TThe smallest fluid on earth
Bj¨orn Schenke
Physics Department, Brookhaven National LaboratoryBldg. 510A, Upton, NY 11973, USAE-mail: [email protected]
May 2021
Abstract.
High energy heavy ion collisions create quark gluon plasmas that behavelike almost perfect fluids. Very similar features to those that led to this insight havealso been observed in experimental data from collisions of small systems, involvingprotons or other light nuclei. We describe recent developments aimed at understandingwhether, and if so how, systems that produce relatively few particles (orders ofmagnitude less than in typical heavy ion collisions) and are only one to a few timesthe size of a proton, can behave like fluids. This involves a deeper understanding offluid dynamics and its applicability, improvements of our understanding of the initialgeometry of the collisions by considering fluctuations of the proton shape, as well asadvancements in the calculation of initial state effects within an effective theory ofquantum chromodynamics, which can affect the observables that are used to studyfluid behavior. We further address open questions and discuss future directions.
1. Introduction
Matter produced in collisions of heavy ions at high energy as performed at theRelativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) has beenshown to behave like an almost perfect fluid, i.e. a fluid with little to no viscosity.Refs. [1, 2, 3, 4, 5, 6, 7, 8] are some of the first publications from each experimentalcollaboration at RHIC and LHC. Many more detailed studies of more complexobservables have followed since and generally support the fluid interpretation strongly.See [9, 10, 11] for reviews.This fluid is of the size of a nucleus, approximately 10 − meters in diameterand reaches temperatures 100,000 times greater than those in the core of the sun.Measurements in smaller collision systems, namely proton+proton (p+p) [12] andproton+lead (p+Pb) [13, 14, 15] collisions at LHC, as well as proton+gold (p+Au)[16, 17], deuteron+gold (d+Au) [18, 19, 20, 17], and He+Au [21, 22] collisions at RHIC,have shown similar behavior as that observed in heavy ion collisions, and have triggereda variety of new theory developments to understand if we are creating an even smallerfluid (approximately 10 times smaller in diameter) in these collisions, or whether otherphenomena, such as color correlations of dense gluon fields in the incoming projectile a r X i v : . [ nu c l - t h ] F e b he smallest fluid on earth What is a fluid and how small can it be? – The applicability of hydrodynamics
Hydrodynamic simulations with an appropriate initial state model for the fluctuatinggeometry do a surprisingly good job in describing the systematics of the measuredazimuthal anisotropies in the transverse momentum distribution of produced particlesin systems with only 10 or more charged hadrons produced per unit rapidity. Thesemomentum anisotropies are the simplest and cleanest observables that have revealedfluid like behavior in heavy ion collisions. Consequently, a lot of recent effort hasbeen invested into understanding how hydrodynamics can be a good description ofrather dilute systems, and how quickly a system can approach hydrodynamization ,i.e., can reach a state in which hydrodynamics provides a proper description. Thisinvolves the study of hydrodynamic vs. non-hydrodynamic modes, hydrodynamicexpansions in strong and weak coupling calculations, as well as explicit simulationsof the Boltzmann equation or dynamics of shockwave collisions within the framework ofAnti-de Sitter/Conformal Field theory correspondence (AdS/CFT). The upshot fromthese studies is that hydrodynamics provides a good description of a system for ratherlow numbers of particles and at times that are significantly earlier than isotropizationor equilibration times.
What is the shape of the smallest fluid? – Subnucleonic structure
A variety ofexperimental data, ranging from diffractive vector meson production in electron+proton(e+p) collisions, to p+p collisions, to other small system and even heavy ion collisions,indicate that the proton’s shape itself fluctuates from event to event. We will reviewwork that describes proton+nucleus (p+A) collisions using the Color Glass Condensate(CGC) Effective Field Theory (EFT) coupled to fluid dynamic simulations, which canonly get close to the experimentally observed azimuthal momentum anisotropies of theproduced charged hadrons when fluctuating subnucleonic structures of the proton areincluded. We will discuss how to constrain these fluctuations using incoherent diffractivevector meson production in e+p collisions, which was measured at the Hadron-ElectronRing Accelerator (HERA) at the Deutsches Elektronen-Synchrotron (DESY). Usingthese constrained fluctuating protons instead of previously assumed approximatelyspherical protons improves agreement with the experimental data significantly. We willfurther review how the subnucleon structure evolves with energy, which is calculablewithin the CGC EFT. This allows comparison to the center of mass energy dependenceof cross sections measured at HERA, and is important to predict the collision energyand rapidity dependence of many observables in small system collisions at RHIC andLHC. We will further discuss other recent calculations that focus on either the role ofcolor charge fluctuations or nucleon position fluctuations in heavy ion collisions.
