriguing manifestation of hydrody-namic behavior has been found: Λ polarization, mea-surable via parity violating decays . It seems tobe aligned to the global vorticity of the ﬂuid and, toan extent, with near-ideal hydrodynamic vorticity be-ing transferred into Polarization via an isentropic tran-sition, respecting angular momentum conservation .As well as a further conﬁrmation of the ﬂuid-like behav- ∗ Corresponding author ior of the medium, this observation opens the door to usepolarization as a tool to study the medium’s dynamics.We propose to use polarization to understand the fateof locally thermalized energy emitted by the jet. Aschematic picture of the physical situation is shown inFig. 1. A hard parton generates a dijet structure andone of these is partially quenched by the quark-gluonplasma, while the other is not. The quenched portion ofthe jet introduces a initial velocity gradient in the ﬂuid.As is known from everyday physics, smoke-rings, ed-dies and so on are ubiquitous in ﬂuids when a velocitygradient is present. This is certainly the case when afast parton deposits energy into a medium. The onlydi ﬃ culty is, of course, that the jet’s direction ﬂuctuatesevent-by-event which vanishes after the event averag-ing.This is, however, easily surmountable: As argued in, the interplay between vorticity and transverse ex-pansion can be used to deﬁne a “jet production plane”.This insight can be sharpened into the deﬁnition of anexperimental observable that ties the polarization direc-tion, the angular momentum and a desired referencevector, which can be deﬁned event-by-event. In thiswork, we shall focus on deﬁning the reference vectoras a high- p T trigger particle. This observable, if mea-sured to be non-zero in classes of events where jet sup-pression exists, would provide unique and compellingevidence that the energy lost by the jet is indeed ther-malized. Moreover, it can be used to infer the medium’sviscosity, provided the initial velocity gradients gener-ated by the jet are quantiﬁed. Preprint submitted to Elsevier February 2021 a r X i v : . [ h e p - ph ] F e b artiallyquenched jetUnquenched jet Hardinteraction Vortical structure 𝜔 Figure 1: Schematic representation of the physical situation proposed.A hard parton generates a dijet structure and one of these jets is par-tially quenched by the quark-gluon plasma, while the other is not.The quenched portion of the jet introduces a momentum gradient inthe ﬂuid which in turn will generate a vortex ring.
2. A model for the jet thermalization
Our ﬁrst step is to choose a suitable model for themedium in which the jet will deposit (part of) its en-ergy. We choose a model which incorporates three di-mensional features, since the Λ polarization calculationwe will perform later on will depend on the dynamicsin all dimensions. The need to perform (3 + R ENTo 3D conﬁgured for simulations of Pb–Pb collisions at √ s NN = .
76 TeV, all of them with impact parameter b = R ENTo) and Ref. (for parameters exclusive to 3D T R ENTo). These aresummarized in Table 1. All computations are made ina grid with spacing equal to 0 . x and y direc-tions and 0 . η s ) direction.We expect the event-averaged ﬂuid background togive a good estimation on the polarization ﬁnal observ-able. Karpenko and Becattini  showed that the dif-ference between event-by-event simulations and an av-eraged initial condition to be small, albeit the source of Λ polarization in their work is di ﬀ erent from ours.Now we turn our attention to the jet thermaliza-tion. We consider a scenario of dijet creation inside themedium, where one jet will lose a negligible amount ofenergy and momentum while the other will be partiallyquenched, causing an asymmetry in jet emission. This We attempted halving the grid spacing in x and y directions andour main results changed by only 1%, at the expense of a much greatercomputational e ﬀ ort. Table 1: Input parameters for T R ENTo 3D.
Parameter ValueRapidity mean coe ﬃ cient 0 . ﬃ cient 2 . ﬃ cient 7 . . .
956 fmNucleon minimum distance 1 .
27 fmis measured experimentally using the jet asymmetry ob-servables A J and x J , deﬁned as [4, 21, 22, 23] x J ≡ p T / p T , (1) A J ≡ ( E T − E T ) / ( E T + E T ) . (2)The index “1” denote the trigger jet (the one that doesnot deposit energy and momentum in the medium)while the index “2” refers to the partially quenched jet.From Eqs. (1) and (2), one can obtain the momentum(energy) of the quenched jet from the values of x J ( A J )and the momentum (energy) of the trigger jet. Once E T and p T are determined, one may get the energy andmomentum deposited in the medium as p th = p T − p T , E th = E T − E T . (3)We will use the data from [4, Fig. 3] and [22, Fig. 8]to determine the values of p th and E th . These are thedistribution of dN / dA J and dN / dx J for central Pb–Pbcollisions at √ s NN = .
