Strongly Interacting Matter Under Rotation: An Introduction
SStrongly Interacting Matter Under Rotation: AnIntroduction
Francesco Becattini, Jinfeng Liao, and Michael Lisa
Abstract
Ultrarelativistic collisions between heavy nuclei briefly generate thequark-gluon plasma (QGP), a new state of matter characterized by deconfined par-tons last seen microseconds after the Big Bang. The properties of the QGP are ofintense interest, and a large community has developed over several decades, to pro-duce, measure and understand this primordial plasma. The plasma is now recog-nized to be a strongly-coupled fluid with remarkable properties, and hydrodynamicsis commonly used to quantify and model the system. An important feature of anyfluid is its vorticity, related to the local angular momentum density; however, thisdegree of freedom has received relatively little attention because no experimentalsignals of vorticity had been detected. Thanks to recent high-statistics datasets fromexperiments with precision tracking and complete kinemetic coverage at colliderenergies, hyperon spin polarization measurements have begun to uncover the vor-ticity of the QGP created at the Relativistic Heavy Ion Collider. The injection ofthis new degree of freedom into a relatively mature field of research represents anenormous opportunity to generate new insights into the physics of the QGP. Thecommunity has responded with enthusiasm, and this book (to be published as a vol-ume of Lecture Notes in Physics series by Springer) represents some of the diverselines of inquiry into aspects of strongly interacting matter under rotation.
Francesco BecattiniUniversity of Florencee-mail: [email protected]
Jinfeng LiaoPhysics Department and Center for Exploration of Energy and Matter, Indiana University, 2401 NMilo B. Sampson Lane, Bloomington, IN 47408, USA,e-mail: [email protected]
Michael LisaDepartment of Physics, The Ohio State University, 191 West Woodru ff Avenue, Columbus, OH43210 USA.e-mail: [email protected] a r X i v : . [ nu c l - t h ] F e b Francesco Becattini, Jinfeng Liao, and Michael Lisa
In 2005, Liang and Wang [1] predicted that spin-orbit coupling would polarizestrange quarks created in non-central heavy ion collisions, resulting in emitted Λ hyperons globally polarized along the direction of the collision angular momentum.The magnitude and momentum-dependence of the predicted polarization dependedon details of specific models of quark-quark potentials, small-angle scattering ap-proximations, and details of hadronization mechanisms.In 2008, Becattini and collaborators [2] noted that in a hydrodynamic picture, lo-cal thermodynamic equilibrium implies a relation between the spin polarization andthe rotational flow structure (vorticity). In the hydrodynamic model, vorticity canbe extracted directly from the evolution, with no need to appeal to specific micro-scopic processes. In 2013, an equation relating the polarization of Λ hyperons andthermal vorticity was derived [3] and such polarization was predicted to be at thelevel of a few percent. The first result regarding the systematic dependence of thise ff ect on the collision beam energy, particularly in the range relevant to the beamenergy scan program at the Relativistic Heavy Ion Collider (RHIC), was reportedin a 2016 paper [4], providing a highly relevant insight for the later experimentalmeasurements.In 2017, the STAR Collaboration published [5] the first observation of global Λ polarization from noncentral heavy ion collisions. As discussed below and through-out this Volume, most theoretical interpretations of these observations are basedupon this hydrodynamic approach.While the phenomenon of global polarization was predicted based on particle-particle interaction, the success of quantitative predictions of the hydrodynamicmodel to reproduce experimental observations (discussed below) seem to confirmthat for spin, as for many other observables, microscopic details are less importantthan bulk thermodynamic properties. Below, we discuss the hydrodynamic approachto vorticity and polarization, followed by experimental observations.We will briefly discuss the related phenomenon of vector meson spin alignment,also predicted by Liang and Wang [6] in 2005. As of now, measurements of spinalignment at the Large Hadron Collider (LHC) and RHIC are di ffi cult to explain inany theoretical approach. Twenty years ago, the world’s first nuclear collider began producing heavy ion col-lisions at energies far surpassing those previously achievable