More than a fluid – Initial state momentum anisotropies
While the case for theimportant role of final state effects in small system collisions that produce many particlesis very strong, other sources of anisotropies have been increasingly discussed after he smallest fluid on earth
2. What is a fluid and how small can it be? – The applicability ofhydrodynamics
In this review we shall call a fluid any system that is well described by (viscous) fluiddynamics (or more colloquially hydrodynamics). Hydrodynamics has been found todescribe particle production in small system collisions well [30, 31, 32, 33, 33, 34, 35, 36,37, 38, 39, 40, 41]. This includes particle spectra as functions of transverse momenta,as well as more detailed observables, such as the so called flow harmonics v n , whichmeasure the azimuthal anisotropy of particle production in the plane transverse to thebeam line.However, one is naturally driven to ask whether it makes sense to believe thathydrodynamic behavior is achieved in a system that produces as little as 10 chargedhadrons per unit rapidity. Considering that the applicability of hydrodynamics isdebatable even in heavy ion collisions, the presence of a significant hydrodynamicphase in small system collisions may seem unlikely. In heavy ion collisions, the issueis our lack of a detailed understanding of how an initially extremely anisotropic andrapidly expanding system can approach local isotropy in momentum space [42] (oreven local thermal equilibrium) on a time scale that is at least an order of magnitudeshorter ( O (1 fm /c ≈ . × − s)) than the total lifetime of the system ( O (10 fm /c )).Calculations relying on microscopic theories in both the weak and strong couplinglimits predict pressure anisotropies of order 50%, meaning that the transverse pressure(perpendicular to the beam line) is on the order of two times larger than the longitudinal(along the beam direction) pressure, at a time of approximately 1 fm /c after the collision he smallest fluid on earth hydrodynamizes , i.e., can be described byhydrodynamics. In other words, there exists a non-equilibrium attractor for the energymomentum tensor (i.e., the energy momentum tensor evolves toward this attractorsolution for a wide variety of initial conditions), which is well described by viscoushydrodynamics after a time of approximately 1 fm /c or less [49, 50, 51, 52, 53, 54, 55,56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. For a detailedreview see [54].A way to understand the phenomenon of hydrodynamization, i.e., the applicabilityof hydrodynamics in systems far from equilibrium, is to consider the various modes,which can be either transient modes or long-lived hydrodynamic modes. In a typicalsituation, as time evolves transient modes decay exponentially fast (on a time scalethat depends on the details of the microscopic theory), hydrodynamic modes beginto dominate, and one approaches a quasi-universal attractor behavior (We note thatthere are also far-from-equilibrium, early-time attractors, that can be reached prior tothe late time hydrodynamic ones [72, 76]). So it is likely the case (the universality ofthis statement has not yet been proven) that approaching the attractor is equivalentto achieving “hydrodynamic behavior” [54]. This means that determination of theapplicability of hydrodynamics should not rely on the smallness of subsequent terms in agradient expansion, because it is divergent (because of the presence of non-hydrodynamicmodes) [49, 50], but instead the dominance of hydrodynamic modes.Based on the momentum scales at which hydrodynamic modes vanish completelyfrom the spectrum, one can even attempt to estimate the minimal size of a droplet, forwhich hydrodynamics can apply (i.e., one estimate for ”the smallest fluid on earth”).It was argued to be ∼ .
15 fm = 0 . × − m in [43]. In terms of the global size ofsystems produced in realistic small system collisions (typically of the proton size scaleand larger), this is a small scale, however, one should keep in mind that when includingsubnucleonic structure (see Sec. 3), local hot spots can be on the order of this size scale.One should note that different viscous hydrodynamic schemes, such as Mueller IsraelStewart (M-IS) [77, 78, 79, 80], Baier Romatschke Son Starinets Stephanov (BRSSS)[81], Denicol Niemi Molnar Rischke (DNMR) [82], or anisotropic hydrodynamics[83, 84, 85, 86, 87, 88, 89] (see [90, 91] for reviews and more references), all havedifferent transient modes. How well they describe a given microscopic theory onshort time scales depends on how well they reproduce the transient modes of thattheory. In many cases, anisotropic hydrodynamics (both the leading order kind, whichassumes a spheroidal particle distribution function, and even more so the next toleading order implementation, which allows additional arbitrary viscous corrections to he smallest fluid on earth η/s = 1 / π ) [86, 87, 88, 92, 93].The differences between different hydrodynamic schemes could well be important insmall collision systems, where the total lifetime of the fireball is often less than 3 fm /c .However, before one can make any conclusive statements about which hydrodynamicscheme is superior for describing nuclear collisions, one needs to better understand theunderlying microscopic non-equilibrium theory in the first place. Recent progress onthat front includes the study of non-equilibrium dynamics in weak coupling asymptotics[94], which shows consistency with the bottom-up thermalization scenario [95], and theimplementation of effective QCD kinetic theory [96] to describe the early time evolutionbefore hydrodynamization [97, 98]. The latter is a good description for a more dilutesituation, where quantum effects are important, and which is reached rather rapidlyvia expansion, even when starting with an overoccupied system described by classicalfields and the Yang-Mills equations (like in the IP-Glasma model [99, 100]). Goingfrom one limit to the other smoothly is possible because there is a regime of occupationwhere both descriptions (classical Yang-Mills fields and distribution functions of kinetictheory) are valid [101].On a microscopic level, the coupling in the kinetic theory (or alternative) descriptionmust be strong enough to overcome the rapid expansion of the system and approach thehydrodynamic attractor for a system with a realistic small shear viscosity to entropydensity ratio of several times 1 / π . Therefore, calculations in kinetic theory areperformed at weak coupling and then usually extrapolated to values of the couplingthat are compatible with the proper hydrodynamic description [97, 98, 102]. There arealso indications [103, 104] that different underlying theories produce similar backgroundevolution and response functions as used in the effective kinetic theory description of[97, 98], such that the approach to the attractor may be well approximated even if thetrue underlying theory is not perfectly captured by the effective kinetic model.Consequently, one strategy for a complete description of a heavy ion (or smallsystem) collision is to start with a classical framework, such as the IP-Glasma model,couple it to an effective kinetic theory description, and then transition to viscoushydrodynamics (followed eventually by microscopic hadronic transport as the systembecomes dilute again at late times). Including the proper early time non-equilibriumtransport phase could have a strong effect on the production of electromagnetic probes insmall systems, as they are produced throughout the entire evolution, and once producedare no longer modified. Investigations of non-equilibrium photon production is ongoing[105, 106, 107] and could shed more light on the details of the complex early timedynamics, especially in small systems. Furthermore, photons and dileptons should alsobe sensitive to chemical equilibration (between quark and gluon degrees of freedom)[108, 109, 110, 111], as delayed equilibration would suppress the early time photon anddilepton yield (gluons will dominate for longer), affecting both spectra and v n of these he smallest fluid on earth
3. What is the shape of the smallest fluid? – Subnucleonic structure
The anisotropy in the particle production transverse to the beam line is characterizedby Fourier coefficients v n , which represent the amplitudes of the cos( nφ ) modulation,with φ being the azimuthal angle and n an integer. In the hydrodynamic framework,these anisotropies are generated by the response of the strongly interacting system tothe initial geometry in the transverse plane of the collision. In heavy ion collisions, oddflow harmonics, such as v , v , etc., ‡ along with the details of the even harmonics,are driven by fluctuations in the initial geometry [122] (the average geometry has asymmetry that allows only even harmonics). The details of these fluctuations modifythe event by event distribution of the flow harmonics, and by that their cumulants,which are accessible by studying multiparticle correlations, e.g. of the produced charged ‡ Directed flow v has a rapidity even component, that is entirely driven by fluctuations, but it alsohas a rapidity odd component, that is finite even in the case of no fluctuations. he smallest fluid on earth − − − − x [fm] − − − − y [ f m ] a) − − − − x [fm] − − − − y [ f m ] b) Figure 1.
Profile of the energy density distribution (normalization arbitrary) inthe transverse plane in a single 5 .
02 TeV p+Pb collision computed in a) a Monte-Carlo Glauber model using nucleon degrees of freedom, where the energy density isproportional to the sum of thickness functions of all wounded (participating) nucleons,and b) the IP-Glasma model assuming round nucleons. Thin circles represent thenucleon positions in the Pb nucleus, the thick circle is the proton projectile position.Nucleon positions are the same for both cases a) and b). hadrons [123, 124, 125]. Similarly, if dominated by the same process, namely the finalstate response to the initial geometry, v n should also be driven by fluctuations in smallsystems. In this case, considering for example p+A collisions, also the even harmonicsare strongly dominated by event by event fluctuations, as the average ellipticity is closeto zero for central collisions.When using fluctuating Monte Carlo Glauber type models [126, 127] to initializesmall system collisions, nucleon degrees of freedom (i.e., using nucleons without anysubstructure) have been shown to provide sufficient fluctuations to produce azimuthalanisotropies close to those measured in experiments at RHIC and LHC [30, 31, 32, 33,33, 34, 35, 36, 37, 38, 39, 40, 41, 128]. This is possible only for certain choices for theenergy or entropy deposition. With a proton projectile, assuming a spherical proton,fluctuations can only originate from the fluctuating positions of nucleons in the heavyion target. Thus, energy (or entropy) deposition that is proportional to a sum of theprojectile ( A ) and target’s ( B ) wounded nucleon nuclear matter densities, characterizedby the thickness functions ( ∼ T A + T B ) § is able to produce rather lumpy structures inthe transverse plane of the collision (with fluctuations on the nucleon size scale), whichis necessary to produce the large v n observed experimentally (See Fig. 1 a)).However, several arguments disfavor this type of energy deposition. First, § The thickness function T A is defined as the integral over the three dimensional nuclear densitydistribution along the direction of the beam line. he smallest fluid on earth − − x [fm] − − y [ f m ] Figure 2.
The color map shows the energy density distribution (arbitrary units) inthe plane transverse to the beam direction for a p+Pb collision. The contour linesindicate the shape of the projectile proton (quantified using a measure of the gluondensity in the proton). in a model that parametrizes the functional dependence of the deposited entropydistribution on the thickness functions (TRENTo) [129], the consistent outcome of arange of Bayesian analyses has been that the experimental data prefers an entropydensity proportional to the square root of the product of the two thickness functions[130, 131, 132, 133]. In small systems, in particular p+A collisions, this would lead to toolittle fluctuations in the transverse geometry, such that subnucleon structure was arguedto be essential in order to describe both p+Pb and Pb+Pb systems simultaneously usingthis initial state model [134].Furthermore, the IP-Glasma initial state model [99, 100], that is based on an actualeffective theory of QCD, the CGC [135, 136, 137, 138], predicts that the initial energydensity is proportional to the product of the two thickness functions. While slightlydifferent from the TRENTo model result, such a dependence will also lead to extremelysmall spatial fluctuations in the initial geometry when one assumes a round proton,see Fig. 