aa r X i v : . [ h e p - t h ] M a y Holography of the QGP Reynolds Number
Brett McInnes
National University of Singaporeemail: [email protected]
The viscosity of the Quark-Gluon Plasma (QGP) is usually described holographicallyby the entropy-normalized dynamic viscosity η/s . However, other measures of viscosity,such as the kinematic viscosity ν and the Reynolds number Re , are often useful, and theytoo should be investigated from a holographic point of view. We show that a simple modelof this kind puts an upper bound on Re for nearly central collisions at a given temperature;this upper bound is in very good agreement with the observational lower bound (fromthe RHIC facility). Furthermore, in a holographic approach using only Einstein gravity, η/s does not respond to variations of other physical parameters, while ν and Re can doso. In particular, it is known that the magnetic fields arising in peripheral heavy-ioncollisions vary strongly with the impact parameter b , and we find that the holographicmodel predicts that ν and Re can also be expected to vary substantially with the magneticfield and therefore with b . . Two Measures of the QGP Viscosity The key result in holography [1, 2], as applied to the Quark-Gluon Plasma (QGP) [3–6],is the Kovtun-Son-Starinets [7–9] computation of the ratio of the boundary field theoryshear (or dynamic) viscosity η to its entropy density s : what one might call the entropy-normalized dynamic viscosity, denoted henceforth by η s , is given in natural units by η s ≡ ηs = 14 π . (1)This relation is computed at arbitrarily large coupling; at finite but large coupling oneexpects (with various subsequently discovered exceptions, some of which will be mentionedbelow) that this equality becomes an inequality, with / π as the lower bound; there isreason to hope that the rate of variation upwards is not large, so the actual plasma shouldexhibit a value not too far above this.Part of the reason for the attention generated by this result is that the actual QGPhas indeed a value for η s which is surprisingly small, that is, not very far above / π [10];another reason is that (1) is valid for a remarkable range of boundary field theories, in factfor all those which are isotropic and dual to Einstein gravity in the bulk. This universality is important for a range of reasons, discussed in very clear detail in [1].However, while it is true that, in many contexts, the viscosity only appears through η s , in many other physical applications one is also interested in (some measure of) theviscosity itself — and this includes discussions of the QGP, see for example [11, 12].Whether the actual viscosity of the QGP is “small” is a somewhat delicate question.In classical hydrodynamics, there are two distinct measures of viscosity: the dynamicor shear viscosity η (commonly measured in units of millipascal seconds or centipoise,cP = − Pa · s ≈ . × eV in natural units) and the kinematic viscosity ν ,defined as the ratio of η to the liquid density (commonly measured in units of centistokes,cSt = − m / s ≈ . × − eV − in natural units). The kinematic viscosity allowsfor the fact that certain liquids resist changes to their flow simply by being very dense.It controls the rate of diffusion of momentum in a moving liquid, and is arguably atleast as important as η . (In relativistic hydrodynamics, the same remarks apply, thoughthe relativistic versions of these quantities may behave in unusual ways, for example inresponse to variations of temperature.)Which is more viscous, water or mercury? There is no conclusive answer: the respec-tive dynamic viscosities of water and mercury [13] are (at around 290 kelvin) 1.002 cPand 1.53 cP, but their respective kinematic viscosities (at that temperature) are 1.004 cStand 0.12 cSt. Similarly, while it is often stated that “the viscosity” of the QGP is “small”,this invariably refers to η s . By everyday standards, η itself for the QGP is far from small,being probably just under × cP ≈ . × eV [14] for the plasmas produced incollisions at the RHIC facility, and even more for LHC plasmas. On the other hand, the kinematic viscosity of this plasma is small compared to that of most “ordinary” liquids,being (see below for the calculation) just under 0.07 cSt — about the same as that ofmercury at 600 kelvin [13], this being one of the lowest measured values for any liquid. η s is dimensionless in natural units. In SI units it has units of kelvin · seconds, and the KSS relationtakes the form η s = ~ πk B ≈ . × − K · s [7].