76 TeV. The energy and mo-mentum of the trigger jet in these measurements were E >
100 GeV and p T = . / c . For the valuesof A J and x J , we choose the ones that have the high-est value of multiplicity, i.e. A J = .
425 and x J = . E th = . p th =
43 GeV / c . Thisimplies that the situation studied in what follows cor-responds to a dijet structure with a momentum of89 . / c for the unquenched jet and 59 . / c forthe partially quenched jet, noting that it is the latter thatdeﬁnes the direction in which lambda polarization willbe studied.The measurements that will be proposed later willbe shown as a function of the di ﬀ erence betweenthe azimuthal angle of the partially quenched jet andthe emitted Λ . For simplicity, we choose the jetin the x -direction without loss of generality. Withthis choice, we may write the thermalized four-momentum as p µ th = (cid:16) E th p th (cid:17) and build an2 − x (fm) − . − . − . − . . . . . . y ( f m ) ( u x , u y ) = (0 . , . η s = 0 , τ = 1 . fm/c -1.00-0.75-0.50-0.250.000.250.500.751.00 ω z − − η s − . − . − . − . . . . . . y ( f m ) ( ω z , ω z ) = (0 . , . x = 0 . fm, τ = 1 . fm/c | ~ ω | Figure 2: Vortex ring formed by the thermalized jet after ∆ τ = .
00 fm / c of hydrodynamic evolution. The jet deposited momentum in the ˆ x direction, i.e. to the right in the left panel and away from the viewer in the right panel. In the left panel, it is shown a slice of the system at η s = z -component of vorticity vector deﬁned in Eq. (8). The arrows shows the x and y components of the ﬂuid’s four-velocities.The dots marks the local maxima of | ω z | . On the right panel, the system is sliced along the position x = . | (cid:126)ω | and thearrows shows the y and z components of the vorticity vector. energy-momentum tensor T µν following T µν = V p µ th p ν th E th , (4)where V is the volume over which the energy and mo-mentum is deposited. The volume is chosen to be anoblate spheroid centered on the origin of the system,with axis size equal to 0 . x and y directionsand ≈ .
29 fm in the z -direction (which equates to η s (cid:39) τ = .
25 fm / c ).We apply the Landau matching procedure T µν u ν = ε u µ to solve for the local energy density and ﬂow veloc-ity from the energy-momentum tensor in Eq. (4) ε = V E th − p th E th , (5) u x = p th (cid:113) E th − p th . (6)The remaining spatial components of u µ are zero and u τ is obtained by imposing the condition u µ u µ = E th and p th obtained above, we obtain ε V =
29 GeV and v x = . c , where V is the volume over which the en-ergy density will be deposited. In our simulations, werounded these values to ε V =
30 GeV and v x = . c .We veriﬁed that the injected energy-momentum gener-ates on average 1% more ﬁnal state particles per unit ofpseudo-rapidity.
3. Fluid vorticity and polarization measurements Λ ’s polarization The described initial condition is evolved with 3Dviscous hydrodynamics [25, 26, 27]. We use the lattice-QCD based equation of state from the HotQCD Collab-oration  and start the evolution at τ = .
25 fm / c .The six independent components of the vorticity tensorare then saved over a hypersurface of T =
151 MeV. Wethen compute the mean spin of Λ following Eq. (2) ofRef. , which we reproduce below for completeness. P µ ( p ) = − m ε µρστ p τ (cid:82) d Σ λ p λ n F (1 − n F ) ω ρσ (cid:82) d Σ λ p λ n F , n F = + exp (cid:16) β µ p µ − µ Q / T (cid:17) ,ω µν = −
12 ( ∂ µ β ν − ∂ ν β µ ) and β µ = u µ T . (7)In our case, we do not consider baryon density andbaryon currents and thus µ = ω µν we calculate a vorticity vector ω µ (inspired onthe Pauli–Lubanski pseudovector), which will act as aproxy for the local spin polarization, ω µ ≡ ε µνρ(cid:15) u ν ω ρ(cid:15) . (8)In Figure 2, we show the spatial distributions of ω z (along a slice of η s =
0) and | (cid:126)ω | (along a slice of x = . τ = .