in fixed-target exper-iments. The goal was to produce the quark-gluon plasma (QGP)– a state of mattercharacterized by partonic (rather than hadronic) degrees of freedom. For decades,production and study of the QGP had long been the focus driving the field of rel-ativistic heavy ion physics, as it holds the promise of shedding light on the non- trongly Interacting Matter Under Rotation: An Introduction 3 perturbative region of quantum chromodynamics (QCD), the most poorly under-stood of the fundamental interactions in the Standard Model.In 2005 [7, 8, 9, 10], based on a systematic and comprehensive analysis of avail-able data, the experimental collaborations at the Relativistic Heavy Ion Collider(RHIC) confirmed that QGP is indeed created in ultra-high energy collisions. Fur-thermore, the data clearly indicated that the QGP was a strongly coupled fluid, con-trary to some expectations that the plasma would be weakly coupled due to thecombination of high temperatures and the running of the QCD coupling constant.The evidence driving this conclusion was the collective anisotropic emission dis-tribution of hadrons from the collision– the so-called “elliptic flow.” These verystrong anisotropies (and the dependence upon mass and momentum) were nearlyquantitatively consistent with expectations based on relativistic inviscid (ideal) hy-drodynamics.The discovery of nearly “perfect fluid” behavior had two major outcomes. Firstly,it prompted a re-evaluation of numerical QCD calculations performed on a lattice,the most reliable ab initio calculations of the strong interaction. While numericallycorrect, lattice calculations could be misinterpreted to suggest that a weakly cou-pled gas of quarks and gluons was the proper paradigm for modeling collisions atRHIC. It was also realized that the QGP near the pseudo-critical transition temper-ature is a peculiar system: unlike ordinary matter, its microscopic interaction lengthis comparable to the thermal de Broglie wavelength, making the kinetic collisionaldescription inappropriate. Nevertheless, even under such unusual conditions, the lo-cal thermodynamic equilibrium concept and hydrodynamics are still valid,. Hence,the discovery established relativistic fluid dynamics as the new paradigm for thebulk evolution of the system. Confronting increasingly sophisticated hydrodynamiccalculations with data has produced valuable estimates of transport coe ffi cients, ini-tial parton distributions, and the QCD equation of state. Triangular and higher-orderazimuthal correlations have probed the substructure of the fluid flow fields at everfiner scale.A relativistic collision between heavy nuclei at finite impact parameter can in-volve angular momentum of order 10 ∼ (cid:126) . In a fluid, angular momentum can man-ifest as vorticity, rotational gradients of the flow and temperature fields [2]. Untilrecently, this aspect of the plasma had been largely ignored, as there had been noexperimental observation of its e ff ects.In 2017, the STAR Collaboration published an observation of global hyperonpolarization in Au + Au collisions at RHIC, opening the potential to probe novelsubstructures of the QGP fluid at the finest possible scale. This is a rare case in whichan entirely new direction is introduced to a mature field. It is especially excitingbecause the natural language for discussing vorticity– three-dimensional relativisticviscous hydrodynamics– has been developed to a high degree of sophistication by alarge community of theorists. It is an opportunity for new insights into the physics ofdeconfined QCD matter, and the heavy ion community has responded with intensefocus on the topic. This book represents a broad sampling of directions of inquiryinto this new area of research.
Francesco Becattini, Jinfeng Liao, and Michael Lisa “Lumpy” azimuthal fluid flow patterns (elliptic flow, triangular flow, etc) may bemeasured by azimuthal correlations between the momenta of emitted particles; thisis experimentally straight-forward. Orbital angular momentum in heavy ion colli-sions, on the other hand, is experimentally inaccessible. Instead, one relies on cou-pling between the orbital (“mechanical”) angular momentum of the fluid and spinof the emitted particles. The first observation of such an e ff ect was reported morethan a century ago by S. Barnett [11], in which an uncharged and un-magnetizedsolid metal object, when set spinning, spontaneously magnetizes .The analogous e ff ect in a fluid , coupling mechanical vorticity of the bulk fluidand