2t seems that these comparisons are not very helpful. In any case they are not offundamental significance; nor are they really relevant, in view of the fact that the QGPitself only exists under conditions which are far indeed from being “everyday” or “ordinary”:we are dealing with a liquid moving, at a speed equal to a substantial fraction of the speedof light — and one knows, as above, that relativistic hydrodynamics differs in some waysfrom classical hydrodynamics — through a “tube” with a radius measured in femtometres.In hydrodynamics, the magnitude of the viscosity of a liquid is usually assessed in adifferent way, which we propose, following Csernai et al. [15], to adopt here: it is done byevaluating the dimensionless
Reynolds number , defined as Re = uδν , (2)where u and δ are the characteristic velocity and transverse dimension of the flow, and ν isthe kinematic viscosity as above. (In natural units, u is dimensionless, dynamic viscosityhas units of eV , energy density has units of eV , so Re is indeed dimensionless since ν hasthe same units as δ , namely eV − .) We can think of Re as a dimensionless version of (thereciprocal of) the kinematic viscosity. It measures kinematic viscosity not by comparingit with other liquids but by including the circumstances in which the liquid in questionfinds itself.As is well known, the Reynolds number has a fundamental role in hydrodynamics:most notably, it provides a remarkably useful indicator of situations in which laminarflow makes the transition to turbulence . This happens when Re is much larger thanunity, typically in the thousands, and such values for Re are commonly encountered evenin laminar flows. That is, the kinematic viscosity is often very small relative to uδ .From a physical point of view, then, one can make the case that “small viscosity” shouldgenerically mean that the kinematic viscosity is extremely small relative to the productof the characteristic velocity and size of the system, so that Re is large, in the range ofthousands or more.Now in fact the Reynolds number of the actual QGP (at, for example, the RHICfacility) is expected to be very small by these standards. Csernai et al. [15] estimatevalues in the range − . We shall use an upper bound corresponding to η ≈ . × eV as given in [14], which, combined with the standard [16] estimate of the energy densityof the RHIC plasma at the relevant time, ≈ GeV/fm ≈ . × eV , yields ν ( QGP ) ≤ ≈ . × − eV − ; (3)this is the value 0.07 cSt mentioned above. With the estimates of u and δ used in [15], u ≈ . and δ ≈ fm, we have Re ( QGP ) ≥ ≈ . , (4)which is indeed small (and compatible with the range given in [15]). Thus, in this objectivesense, the QGP should be considered to be very viscous . The Reynolds number is by no means the only indicator of hydrodynamic instability, so its smallnesshere does not establish that the flow of the QGP is fully stable: see [15] for a detailed phenomenologicaldiscussion of the QGP Reynolds number in this context. The − range cited there does not, of course,take into account the unusual effects predicted in the present work: for the sake of clarity, we will takethese values to apply to central or near-central collisions only . We will argue that values as much as 8times higher are possible for certain peripheral collisions. But these are still “small” Reynolds numbers. Re for the QGP, and its variation in response to changes ofother physical parameters, need to be accounted for in a holographic description, no lessthan the small value of η s . For ν clearly represents the viscosity of the plasma at least aswell as η s . Henceforth our focus will be on ν and Re (though of course η can be computedat any point in the discussion if the energy density is known).This is not to say that Re or ν should replace η s as a quantity of basic interest in thestudy of the QGP; we will argue instead that it can complement η s in some extremelyuseful ways.In particular, the universality of (1) is a two-edged sword: while it encourages the beliefthat (1) is no mere peculiarity of some specific (and inevitably not fully realistic) model,it also means that completely unrealistic holographic models can correctly “predict” η s .In other words, a bad prediction for η (and therefore for ν ) can be compensated by anequally bad prediction for the entropy density. One expects that a holographic model of ν can detect this. In this work we will be concerned precisely with just such a case; wenow discuss general aspects of such a model, and the results we have obtained from it.