25 fm / c . The external energy-momentum from the jet induces a ring-shaped concen-tration of vorticity around the jet axis during the hydro-dynamic evolution.To verify the vortical structures in the ﬂuid velocityﬁeld are mapped to the spin polarization of emitted Λ ,we compare the averaged ω z on the particlization hyper-surface in the region | η s | < . Λ ’s P z , averagedover the region | y | < . p T < . / c in Fig. 3.To obtain the azimuthal angle of each cell on the par-ticlization hypersurface, we use the cell’s four-velocity,i.e. ϕ = arctan( u y / u x ). Since the ﬂuid is expanding ina mostly radial way, the velocity angle ϕ is close to thespatial azimuthal angle of the cell. Figure 3 shows thatthe sign of Λ polarization correlates well with that ofthe ﬂuid vorticity vector ω µ in Eq. (8). − − − ϕ − ϕ ˆ J ) (rad) − . − . − . . . . . . ¯ hc h ω z i , | η s | < . P z × , | y | < . Figure 3: Comparison between the weighted average of the z -component of the vorticity vector (see Eq. 8) and the weighted averageof the z -component of the Λ -polarization (see Eq. 7) at mid-rapidity. Furthermore, we investigated the dependence of the z -component of the Λ -polarization ( P z ) with transversemomentum and the angular distance (in the transverseplane) from the partially quenched jet, which we present in Fig. 4 as a color map. The markers indicate the po-sitions of the | P z | ’s maxima in each p T -bin. The | P z | ’smaxima are closer to the jet axis at high p T than thoseat low p T bins. − − − ϕ − ϕ ˆ J ) (rad) . . . . . . . p T ( G e V / c ) | y | < . -0.12-0.08-0.0400.040.080.12 P z ( % ) Figure 4: Distribution of the weighted average of the z -component ofthe polarization ( P z ), using Λ -multiplicity as weight and as functionof p T and the angular distance in the transverse place. The averageconsiders only data in the range | y | < .
5. The orange / blue dots marksthe bins where | P z | is highest for that p T bin. We focused on the longitudinal component of po-larization / vorticity for a jet that travels along the + ˆ x direction. Since the transverse components are anti-symmetric with respect to rapidity / spatial-rapidity (seeFig. 2, right panel), they will average to zero in theabove calculations and we lose information about them.However, the formation of a vortex ring due to ourchoice of initial condition has similarities with the vor-tex rings present in p + A collisions which were studiedin Ref. . There we introduced the ring observable R ˆ t Λ , which we replicate below for completeness R ˆ t Λ ≡ (cid:42) (cid:126) P Λ · (cid:0) ˆ t × (cid:126) p Λ (cid:1) | ˆ t × (cid:126) p Λ | (cid:43) p T , y . (9)Here, ˆ t = ˆ J is the axis direction of the jet , and (cid:104)·(cid:105) p T , y denotes an weighted average over transverse momen-tum (in the range 0 . / c < p T < . / c ) andrapidity (in the range | y | < . Λ multiplicity as on our calculation, ˆ J = ˆ x R ˆ J Λ will ﬁlter most contributions tothe polarization which were not induced by the jet ther-malization while allowing us to take into account e ﬀ ectsin the direction besides ˆ z . We will focus on R ˆ t Λ fromnow on.The use of thermal vorticity, as shown in Eq. 7,has been debated in the literature [30, 31, 32]. Thereare three other deﬁnitions of vorticity which are pop-ularly employed. The “kinetic vorticity” consists ofthe replacement β µ → u µ and is appealing becauseit can be more intuitively interpreted. The “tempera-ture vorticity” or “T-vorticity” relies on the replacement β µ → T u µ and also allows vorticity generation by tem-perature gradients. Finally, there is the “spatially pro-jected kinetic vorticity” which replaces the derivative ∂ µ by ∇ µ = ( g µν − u µ u ν ) ∂ ν . This has the e ﬀ ect of removinglocal acceleration terms from the kinectic vorticity. Italso has a direct connection to the ﬂuid vorticity in thenon-relativistic limit. We show a comparison betweenthe polarization results using these four di ﬀ erent vor-ticity values in Fig. 5. The fact that polarization fromkinetic, thermal, and temperature vorticities are essen-tially equal implies that in this case the vorticity is pre-dominately generated by gradients in velocity, not intemperature. The higher value for the polarization fromthe spatially projected kinetic vorticity implies that lo-cal acceleration (caused mostly by the ﬂuid expansion)has the e ﬀ ect of reducing the ﬁnal Λ polarization. − − − ϕ − ϕ ˆ J ) (rad) . . . . . . . R ˆ J Λ ( % ) | y | < . , . < p T < . GeV/c
KineticThermal Kinetic w/spatial proj.Temperature
Figure 5: R ˆ t Λ (see Eq. 9) computed from Λ -polarization calculationsusing four types of vorticity tensor. We study the sensitivity of the ring observable R ˆ t Λ on medium’s speciﬁc shear viscosity. In addition to η/ s = .
08, we perform calculations with η/ s = . .
01, 0 .
16 and 0 .
24. Figure 6 shows that the medium’sshear viscosity suppresses the ring observable R ˆ t Λ . Weobserve a higher sensitivity of R ˆ t Λ to small viscosity val-ues η/ s < .
08 than η/ s > .