quantum spin polarization, was first reported by Takahashi et al, in 2016 [12]. Intheir experiment, liquid mercury flowing through a channel acquired local vorticitydue to viscous friction with the wall. Spin-vorticity coupling produced a polarizationgradient that could then be detected directly through the inverse spin Hall e ff ect. Theresults could be understood by expanding angular momentum conservation in fluiddynamics, to include angular momentum transfer between the liquid and electronspin [12].In the Barnett and Takahashi experiments, the macroscopic rotational motion wasa controlled variable and the spin polarization straightfoward to measure. In high-energy nuclear collisions, the magnitude and direction of the angular momentumfluctuates from one event to the next, and a statistically significant measurement re-quires combining ∼ − events. Furthermore, the particles whose polarizationis to be measured are emitted at all angles at speeds approaching that of light.These challenges are addressed by precision tracking and correlating detectorsubsystems in di ff erent regions of the experiment. In particular, the angular momen-tum in a collision is given by J = b × p beam , (1)where the impact parameter, b , is the transverse (to the beam direction) vector con-necting the center of the target nucleus to that of the beam nucleus (where attentionto the designation of beam and target is important [13]), and p beam is the momentumof the beam in the collision center-of-momentum (c.o.m.) frame. The magnitudeof the impact parameter, | b | , is estimated by the total number of charged particlesemitted roughly perpendicular to the beam in the collision c.o.m. frame, while itsdirection, ˆ b , is estimated by the sidewards deflection of particles emitted close to thebeam direction. See figure 1 for an illustration.The flow pattern of the QGP fluid is complex and any local vorticity may fluctuateas a function of position within each droplet; however, the average vorticity mustbe parallel to J which is event-specific. For this reason, spin polarization projectionalong ˆ J is termed the “global” polarization. That the magnetization arose from spin polarization of the electrons was not known to Barnettand his contemporaries in 1915, as the concept of quantum spin was not introduced until nearly adecade later.trongly Interacting Matter Under Rotation: An Introduction 5
Fig. 1
The geometry of a collision. (a) Before collision: the angular momentum is determined bythe impact parameter, b , an uncontrolled variable that fluctuates from one collision to the next.(b) In a non-central ( | b | (cid:44)
0) collision, parts of the nuclei overlap, producing the QGP, while theso-called “spectators” continue to travel forward, experiencing only a slight impulse directed awayfrom the collision. (c) One reconstructed event in two subsystems in STAR experiment. The TimeProjection Chamber (TPC) [14] records ∼ charged particles emitted from the QGP createdin the collision, while the Event Plane Detector (EPD) [15] measures spectator fragments. Themagnitude and direction of b are determined, respectively, by the number of charged particlesmeasured in the TPC and the anisotropic hit pattern in the EPD. Having determined the direction of the average vorticity, the second challengeis to measure the spin polarization along that direction.If the QGP fluid does indeed have non-vanishing vorticity, and if thermalization(complete or partial) of orbital and spin degrees of freedom does occur, then presum-ably all particles emitted in the collision will have their average spins aligned withˆ J . Of the zoo of particle types emitted in a heavy ion collision, the spin directions ofonly a few are easily measurable. In particular, particles undergoing parity-violatingweak decay betray their spin direction through asymmetries in the momentum dis-tribution of their daughters. Of this already restricted subset of particles, only afew are created in reasonable numbers to allow a significant measurement. The bestcandidate is the Λ hyperon, which can be cleanly measured by its p + π − decay inthe TPC, as seen in panel (a) of figure 2. The decay topology is sketched in panel(b) of figure 2. An ensemble with polarization P Λ will preferentially emit daughterprotons along the direction of polarization according to dNd cos θ ∗ = (cid:16) + α Λ P Λ · ˆ p ∗ p (cid:17) , (2)where θ ∗ is the angle between the polarization and daughter proton momentum p ∗ p in the hyperon rest frame. The decay parameter α Λ = .