2. Holographic Model of ν (QGP): Generalities and Results Recently, it has become clear that any holographic model of the QGP that ignores mag-netic fields must be considered inadequate: for extremely intense magnetic fields do per-meate the plasma in all but the most central collisions of heavy ions [17–21]. The fieldexperienced by the plasma in any given case depends on the impact parameter, b ; in aspecific beam it varies with b from zero up to some maximum, which can be computed.The magnetic field is incorporated in a holographic model simply by attaching a mag-netic charge to the bulk black hole responsible for the thermal properties of the bulktheory, and of course one need not go beyond Einstein gravity to implement this. How-ever, because the magnetic charge generates a bulk field that back-reacts on the blackhole geometry, this procedure affects the position of the event horizon, thereby alteringthe relation between the mass of the black hole and its entropy, as well as the relationbetween these parameters and the Hawking temperature. Therefore, the holographicallydual versions of all of these quantities will be sensitive to the magnetic field. In oneextraordinary case, η/s , the effects cancel (as long as we retain Einstein gravity in thebulk), but one must not expect this to occur for other combinations such as ν and Re ,and indeed it does not.In short, models that ignore magnetic fields may well predict Re badly, while never-theless being fully compatible with (1). The question is this: in concrete cases, does theholographically predicted value of Re vary significantly with the magnetic field, at a givenvalue of the temperature? The results will be model-dependent, but this just means thatwe should, as always, begin with a simple model and work towards more complex ones,hoping that qualitative features of the former will persist into the latter.Simplicity in this case means that we wish to keep η s fixed at / π , so that we have aclean indication of the effects of the magnetic field on Re , and because holography itselfpredicts (see below) that, even if it is allowed to vary, η s does so extremely slowly evenfor very large values of the field.We can achieve this by using an explicit bulk black hole metric, which solves the4instein equations exactly , and which incorporates the minimal set of thermodynamicand other variables needed here. We therefore exclude, in the present work, all higher-derivative corrections to the KSS bound; thus, for example, we do not allow for thetheoretically indicated possible slow increase [24] of η s with temperatures from the lowerend of the hadronic/plasma crossover up to temperatures characteristic of the QGP ; forthe same reason we are excluding variations of η s with the baryonic chemical potential [29]:that is, we consider only the case µ B = 0 .Simplicity also means that we focus attention purely on viscosity in the reaction plane of a given heavy-ion collision: in the conventional coordinates, this means that we consideronly the two-dimensional ( x, z ) plane, where z corresponds to the axis of the collision, andnot viscous flow in the perpendicular y direction — the direction of the magnetic field.Apart from the usual benefits [30, 31] of restricting the dimensionality in a holographictreatment, this is particularly important here because recent work has suggested thatmagnetic fields can give rise to violations of the KSS bound, if one attempts to apply itto shearing in the y direction. (The point is that the KSS bound was originally derivedin the context of full ( SO (3) ) symmetry, whereas the magnetic field preserves only the SO (2) symmetry of the reaction plane: see [32–36].) We will maintain SO (2) symmetry(and translation symmetry [37–39]) in the reaction plane here, so the KSS bound is notaffected by the presence of a strong magnetic field .Thus, while in the most general case η s can vary with temperature, direction, andperhaps other parameters, we confine ourselves here to the simplest cases, in which it doesnot: so we have equation (1) and the usual corresponding inequality at finite coupling.We will then use this to estimate or put a lower bound on the kinematic viscosity, or anupper bound on the plasma Reynolds number, by means of a holographic computation ofthe ratio of the entropy density to the energy density. This can be done by means of arelatively straightforward investigation of a suitable black hole geometry in the bulk. It isreasonable to expect that qualitative results obtained in this manner will persist in morecomplex, less tractable models.We find that holography imposes, in the absence of a magnetic field (that is, for centralor near-central collisions), a surprisingly strong upper bound on the Reynolds number ofthe QGP studied at the RHIC facility: using the same values for u and δ as in [15] (leadingto the phenomenological estimate given there, Re ≈ − ), we find a holographic boundof the form Re fc0 (Hol) < Re (Hol) ≈ , (5)where the subscript indicates a zero magnetic field and the superscript fc denotes the valueat finite coupling . Thus, holography correctly encodes the smallness of the Reynolds Far more sophisticated models are available: see for