08. This trend is consistentwith the vorticity ring being quenched by the medium,an e ﬀ ect which will be stronger for higher viscosity, butthat eventually gets saturated. This is in contrast to ellip-tic ﬂow, which has a more or less uniform dependencewith viscosity . − − − ϕ − ϕ ˆ J ) (rad) . . . . . . . R ˆ J Λ ( % ) | y | < . , . < p T < . GeV/c
Ideal hydro η/s = 0 . η/s = 0 . η/s = 0 . η/s = 0 . Figure 6: Distribution of R ˆ J Λ (see Eq. 9) for di ﬀ erent speciﬁc shearviscosities. It is possible to argue that a jet which is quenchedat the center of the system will not be accompaniedby an unquenched jet. Instead, there would be a pairof quenched jets, inducing a pair back-to-back vortexrings. One could approximately treat the medium ex-citation from the two quenched jets as independent su-perposition (after rotating one of them by π rad). How-ever, this would neglect the possibility of interactionsbetween the two vortexes during the hydrodynamic evo-lution. We investigate the possibility of a double-quenched jet by displacing the energy-momentum de-position to x = . x = − . The angle where the signal is strong has a small dependence onviscosity as well.
5o see that the superposition scenario has a polarizationwhich is almost double the one where we evolve the twoquenched jets, indicating the interaction between themduring hydrodynamic evolution is crucial and has a self-canceling e ﬀ ect. − − − ϕ − ϕ ˆ J ) (rad) − . − . . . . . . . . R ˆ J Λ ( % ) | y | < . , . < p T < . GeV/c
Doublequenched jetSuperimposedquenched jets Singlequenched jet
Figure 7: Comparison between the R ˆ J Λ in a double-quenched jet sce-nario versus the single-quenched jet case. The blue curve shows theresult from the simulation and the red one by superimposing twosingle-quenched jets (shown in green).
We modeled the thermalization of the energy-momentum from a hard parton as a “hot spot” whichpropagates inside ﬂuid dynamic simulations. Suchconﬁguration of velocities will generate a vortex ring,which can be quantiﬁed by the vorticity of the ﬂuid.The vorticity will lead to the emission of polarizedhadrons on the particlization hypersurface as describedin [30, 15].To obtain the energy and momentum deposited in themedium by the jet thermalization, we assumed a jet witha transverse momentum of 89 . / c that would de-posit approximately 40% of its energy in the medium,motivated by [4, Fig. 3] and [22, Fig. 8]. The polarizedhadron emission would accompany a partially quenchedjet, meaning that experimentally any analysis aiming tomeasure this e ﬀ ect would have to focus on an asym-metric jet pair, with the higher momentum jet havingmomentum of the order of 90 GeV / c and the lower mo-mentum being of order 60 GeV / c . Other options, such as using high-momentum trigger particles, will also beinvestigated in future work.We computed the polarization of the Λ hyperon dueto the vorticity caused by our model of jet thermaliza-tion. We showed that, for this speciﬁc case, the e ﬀ ectsare dominated by velocity gradients and thus there islittle di ﬀ erence in using thermal vorticity versus otherdeﬁnitions which are often suggested in the literature.We also showed that the strength of the signal is highlysensitive to the ﬂuid’s shear viscosity.The angular distribution of the ring observable R ˆ t Λ in the transverse plane with respect to the quenched jetpeaks in the range 0 . . ﬀ ectthe region where R ˆ J Λ peaks. Instead, it will only dampenthe overall magnitude in addition of an expected addi-tional lobe in the opposite direction.We point out that, despite the e ﬀ ect being of the orderof only a few tenth of a percentiles, the proposed ringobservable R ˆ t Λ should be measurable by experiments,since it has the same of magnitude as reported per AL-ICE and STAR for the global Λ -polarization [16, 34].We also inspected the typical maximum value found for R ˆ J Λ . We found that R ˆ J Λ < .
25% always, peaking in the p T range of 0 . / c < p T < . / c .We devote a future study to quantify the e ﬀ ects ofevent-by-event ﬂuctuations in the ﬂuid on R ˆ t Λ .We note that the discussed jet induced polarizatione ﬀ ect requires both color opacity and rapid thermaliza-tion. Thus, it is very likely present in AA and might dis-appear in pp and pA collisions (which may have rapidthermalization, but very small opacity). Since the ref-erence is a high momentum trigger rather than a globalquantity like the reaction plane, it should be possible forexperiments to examine events with one Λ and one highmomentum triggered hadron to verify this e ﬀ ect. If itturns out that indeed R ˆ J Λ is non-zero for AA events, onecould proceed to do more detailed model-data compar-isons as a way to constrain viscosity and jet energy loss. Acknowledgments
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