732 determines the strengthof the e ff ect.The global polarization is then measured by correlating information from bothdetector subsystems: In principle, the magnitude | J | of the collision’s angular momentum may be estimated as well.However, not all of this angular momentum is transferred to the plasma at midrapidity [4], sousually only the direction ˆ J is of interest. This quantity is the only important ingredient to estimatevorticity in any event. Francesco Becattini, Jinfeng Liao, and Michael Lisa Fig. 2 (a) A Λ hyperon detected in the STAR TPC by combining its charged proton and piondaughters. Inset: the invariant mass of daughter pairs shows a clear peak at the Λ mass. (b) In theparity-violating decay topology, the daughter proton tends to be emitted in the direction of theparent Λ hyperon, in the Λ center of mass frame. (cid:68) P Λ · ˆ J (cid:69) = πα H R (1)EP (cid:68) sin (cid:16) Ψ EP , − φ ∗ p (cid:17)(cid:69) , (3)where φ ∗ p is the azimuthal angle of the daughter proton in the parent hyperon frame.In equation 3, Ψ EP , is the first-order event plane angle, an estimator of the azimuthalangle of the impact parameter b ; the resolution of this estimation is R (1) EP . Standardmethods have been developed to extract both the event plane and the resolution fromanisotropic particle distributions in the EPD.The discussion thus far has described the global Λ hyperon polarization mea-surement in the STAR experiment at RHIC. The ALICE experiment at the LHCperformed a similar analysis, tracking charged hyperon daughters with a gas-filledTPC at midrapidity, and measuring Ψ EP , with segmented detectors at forward ra-pidity. The STAR and ALICE measurements thus far comprise the world’s dataseton global polarization, and are shown in figure 3.We o ff er some general remarks on figure 3 in the next section, but at the ex-perimental level, we note that the statistical uncertainties at low √ s NN are large.These uncertainties are determined by (1) the number of collision events recorded bythe experiment; (2) the per-event hyperon yield; (3) the event-plane resolution R (1)EP .Measurements by the STAR Collaboration in the second phase of the RHIC BeamEnergy Scan (BES-II) [17] will have an order of magnitude better statistics [18] andbetter event plane resolution [15]; overall, the precision should increase roughlyeight-fold, allowing important systematic studies [16] not currently possible.The average “global” polarization vector must point along the direction of ˆ J . Onthe other hand, the mean spin polarization vector for particles with specific momen-tum have three components which can be also measured. The component along thebeam (longitudinal component) is expected to show a 2 nd -order azimiuthal oscilla-tion relative to the event plane. The amplitude and phase of this oscillation has beenmeasured for Au + Au collisions at √ s NN =
200 GeV by the STAR Collaboration.Figure 4 shows the transverse momentum dependence of the 2 nd Fourier compo-nent for non-central collisions. Thanks to the excellent tracking, good event plane trongly Interacting Matter Under Rotation: An Introduction 7
10 (GeV) NN s0246 ( % ) æ J (cid:215) P Æ STARALICE
L L = 0.732 L a scaled using L and L Average of hydrodynamicsparton cascade (AMPT)hadron cascade (UrQMD) 3-fluid dynamicschiral kinetic
Fig. 3
The world dataset of global Λ hyperon polarization in relativistic heavy ion collisions com-pared to expectations from hydrodynamic and transport simulations. Figure from [16]. resolution, and a high statistics dataset available at RHIC top energy, an oscillatingsub-percent polarization signal is easily measured. As we discussed in section 1, the original idea for global Λ polarization in heavyion collisions was based on microscopic processes that drove the initial state, thetransfer of angular momentum from orbital to spin degrees of freedom, and thesubsequent hadronization mechanism. Assumptions and parameters were requiredto compute each of these components of the calculation. A detailed discussion alongthis line can be found in Chapter 7 [20].The tremendous success of hydrodynamics to heavy ion physics suggests thatthe myriad details of microscopic processes undoubtedly at play in these complexcollisions are eventually unimportant, as the system approaches local equilibriumquickly. In the earliest days of RHIC, ideal (inviscid), boost-invariant hydrodynamiccalculations with simple initial conditions largely reproduced– nearly “out of thebox”– the multiplicity, p T and mass systematics of measured elliptic flow. This suc-cess gave some confidence that equilibrium hydrodynamics was a good paradigm tounderstanding the collective physics of heavy ion collisions. Francesco Becattini, Jinfeng Liao, and Michael Lisa [GeV/c] T p (cid:60) [ % ] (cid:156) ) (cid:94) - (cid:113) s i n ( z P (cid:157) (cid:82) + (cid:82) hydro (x 0.2) 20%-50%) 20%-60% BW (spectra+v +HBT) 20%-80% BW (spectra+vSTAR = 200 GeV NN sAu+Au 20%-60% Fig. 4