example [22, 23] and references therein. See [25–27] for recent (non-holographic) theoretical perspectives on this question. Results from theALICE experiment [28] indicate however that, for the range of temperatures we shall consider here, thisvariation is not significant. In any case, even in the direction of the field, the holographically predicted effect of the magneticfield on η s is small: see [34]. We are assuming here that the effect of finite coupling is always to increase the kinematic viscosity.The (admittedly heuristic) reasoning here is based on simple concrete examples, such as [40] the caseof the quartically coupled scalar, with coupling term gφ . Here (see [1], Section 12.1.2) one finds that η scales with T /g , and the energy density with T ; so both η and ν should be expected to be larger umber in this case; and the holographic bound is in very good agreement with theestimated lower bound on the Reynolds number of the actual QGP given by the inequality(4). (Bear in mind that “typical” Reynolds numbers for laminar flows are in the hundredsor thousands.)Next we turn to the case of peripheral collisions, in which the magnetic field can bestrong. But before discussing our results, let us first ask: what do we expect?Qualitatively, we expect the kinematic viscosity to decline as the magnetic field in-creases: kinematic viscosity corresponds to a momentum transfer (in fact, ν is sometimescalled the “momentum diffusivity”), and this transfer will be suppressed by a magneticfield in directions perpendicular to the field [41].More interestingly, a decline is also expected in connection with the remarkable phe-nomenon of paramagnetic squeezing [42]. Briefly, the magnetic fields exerted on the plasmain peripheral collisions will tend to compress the plasma in the reaction plane directions(and to elongate it in the y direction, the direction of the field). This directly affects thepressure gradients in the plasma and may well have very important consequences for thevalue of the elliptic flow parameter v . However, it can also be expected to have an effectrelevant here, by reducing the dominant outward pressure in the reaction plane; and sucha reduction will normally result in a reduced viscosity.However, it is a general rule in hydrodynamics that variations of viscosity with vari-ables other than temperature are extremely small, even for large changes in such variables.Thus, while a decrease in the kinematic viscosity of the QGP is to be expected (for eitherof the above reasons) for collisions involving increasing magnetic field values, it is veryunclear whether this decrease will be significant.We find that holography predicts that the kinematic viscosity of the QGP, at fixedtemperature, does decline with increasing magnetic fields: so at least the direction ofthe predicted variation is correct. Equally important, the variation is quite large, evenfor relatively small values of the magnetic field : for example, for plasmas produced atthe RHIC facility, an increase of the field from zero to eB ≈ × m π (where m π is theconventional pion mass) causes a change in the predicted kinematic viscosity of almost25 % ; for eB ≈ × m π , the change is nearly 50 % . (Of course, the corresponding Reynoldsnumbers increase accordingly.) Such magnetic fields are well below the latest estimatesfor the maximal fields produced in peripheral collisions at the RHIC, which range upto eB ≈ m π [44–48], and are therefore quite realistic, in the sense that they can beattained for a wide variety of peripheral collisions (and not merely those with impactparameter carefully selected to achieve the maximal value). The predicted variations ofthe kinematic viscosity and Reynolds number in the case of the plasmas observed by theALICE [49] and other experiments at the LHC are still more dramatic. for finite than for infinite coupling. This argument certainly should be improved; it does not show, forexample, that our bound will be as realistic as that of KSS. One should be somewhat cautious here, because there are anomalous liquids for which, in certainregimes, a reduction of pressure gives rise to an increase in viscosity. A perhaps surprising exampleof such a liquid is water : for temperatures below +32 ◦ C, water’s viscosity does indeed increase as thepressure drops from around 20 megapascals. However, this behaviour is associated (in a very complexway, see [43]) with a better-known anomaly of water, its negative thermal expansion coefficient in certainregimes. One does not expect such behaviour in the QGP. Of course, the reader should bear in mind that the “relatively small” magnetic fields here are measuredin units of m π ≈ gauss. ν does in fact usefully complement the more familiar holographic universality of theentropy-normalized viscosity η s . For in a simple but very explicit holographic model,one finds an upper bound on Re for central or near-central collisions which is in goodagreement with the observational lower bound, and that, whereas η s does not vary withthe collisional impact parameter (that is, with the magnetic field), ν does vary in a sensewhich is physically reasonable but with a surprisingly large magnitude.We now explain the details.