1 b). This, along with the resulting small azimuthal anisotropies, was shownexplicitly in [139]. Interestingly, a similar dependence on the thickness functions as inthis weak coupling limit calculation is also obtained in the extremely strong couplinglimit [140, 141] using AdS/CFT correspondence [142].If the energy deposition is proportional to the product of thickness functions,geometric fluctuations large enough to generate as much anisotropic flow as observed he smallest fluid on earth N part andnumber of nucleon-nucleon collisions N coll in the well known form [(1 − x ) (cid:104) N part (cid:105) / x (cid:104) N coll (cid:105) ] (with a free parameter x ) only works well to describe particle production insymmetric systems [143]. Additionally, because particle production in the constituentquark picture does not explicitly depend on N coll , it leads to a better description of thecorrelation of v and N ch in ultracentral U+U and Au+Au collisions, compared to thatobtained with the two-component Glauber ansatz [146].For p+p collisions, it was shown [147] that the hollowness effect [148, 149, 150, 151],which refers to the inelasticity density of the collision not reaching its maximum at zeroimpact parameter and was observed in p+p collisions at √ s = 7 TeV [152], can alsobe explained when considering subnucleonic hot spots in the proton. Finally, there areindications for size fluctuations of the proton in jet measurements in p+Pb or d+Aucollisions as a function of centrality [153].So there are plenty of indications for the relevance of a fluctuating nucleonsubstructure, but the details of what this structure looks like are less clear. As thereare no first principles calculations available yet, that would determine the fluctuatinggeometric structure of a nucleon, there are two possible ways to proceed. First, one canparametrize the substructure, typically as a combination of N hot spots with variabledistributions of their width and position within the nucleon. Then, to constrain theparameters one can perform a Bayesian analysis of heavy ion and small system collisionsand this way optimize the agreement with experimental data. This procedure wasadopted in [134] (also see [133]). In this particular work, the number of subnucleonichot spots could not be very well constrained, but a number greater than one (onecorresponds to no substructure) was favored. The widths of the subnucleonic hot spotswas on the other hand tightly constrained to approximately 0 . nucleon width in the IP-Glasma model.Alternatively, one can attempt to constrain the subnucleon structure usingindependent measurements, e.g. from e+p collisions. As it was demonstrated in [154],there is an exclusive process in e+p collisions that is particularly sensitive to geometricalfluctuations of the gluon distribution in the proton, namely the incoherent diffractive he smallest fluid on earth | t | (where | t | is the transverse momentum transfersquared) contains information on the average size of the target, but more interestingfor our purpose, the incoherent differential cross section is proportional to the varianceof the scattering amplitude, making it sensitive to its fluctuations, including those ofgeometric nature.Motivated by discussions in [155, 156, 157, 158, 159], it was thus suggested in [154]to use data on diffractive J/ψ production in e+p collisions at HERA [160, 161, 162, 163,164] to constrain the proton average shape and fluctuating substructure within the IP-Glasma model (see also [165]). The sensitivity of the incoherent differential cross sectionto the substructure turned out to be rather dramatic. As shown in Fig. 3 (left), while thecoherent diffractive cross section can always be approximately described by adjustingthe average shape, describing the experimentally determined incoherent diffractive crosssection requires the presence of substantial geometric fluctuations. Comparing to theresult that assumes a Gaussian thickness function of the proton and includes only colorcharge fluctuations, the result obtained using three Gaussian hot spots (whose radius isapproximately three times smaller than the proton radius) produces an incoherent crosssection that is significantly larger, and also has a shape in | t | that is much closer tothat of the experimental data. (cid:107) Three hot spots are motivated by the presence of threevalence quarks, around which one assumes the gluons to be clustered. Other numbers ofhot spots are certainly conceivable, as there could be more large x degrees of freedom,such as large x gluons or sea quarks, around which smaller x gluons can cluster. Anadditional fluctuation of the normalization for each hot spot is also included, whichmainly affects the low | t | part of the incoherent spectrum.As an illustration of the degree of fluctuations required by the incoherent diffractiveHERA data, we also show four example protons in Fig. 3, visualized by plotting the realpart of the trace of the gluon Wilson lines in the transverse plane (divided by N c ),which can be loosely taken to represent the density of gluons. The pictures reveal thethree hot spots that have both fluctuating intensity and positions. We stress that thelength scale for the subnucleon hot spots is not derived from theory, but extracted fromexperimental data. It is not clear exactly what sets this intermediate scale (between1 / Λ QCD and 1 /Q s , where Q s is the saturation scale), but it must emerge from thedynamics of quarks and gluons at intermediate x . Perhaps lattice QCD calculationswill be able to address this question in the future.One can now explore the effect of including these constrained subnucleonfluctuations in the initial condition of a hydrodynamic simulation of p+Pb collisionsfor example. Since above calculations of diffractive J/ψ production were done in aframework that is identical to the pre-collision stage of the IP-Glasma model, the (cid:107)
Many different models for subnucleonic spatial fluctuations in the proton are conceivable. Forexample, one could base a model on the spin fluctuations in the proton, as done in [166, 167]. he smallest fluid on earth . . . . . . | t | [GeV ]10 − − d σ / d t [ nb / G e V ] Coherent w fluctuating geometryIncoherent w fluctuating geometryCoherent w/o fluctuating geometryIncoherent w/o fluctuating geometryH1 coherentH1 incoherent − . − . . . . y [ f m ] − . − . . . . x [fm] − . − . . . . y [ f m ] − . − . . . . x [fm] Figure 3.