The second-order Fourier coe ffi cient of the azimuthal oscillation of the longitudinal com-ponent of the Λ hyperon polarization. Figure from [19]. In the subsequent decades, several important insights have been achieved byworking within this framework, using details in the data to probe the partonic struc-ture of the initial state, transport coe ffi cients, and hadronization mechanisms. Theseinsights required considerable elaboration of the initial simple models, incorporat-ing viscosity, baryochemical currents, vorticity, three-dimensional dynamics, andevent-by-event fluctuations in the initial state. However, the close resemblance ofthe initial simple calculations with observations set this fruitful enterprise on firmground.Figure 3 suggests that the same situation exists in the study of global hyperonpolarization. Theoretical curves show predictions from hydrodynamic and transportcalculations, in which fluid vorticity is assumed to equilibrate with Λ spin degreesof freedom to produce the polarization. Vorticity – more properly, thermal vorticity– trongly Interacting Matter Under Rotation: An Introduction 9 is calculated directly from the flow field in the hydrodynamic calculations, as dis-cussed in detail in Chapter 8 [21]. On the other hand, in the transport calculations,flow and temperature fields are calculated from the motion of multiple particles incoarse-grained spatial cells; this implicitly assumed local thermalization; see de-tailed discussions in Chapter 9 [22]. Eventually, in both methods, polarization of aspin 1 / (cid:36) at the leading order [3]: S µ ( p ) = − m (cid:15) µνρσ p σ (cid:82) Σ d Σ · p (cid:36) νρ n F (1 − n F ) (cid:82) Σ d Σ · pn − F (4)where S µ ( p ) is the mean spin vector and n F is the covariant Fermi-Dirac distributionfunction. The thermal vorticity is defined as the antisymmetric derivative of the four-temperature vector field, that is: (cid:36) µν = (cid:34) ∂ ν (cid:32) T u µ (cid:33) − ∂ µ (cid:32) T u ν (cid:33)(cid:35) (5)where T and u µ are local temperature field and flow velocity field, respectively. Theintegration in equation (4) is performed on the 3-D hadronization hypersurface Σ .The polarization vector P µ is simply S µ / | S | and its global, momentum integrated,value in the particle rest frame turns out to be directed along the angular momentumvector, so that, approximately one has: (cid:104) P (cid:105) · ˆ J ≈ (cid:104) (cid:36) (cid:105) · ˆ J (6)where the (cid:104) (cid:36) (cid:105) is the mean thermal vorticity value over the hadronization hypersur-face. The above relation is a direct manifestation for the rotational polarization ofmicroscopic spin. The theoretical underpinning of this phenomenon is to be fullyelaborated through a variety of approaches such as quantum field theory (in Chap-ters 2 [23], 3 [24] and 4 [25]) and relativistic kinetic theory (in Chapters 5 [26] and6 [27]).For the most part, these models have been used to understand other observationsfrom heavy ion collisions, and the results in figure 3 are obtained largely “out ofthe box”. The quantitative agreement, as well as the universal decreasing trend ofpolarization with √ s NN (despite the fact that | J | in creases with increasing collisionenergy) is a clear indication that we have at hand a paradigm to understand hyperonpolarization.That said, there are strong tensions with the existing theoretical expectations incertain observables. One is seen in figure 4; the same hydrodynamic calculation thatreproduced (cid:68) P Λ · ˆ J (cid:69) with no special tuning, predicts the wrong sign of the longi-tudinal polarization, (cid:104) P Λ · ˆ z (cid:105) . Hence it seems that, similar to the early collectiveflow studies, the framework is well-grounded, while there is much to learn from thedetails. Chapters 8 [21], 9 [22] and 10 [28] in this Volume provide an in-depth dis-cussion on the phenomenology study based on this framework. More broadly, the Fig. 5
Vector meson alignment in Au + Au collisions at √ s NN =
200 GeV, measured by the STARCollaboration at RHIC [31]. Left: ρ , for K ∗ mesons as a function of transverse momentum formid-central collisions. Right: ρ , for φ mesons as a function of centrality. Dashed lines indicate ρ , = , corresponding to no alignment with the normal to the event plane. presence of global rotation has opened a new dimension for investigating its non-trivial e ff ects, for example, on the phase structures of matter (see chapter 11 [29]) oron the interplay between orbital and spin angular momentum (c.f. chapter 12 [30]). In an equilibrium picture, the spins of all emitted particles will be aligned with thetotal angular momentum of the system. In addition to Λ and Λ hyperons discussedabove, preliminary results from the STAR collaboration indicate consistent polar-ization of Ξ , Ξ and Ω baryons [32]. Besides baryons, in principle polarization couldbe detected for vector mesons such as K ∗ or φ .The spin of a vector meson is quantified by the 3 × ρ i , j .Becattini discusses the coupling of this quantity to fluid vorticity in Chapter 2 [23].Due to the parity-conserving nature of their strong decay, the elements ρ , and ρ − , − cannot be separately determined. Because the trace is unity, there is only oneindependent diagonal element, ρ , which quantifies the component of the mesonspin perpendicular to the quantization axis. As with the baryons, for the averagespin, the axis of interest is ˆ J , perpendicular to the event plane. Random alignmentof spins would yield ρ − , − = ρ , = ρ , = . Given only experimental access to ρ , , it is impossible to determine whether the meson spin is parallel or anti-parallelto ˆ J , but in either case, spin alignment would imply ρ , < [33].The 2-particle decay topology of a vector meson is related to the alignment ac-cording to [34]: d N d cos θ ∗ = (cid:104) − ρ + (3 ρ −
1) cos θ ∗ (cid:105) , (7)where θ ∗ is the angle between the parent spin and a daughter momentum in theparent’s rest frame. At local thermodynamic equilibrium, the alignment is quadratic trongly Interacting Matter Under Rotation: An Introduction 11 Fig. 6
Vector meson alignment, for Pb + Pb collisions at √ s NN = .
76 TeV, measured by the AL-ICE Collaboration at the LHC [36]. Left (right) panel shows ρ , for K ∗ ( φ ) mesons as a functionof collision centrality, for two ranges in transverse momentum. Dashed lines indicate ρ , = ,corresponding to no alignment with the normal to the event plane. in thermal vorticity to first order [35, 16]: − ρ ≈ (cid:36) . (8)Therefore, consistency with the hyperon results would lead to the expectation − ρ , ≈ − Experimental results deviate strongly from that expectation. Figures 5 and 6 show K ∗ and φ alignment measurements from the STAR and ALICE experiments atRHIC and LHC, respectively. In all cases, | − ρ | ≈ .
1, two orders of mag-nitude larger than expectations based on P Λ and vorticity considerations. Perhapsmore surprisingly, STAR reports ρ > for φ mesons.As discussed above, the hydrodynamic equilibrium ansatz seems a reliable base-line for understanding hyperon polarization, as it is for understanding much elsein heavy ion physics. However, there may be many other e ff ects at play. In theiroriginal paper [6], Liang and Wang considered di ff erent hadronization mechanismsinvolving polarized quarks. If vector mesons are produced by simple coalescence ofa quark and antiquark with polarizations P q and P q , respectively, then ρ meson0 , = − P q P q + P q P q ≈ − (cid:16) P q P q (cid:17) , (9)where the approximation holds for small polarizations. This is consistent with thehydrodynamic equilibrium prediction (equation 8) if P q = P q = (cid:36) . It is not possibleto reconcile the ALICE measurements of very small values of hyperon with largevalues of | − ρ , | in a simple recombination picture [36].Perhaps even more surprising are measurements of the STAR Collaboration atRHIC. For K ∗ , they report [31] values of − ρ , similarly large as those seen atthe LHC, but for φ mesons, ρ , > ; c.f. figure 5. Liang and Wang pointed outthat hadronization via polarized quark fragmentation could result in ρ , > . Thismechanism may be most important at large rapidity or transverse momentum, but could in principle play a role at midrapidity, where these measurements are made.However, naively, if fragmentation is the dominant hadronization mechanism, K ∗ and φ should be a ff ected similarly. Furthermore, it would seem natural that quarkhadronization would be more important at LHC energies than at RHIC.Sheng, Olivia and Wang [37] propose that an entirely new physical e ff ect couldbe at play, in which a hypothetical mean φ field couples to the system angular mo-mentum. Depending on the values of several parameters, ρ φ , could be greater or lessthan . In principle, by fine-tuning [37] the energy dependence of four parameters,this model might accommodate ρ φ , > at RHIC energies and ρ φ , < at the LHC.Because this model is not expected to apply to K ∗ mesons [37], it will be importantto identify independent measurements that can constrain and verify its assumptions.In summary, it is clear that the situation with spin alignment of vector mesons isvery di ff erent than that for hyperon polarization. In the latter case, the equilibriumhydrodynamic paradigm which works well for other aspects of heavy ion collisionsseems a reasonable starting point; polarization then allows a more sensitive probeof the system evolution at the finest scales. For the vector mesons, however, it isclear that observations cannot be explained by this established paradigm. Compet-ing e ff ects from multiple hadronization mechanisms, hadronic e ff ects, and novelmean fields may be at play, di ff erentially a ff ecting the di ff erent particle species anddi ff erent collision energies. See Chapter 7 [20] by Gao, et al in this Volume, foran extensive discussion. The phenomenon of vector meson spin alignment deservescontinued intense theoretical focus; at the moment, the situation is too unclear tosummarize what might be learned. Among the most pressing issues in the field of heavy ion physics is the existenceand consequences of an intense, long-lived magnetic field. Its presence could allowexperimental access to novel e ff ects due to chiral symmetry restoration. Because Λ and Λ have opposite magnetic moments, a strong B -field at hadronization wouldlead to a polarization “splitting” [35, 38, 39]. The magnitude of the splitting remainsbelow the statistical sensitivity of existing measurements, but the ongoing BES-IIcampaign at RHIC is expected to either discover the splitting or set meaningfullimits on possible