Left: Coherent (black) and incoherent (red) production of
J/ψ vector mesons in e+p collisions [154, 168] compared to measurements from theH1 Collaboration [164]. Solid lines represent calculations including a fluctuatinggeometric substructure of the proton with three hot spots, dashed lines assume around geometry and only include color charge fluctuations. Right: Four examples ofproton configurations illustrating the degree of fluctuations necessary to describe theincoherent data on the left. Shown is (one minus) the real part of the trace of theproton’s Wilson lines (normalized by 1 /N c ), which can be interpreted loosely as agluon density. implementation in IP-Glasma is trivial. A full calculation using the constrainedsubnucleonic fluctuations was first done in [169] where significantly larger v n comparedto those for round protons [139] were found. To demonstrate the increase of the v n andthe improvement when comparing to experimental data, we show the results for thecharged hadron v { } and v { } (where the { } indicates a root-mean-square measureobtained from 2-particle correlations) in 5.02 TeV p+Pb collisions from [170], whichuses a fluctuating nucleon structure together with the same calculation but assuminground nucleons in Fig. 4.The difference between the two cases is dramatic, especially for v , where for themost central events shown it is approximately a factor of 2.5. For v the increasefrom including subnucleon structure is approximately a factor of 1.5. In the case of v the experimental data is still underpredicted. One should note that the p+Pb resultis a true prediction, as tuning of the model was only done for Au+Au collisions atRHIC energies. It is thus conceivable that a better description could be achieved whenincluding all systems in a coordinated tune of the model. Nevertheless, the importanceof subnucleon fluctuations is clearly demonstrated here, and, at least when focusing on v , p+Pb collisions at the LHC and diffractive vector meson production in e+p collisionsseem to favor a similar degree of subnucleonic fluctuations in the proton. he smallest fluid on earth N ch ( | η | < . . . . . . . . . . . v { } p+Pb 5.02 TeV a) η/s = 0 . , ζ/s ( T ) proton with substructureround protonALICE N ch ( | η | < . . . . . . . . v { } b)p+Pb 5.02 TeV η/s = 0 . , ζ/s ( T ) proton with substructureround protonALICE − − x [fm] − − y [ f m ] “round proton” − − x [fm] − − y [ f m ] with substructure Figure 4.
The charged hadron azimuthal anisotropy harmonics from two particlecorrelations v { } (a) and v { } (b) computed within the IP-Glasma+ Music +UrQMDhybrid model assuming a round thickness function for the proton (dashed lines) orproton substructure with three hot spots in the gluon distribution (solid lines). Theincrease of the v n , driven by the increased fluctuations and larger eccentricities, isparticularly dramatic for n = 2. Experimental data from the ALICE Collaboration[171]. On the right, we show the energy density distribution in the transverse plane(arbitrary units) for a typical collision with “round” nucleons on the top and oneexample collision using nucleons with substructure on the bottom. As an illustration, Fig. 4 also shows the energy density distribution in the transverseplane for a typical central p+Pb collision with “round” nucleons on the top right and oneexample collision using nucleons with substructure on the bottom right. Eccentricitiesin the latter case are significantly larger compared to the round nucleon calculation,explaining the increase in the v n when including subnucleonic fluctuations.With the Electron Ion Collider (EIC) [29, 173] on the horizon, more precisemeasurements of diffractive vector meson production have the potential to betterconstrain the subnucleonic structure not only of protons, but also light [174] and heavynuclei (c.f. [175], where ultraperipheral Pb+Pb collisions at LHC are discussed and canprovide similar information). For e+p collisions it was recently suggested [176] to alsomeasure the dependence of the | t | differential incoherent diffractive cross section on theazimuthal angle between the produced vector meson and the scattered electron, to getan additional handle on the substructure fluctuations. Of particular interest would alsobe the energy ( W , the center-of-mass energy in the virtual photon-proton scatteringprocess), or Bjorken x dependence of the incoherent (and coherent) cross section, forwhich predictions were made [172] using the CGC framework including Jalilian-MarianIancu McLerran Weigert Leonidov Kovner (JIMWLK) evolution [177, 178, 179, 180].We show the ratio of the incoherent to coherent cross section in Fig. 5 (left), where aclear decrease of the ratio with increasing W is visible, in contrast to a calculation in theIPSat model [181, 182, 183], where the evolution of the spatial structure is neglected.Several physics effects lead to this behavior: The JIMWLK evolution will be faster he smallest fluid on earth W [GeV]0 . . . . . . . . i n c o h e r e n t / c o h e r e n t JIMWLK evolutionIPSatH1 coherent − y [ f m ] ∆ y = 1 . − x [fm] − y [ f m ] ∆ y = 3 . − x [fm] ∆ y = 5 . Figure 5.