magnetic e ff ects.While the average (“global”) polarization must align with the total angular mo-mentum of the collision, hydrodynamic and transport simulations predict a rich flowstructure featuring nontrivial local vorticity. The longitudinal polarization results infigure 4 represent the first observation of such an e ff ect. However, more complicatede ff ects may be present on an event-by-event basis, leading to vorticity “hot spots”that may be revealed by spin-spin correlations [40]. Experimental searches for sucha signal are ongoing at RHIC, but two-particle tracking artifacts make them highlychallenging. trongly Interacting Matter Under Rotation: An Introduction 13 As discussed above, the physics driving vector meson spin alignment is appar-ently much more complicated than that behind Λ polarization. The STAR Collab-oration at RHIC has presented a nearly finalized study [32] of polarization of Ξ and Ω hyperons which appear consistent with the Λ polarizations, with small mass-dependent e ff ects.While the total system angular momentum decreases with reduced collision en-ergy, the largest global polarization is observed at the lowest energy. It will be im-portant to measure polarization at still lower energies, below the energy thresholdfor QGP formation and at energy densities below the limits of applicability of hydro-dynamics. The BES-II program at RHIC includes a fixed-target campaign alreadyproducing results [41] in this regime. Much higher statistics datasets at low energyare expected at the NICA and FAIR facilities soon to commence operation.The energy dependence of the polarization signal may reflect an evolution of ro-tational flow structure away from midrapidity, where measurements have focusedthus far, as the collision energy increases. Experiments with good tracking nearbeam rapidity may probe strong vorticity resulting from the breakdown of longitu-dinal boost-invariance, challenging hydrodynamic and transport simulations morestringently than possible previously. Over several decades, the field of relativistic heavy ion physics has matured andfocused on the creation and study of the quark-gluon plasma. Hydrodynamics andtransport theory have provided a useful paradigm in which to interpret a wide di-versity of experimental results from high-energy collisions at RHIC and the LHC.Theory and models based on this paradigm have become increasingly sophisticated,simulating the entire evolution of the dynamic system and making quantitative con-nection to the initial state and fundamental transport coe ffi cients.The observation of rotational phenomena has opened an exciting new directioninto this well-developed and fertile environment, a rare example of a truly new de-velopment in a mature field. First measurements of global hyperon polarization arelargely consistent with predictions from existing hydrodynamic and transport sim-ulations, indicating that the tools are at hand, to understand the phenomenon. Moredi ff erential measurements, of the azimuthal dependence of global and longitudinalhyperon polarization, are more di ffi cult to understand; the e ff ects have magnitudesin line with standard expectations, but reproducing the sign of the observed oscil-lations may require nontrivial revisions to our current understanding. On the otherhand, vector meson spin alignment– presumably related to hyperon polarization–is quantitatively and qualitatively impossible to understand solely in terms of thehydrodynamic paradigm that successfully explains other observables; here, theremay be numerous competing e ff ects that depend nontrivially on particle species andcollision energy, including a newly-proposed coherent mesonic mean field. Thus, it appears that the new phenomena of strongly interacting QCD matter un-der rotation may be addressed by current theory and models, while at the same timerequiring new insights. The contributions to this book represent a broad sample ofsome of the early theoretical e ff orts– from fundamental theory to phenomenology–to determine the physics behind these phenomena. New insights are bound to resultfrom continued theoretical focus and upcoming experimental results. The followingpages are the first chapters in what will surely be a much longer story. Acknowledgements
This work is supported in part by U.S. Department of Energy grant DE-SC0020651and by U.S. National Science Foundation grant PHY-1913729.
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