Left: The ratio of the W dependent incoherent to coherent cross sectionsfrom [172] compared to H1 data [164]. The solid line is a CGC calculation forfluctuating protons including JIMWLK evolution, the dashed line is the IPSat modelcalculation, which does not include the evolution of the proton size or the fluctuationscale. Right: The energy (rapidity, or x ) evolution of one example configuration of aproton. ∆ y indicates the evolution in rapidity from the initial configuration shown inthe upper left. Shown is (one minus) the real part of the trace of the proton’s Wilsonlines (normalized by 1 /N c ). in local regions with small saturation scale than inside the hot spots, which are closer tothe saturated regime. This leads to an effective growth of the hot spots and consequentlya smoother proton. In addition, as overall Q s values increase with evolution, the sizescale of color charge fluctuations ( ∼ /Q s ) decreases, producing more “color domains”and effectively decreasing geometrical fluctuations. In the extreme black disk limit, oneexpects the coherent cross section to dominate, as it receives contributions from theentire proton area, while the incoherent cross section is only sensitive to the edge of theproton.In Fig. 5 (right) we show the corresponding visualization of the proton as itundergoes JIMWLK evolution, using the same quantity as in Fig. 3 (right). Both thegrowth of the proton and the decrease in the length scale of color charge fluctuations(which are the shortest scales for all rapidities) with increasing rapidity (decreasing x )are clearly visible (also see [184]). The details of this evolution depend on an infraredregulator in the JIMWLK kernel, which is required to avoid violation of the Froissartbound, which based on unitarity arguments puts a constraint on the growth of theinelastic cross section with the collision energy [185, 186]. Work on finding constraintsfor this regulator, for example in the Gribov-Zwanziger approach for confinement, isongoing [187].Implementation of a detailed x dependence of the fluctuating nucleon structure in he smallest fluid on earth x valuesand thereby modify the properties of the projectile and target. For first studies of theimplementation of JIMWLK evolution in heavy ion collisions see [188, 189].It should be noted that an attempt was made within the CGC effective theoryat explaining the azimuthal anisotropies (mainly in large systems) from color chargefluctuations of the energy momentum tensor alone, i.e., without including nucleondegrees of freedom at all [190, 191]. The authors start from the connected two-pointfunction of the glasma energy-momentum tensor derived in Ref. [192], and by meansof an approximation, which neglects all logarithmic corrections to the fluctuations ofthe glasma energy density, they obtain expressions for the eccentricities of the systemwhich turn out to be compatible with experimental data for realistic choices of themodel parameters. However, it has been recently realized [193] that as soon as theapproximations made in the calculation are relaxed, and one makes use of the fullMcLerran Venugopalan (MV) model expressions of Ref. [192], the idea breaks down, asthe eccentricities become smaller by one order of magnitude (in agreement with whatthe numerical calculations within the IP-Glasma model for a smooth nucleus wouldyield) and can no longer be used to describe the experimental data.Reversely, another work, which neglects almost everything but the nucleon positionfluctuations of the CGC calculation, and dubbed “Jazma” [194], reproduces theeccentricities of IP-Glasma in heavy ion collisions well, emphasizing the importanceof nucleon position fluctuations, and the limited importance of color charge fluctuationsfor the initial geometry. However, the model can not produce the full energy momentumtensor of IP-Glasma, which in particular means that it misses the initial state momentumanisotropies that turn out to be important for very small, low multiplicity systems, aswill be discussed in detail in Sec. 4.In summary, small system collisions have contributed significantly to our improvingunderstanding of the subnucleonic structure. Many observables in a wide variety ofexperiments, ranging from e+p and e+A collisions, to p+p, p+A (and other smallsystems), to A+A collisions, prefer a description that includes some level of fluctuatingsubnucleonic structure, whose details are yet to be understood. Future studies at theEIC, along with new results from small system and heavy ion collisions, are likelyto fill in the gaps and provide a much deeper understanding of the fluctuating spatialnucleon and nuclear structure (in addition to the expected advances in understanding theaverage structure via generalized parton distributions (GPDs) [195, 196, 197, 198, 199],generalized transverse momentum dependent parton distributions (GTMDs) [200, 201,202, 203], and Wigner distributions [204, 205]). he smallest fluid on earth
4. More than a fluid – Initial state momentum anisotropies
It was predicted in [206, 207, 208, 209, 210, 211] that multi-gluon production from theCGC leads to long range rapidity correlations that contain azimuthal anisotropies. Withthe experimental discovery of long range momentum anisotropies in p+p collisions andother small systems [12, 13, 14, 15, 19, 21] many more theoretical calculations, mostprominently within the CGC effective theory, were triggered to explain the observations.In these frameworks, no final state effects are necessary to obtain a finite ellipticanisotropy (and in some cases also odd harmonics). We will focus on CGC calculations inthis review, but mention that different (but in principle related) frameworks have alsobeen used to try and explain the long range azimuthal anisotropies in small systemswithout the need for strong final state effects [212, 213].CGC calculations are based on solutions of the Yang-Mills equations, which can beobtained numerically, or, under a variety of simplifying assumptions, analytically. Earlycalculations were based on the glasma graph approximation, which limit the interactionsto maximally two-gluon exchanges and uses Gaussian statistics for the initial colorcharges [207, 208, 209, 214, 215, 216]. Keeping Gaussian statistics but resumming multi-gluon exchanges leads to the non-linear Gaussian approximation [217, 218, 219]. Whentreated fully numerically [220, 221, 222, 99], multi-gluon exchanges are included and onehas the freedom of using any color charge statistics and realistic spatial distributions[223]. Finally, quantum effects can be included to leading logarithmic order in ln(1 /x )by evolving the initial color charge distributions using e.g. the JIMWLK equations [224].Typically, all of the above calculations find non-zero even harmonics in the long-range two particle correlations, as they have a symmetry in (cid:126)k → (cid:126)k , (cid:126)k → − (cid:126)k and (cid:126)k → − (cid:126)k , where (cid:126)k , are the transverse momentum vectors of gluon 1 and 2, respectively.Odd harmonics for gluons generally require an extension of the calculation to includinga finite time of Yang-Mills evolution [225, 223, 226], going beyond dilute and beyondclassical approximations [227, 228] (see a discussion of the dilute approximation in [229]),or including non-eikonal effects [230, 231] (also see [232, 233] - and comments in [227]- on how odd harmonics could emerge from the Bose-Einstein correlations of identicalparticles). Quark production within the CGC as in [234, 235, 236] does not have theabove mentioned symmetries and contains odd harmonics.The first question one is bound to ask is how such anisotropies emerge within theCGC calculations. There are in fact a variety of sources of anisotropy, both of classicaland quantum nature.The first emerges from the fact that gluon fields are correlated within a correlationlength of 1 /Q s , which one can loosely interpret as having color field domains of thatsize. Particles, that scatter (or are produced) from the same domain, are correlated asthey feel the same color field. Also density gradients can contribute to anisotropies fromthe CGC. This is particularly evident in the scattering of a quark-antiquark dipole offa target [237]. The cross section for dipole target scattering will depend on the dipoleorientation if there are significant density gradients in the target. Both effects are purely he smallest fluid on earth /Q s ,which turns the projectile after the scattering into a collection of sources for incoherentemission.Having established that and how correlated gluons are produced within the CGCframework, the second question that emerges is of course whether the resulting initialstate momentum anisotropies can by themselves explain the observed anisotropies in theexperimental data. While early works that studied the associated yield within the nearside ridge in p+p and p+Pb collisions (and combined jet as well as glasma contributionsto the yield) showed good agreement with the experimental data [241, 242], more recentcalculations, comparing both parton level [236] and hadron level results [243] in p+Pbcollisions to experimental data, underestimate the experimentally observed second andthird order harmonics. Hadron level results for azimuthal anisotropies in p+p collisionsalso underestimate the experimental anisotropies [244]. The difference to the earliercalculations may be that a) It is possible that the harmonics reveal more detail thanthe associated yield, and b) that including the jet contribution (in the data and thecalculation) played a non-negligible role.Having to describe hadronization outside of a fluid dynamic description that canmake use of an equation of state, complicates the description of the experimentaldata with purely initial state models (for recent progress see [243]). However, so far,the systematics of the experimental data with multiplicity or collision system (e.g.p+Au, d+Au, He+Au, that were studied at RHIC [245]) could not be reproducedin such frameworks, at least for multiplicities larger than or equal to the minimumbias multiplicity in p+A collisions. This suggests that final state effects are necessaryto describe the data, at least for large enough multiplicities. We note that a varietyof observables, such as multiparticle cumulants [246, 247, 248, 249, 250, 251] or themass ordering of anisotropy coefficients [34, 37, 244, 243] were suggested to distinguishbetween initial state and final state pictures. While the CGC calculations strugglewith getting the right magnitudes, some qualitative behavior, expected from final stateeffects, could also be reproduced from the initial state alone [252, 244, 243].It is conceivable that as the multiplicity of the collision decreases, final state effects,which rely on the production of a sufficiently strongly interacting system, become less he smallest fluid on earth ρ ( v , [ p T ]) = (cid:104) δv δ [ p T ] (cid:105) (cid:112) (cid:104) ( δv ) (cid:105)(cid:104) ( δ [ p T ]) (cid:105) , (1)where [ p T ] indicates the mean transverse momentum in a single event, v is the ( p T -integrated) elliptic anisotropy of the particle spectra, and δ indicates the differencebetween the single event value and the event average. Typically, ρ ( v , [ p T ]) is studiedat fixed multiplicity [255]. It has now been studied experimentally in Pb+Pb andp+Pb collisions by e.g. the ATLAS Collaboration [256, 257, 255] and is under closerinvestigation at both RHIC and LHC in a variety of systems, e.g. U+U collisions, as itis sensitive to the quadrupole deformation of the colliding nuclei [258].In hydrodynamic frameworks, all initial state models that have so far been usedto compute ρ ( v , [ p T ]) lead to similar qualitative results for heavy ion collisions, whenplotted as a function of charged particle multiplicity [253, 257, 258, 255]. These resultsfollow closely predictors for the observable that are based solely on the initial geometry.At small multiplicity (in centrality classes around 70% or greater for Au+Au and Pb+Pbcollisions) the correlator is negative, and turns positive for larger multiplicities. Thiscan be understood on the level of the predictors (which are measures of the systemsize for [ p T ] and eccentricity for v ): At small multiplicity, a small area (and by thata large [ p T ]) is achieved by clustering the participants into a single compact region,which tends to have a smaller eccentricity (resulting in smaller v ). This results in anegative correlation between [ p T ] and v . At larger multiplicity, a smaller area at fixedmultiplicity is achieved by fluctuating to a large eccentricity ε , as the area of an ellipseis given by A = π [ r ] (cid:112) − ε [255], which leads to the observed positive correlationbetween [ p T ] and v .Interestingly, one expects a positive ρ ( v , [ p T ]) at small multiplicities, if theanisotropy has initial state momentum correlations as its dominant source. This can beunderstood as follows. Taking for example the classical source of momentum anisotropymentioned above, the anisotropy decreases with the number of color domains, whichincreases with the system size. With [ p T ] being well predicted by the initial entropyper area (which is expected also in this initial state scenario, as for fixed multiplicity,which is typically considered, the initial entropy is approximately constant, such that Q s of the projectile, which drives [ p T ], decreases with increasing system size; Q s of thetarget is approximately constant), this introduces a positive correlation between [ p T ] he smallest fluid on earth dN ch /dη − . . . . . ˆ ρ ( v , [ p T ] ) p+Au 200 GeV, 0.2 < p T < IP-Glasma+MUSIC+UrQMDIP-Glasma+MUSIC+UrQMD, final state only ˆ ρ est ( (cid:15) p , [ s ]) : initial mom. anisotropy estimator ˆ ρ est ( (cid:15) , [ s ]) : geometry estimator Figure 6.
The correlation ˆ ρ ( v , [ p T ]) of the elliptic flow (squared) with the event-by-event mean transverse momentum [ p T ] at fixed multiplicity in √ s = 200 GeV p+Aucollisions. Circles indicate the simulation result including both initial momentumanisotropy and final state evolution. Squares show the expectation for initialmomentum anisotropy only, triangles the expectation for final state response to theinitial geometry only. Simulation results without initial state momentum anisotropyare shown as stars and follow the expectation for final state effects only. Figure adaptedfrom [259]. and v . This is opposite to the negative correlation produced by geometric effects insmall systems.Whether these expectations materialize can be tested in a model that containsboth initial momentum anisotropies and geometry driven effects, such as the IP-Glasma+ Music +UrQMD model, described in detail in [170] (also see [260, 261, 262]and [263, 264]). Within this model ρ ( v , [ p T ]) at fixed multiplicity was computed forsmall systems as a function of charged hadron multiplicity. The result for 200 GeVp+Au collisions is shown in Fig. 6 as solid circles (connected with solid lines). Onecan see that the sign of ρ ( v , [ p T ]) changes from positive to negative as one increasesthe multiplicity. In fact, studying the geometric predictor (triangles, dotted line) andthe predictor from the initial momentum anisotropy (squares, dashed line), one seesthat the final result moves from one predictor to the other as the multiplicity changes.This indicates that within the simulation the initial state momentum anisotropy beginsto dominate as one decreases the multiplicity to approximately 10 charged hadronsper unit rapidity, as this is where ρ ( v , [ p T ]) changes sign. A simulation with the initialmomentum anisotropy turned off (stars, dash-dotted line) agrees well with the geometric he smallest fluid on earth
5. Conclusions and Outlook
Small system (p+p, p/d/ He+A) collisions at RHIC and LHC have driven theorydevelopments towards understanding conditions for the applicability of (relativistic)fluid dynamics as well as particle production and properties of nuclei and hadrons athigh energy. We have reviewed three major developments.First, our understanding of the applicability of hydrodynamics in systems awayfrom equilibrium, using investigations of hydrodynamic and non-hydrodynamic modes,as well as attractor behavior in kinetic theory or holographic theories, has significantlyimproved over the last several years. Second, the role of subnucleon fluctuations hasbeen increasingly recognized. The substructure of the proton seems to have importantimplications for both e+p as well as p+p and p+A collisions. This connects a widerange of experiments, and new calculations modeling a hot spot substructure withinthe CGC EFT paint a consistent picture among them. Finally, the CGC predictsazimuthal correlations between produced particles even in the absence of final stateeffects. Attempts to systematically describe experimental data using only these initialstate effects have failed. Nevertheless, it is conceivable that both initial and final stateeffects affect the observable azimuthal anisotropies of produced charged particles in smallsystem collisions, with the role of the initial state contribution increasing with decreasingsystem size (or particle multiplicity). We have presented recent developments inidentifying observables that could be used to distinguish the two sources of anisotropies.It would be most exciting if one could show that both fluid behavior is present, i.e., theworld’s smallest fluid is produced in p+A or p+p collisions, but access to complex gluoncorrelations, directly driven by QCD, is also possible.It should be noted that interesting developments are also ongoing within the highenergy physics community to extend event generators such as PYTHIA [266] to be ableto reproduce the momentum anisotropies observed in p+p collisions. This involves forexample the inclusion of a dynamically generated transverse pressure, produced by theexcess energy from overlapping strings [267] in the rope hadronization picture [268], he smallest fluid on earth p T are also veryimportant to gain a complete understanding of small systems [26]. While a significantsuppression of particles at high transverse momentum in A+A collisions relative to p+pcollisions is observed experimentally, p/d+A collisions show almost no modification[275, 276, 277, 278, 279] (and it may not be expected theoretically [280, 281]). Onthe other hand, a reasonably strong elliptic anisotropy is observed at high p T in p+Pbcollisions, whose origin is explained by directionally dependent jet quenching in A+Acollisions, but appears mysterious in p+Pb collisions that show no jet quenching. Also,as centrality selection is problematic in small systems, which complicates the definitionof the nuclear modification factor [282], the proposed study of O+O collisions [283]at LHC and RHIC, could prove useful in exploring jet quenching in small systems.Comparing O+O to p+Pb collisions at a similar number of produced particles couldyield differences in jet quenching as the initial system sizes can differ considerably. Thestudy of O+O along with p+A is also argued to help with separating initial momentumanisotropy from final state effects [283].Finally, measurements of electromagnetic probes could be used to further supportthe interpretation of the formation of a strongly interacting medium in small systemcollisions [284]. Additional photon and dilepton radiation from the medium should bevisible, as it is in heavy ion collisions. While the existing experimental data from RHIC he smallest fluid on earth
6. Acknowledgments
B.P.S. thanks Charles Gale, Giuliano Giacalone, Jiangyong Jia, Heikki M¨antysaari,Aleksas Mazeliauskas, Jean-Fran¸cois Paquet, S¨oren Schlichting, Prithwish Tribedy,Chun Shen, and Raju Venugopalan for useful discussions. Special thanks go to CharlesGale, Giuliano Giacalone, S¨oren Schlichting, Chun Shen, and Michael Strickland forvaluable comments on an early version of the manuscript. B.P.S. is supported underDOE Contract No. DE-SC0012704. [1] Arsene I et al. (BRAHMS) 2005
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