3. Holographic Model of ν (QGP): Specifics For our purposes, the simplest possible bulk black hole [51] has a flat (indicated by a zerosuperscript) planar event horizon and a Euclidean asymptotically AdS magnetic Reissner-Nordström metric g E (AdSP ∗ RN ) , given by g E (AdSP ∗ RN ) = " r L − πM ∗ r + 4 πP ∗ r d t + d r r L − πM ∗ r + 4 πP ∗ r + r (cid:2) d ψ + d ζ (cid:3) ; (6)here L is the asymptotic AdS curvature scale, r and t are the usual radial and “time”coordinates, and M ∗ and P ∗ are parameters such that, if r h is the value of the radialcoordinate at the event horizon, then, if ℓ P is the bulk Planck length, M ∗ /ℓ P r h is themass per unit horizon area, and P ∗ /ℓ P r h is the magnetic charge per unit horizon area;and ψ and ζ are dimensionless coordinates on the plane, related at infinity to the standardcoordinates x = Lψ and z = Lζ in the reaction plane (the y axis being parallel to themagnetic field) of a heavy-ion collision . The Hawking temperature of this black holedefines the temperature T ∞ of the boundary theory, and the parameter P ∗ determinesthe boundary magnetic field B ∞ [53], in the familiar ways. In the bulk, we work exclusively in the Euclidean domain. The Lorentzian case involves, in general,additional subtleties, not relevant here (because magnetic charge is not complexified when passing to theLorentzian domain): see [52] and the discussion below. To understand the significance of this, note that we are entitled to regard ψ and ζ as angularcoordinates on a flat torus (though this is not essential, and it plays no further role here). Clearly, L sets the length scale of the boundary theory once conformal symmetry is broken. We therefore take it tohave a typical nuclear physics value, 10 fm. We will use B ∞ to refer to the “holographic” magnetic field, the field on the boundary of the relevantasymptotically AdS spacetime. Observed or phenomenologically computed magnetic fields will be denoted
7e note that the bulk being four-dimensional, the boundary theory has two-dimensionalspatial sections: these are taken to represent the reaction plane. That is, we are construct-ing a holographic model of the relation between magnetic fields and viscosity in that plane only . Precisely because the magnetic fields we consider are so intense, and because therelative motion between different sections of the form y = constant is relatively negligible,and also (see above) because we take the physics in the reaction plane to be isotropic,we assume that this is a good approximation; note however that in other cases (such asrecently proposed experiments [54] involving collisions of highly non-spherical nuclei), itmight not be.Unfortunately, this model is in fact a little too simple. To see why, recall that thisbulk geometry corresponds to a simplified version of a much more complex string-theoreticconfiguration. This simplification is legitimate provided that it does not cause the morecomplex system to become internally inconsistent. The relevant consistency conditionshave been discussed in [55–58]. One of these conditions (discussed in detail in [57]) is thata certain function S E defined on the bulk geometry (in terms of the areas and volumesof hypersurfaces homologous to the conformal boundary) should never be negative at anypoint . In [62] we translated this inequality into an inequality governing the parametersof the specific metric g E (AdSP ∗ RN ) , and we found that this condition (when the baryonicchemical potential vanishes, as we assume throughout) takes a very simple form: B ∞ ≤ π / T ∞ ≈ . × T ∞ . (7)That is, if this inequality is violated, then S E evaluated on this geometry does take onnegative values and there is a serious mathematical inconsistency.This inequality is in fact satisfied by the plasmas produced in most heavy-ion collisionsat the RHIC facility, but (in view of recent suggestions [44–48] that the magnetic fieldsproduced in these collisions may be even larger than previously thought) probably not inall; and it is still more likely to be violated in collisions at the LHC (again, particularlywith the recent upward revision in estimates of B ). For example, as mentioned above,the latest estimates for the maximal magnetic field produced in RHIC collisions put itaround eB ≈ × m π , or B = 6 . × MeV ; whereas for typical RHIC temperatures( T ≈ MeV), the right side of (7) is ≈ . × MeV .In [63] we argued that the problem here is that the computations in [62] (whichwas concerned with a different application, to cosmological magnetic fields) neglected animportant physical effect: the fact that these collisions involve not just large magneticfields, but also large angular momentum densities, as predicted in [64–69], and possiblyobserved by the STAR collaboration [70]. It turns out that the inclusion of an angularmomentum parameter in the bulk geometry can resolve the consistency problem if it issufficiently large . simply by B . The Lorentzian version of the argument runs as follows [52]: the Lorentzian version of S E is, upto factors involving the brane tension, equal to the action of a BPS brane in the Lorentzian black holegeometry [59]. If this action becomes negative at sufficiently large r , this means that it is smaller thanits value at the event horizon. The black hole will therefore generate arbitrary quantities of branes by apair-production process [60, 61] at such values of r , and these branes will have no tendency to contractback into the hole. The system therefore becomes unstable. There is an issue here: while consistency can always be re-established in this way, it is not clear dilaton into the bulk. This is a very natural move from a string-theoretic point of view; it need not break the SO(2) symmetry in the reaction plane,and it can be done, with a careful choice of potential, in such a way that the geometryremains asymptotically AdS [76]: see [77] for more detail, including the embedding in ahigher-dimensional theory.With a sufficiently strong coupling to the magnetic field , the dilaton has much thesame effect as angular momentum [78], that is, it enforces the consistency condition of [57]even when the inequality (7) is violated. We can think of the dilaton coupling constant α as a sort of proxy for the angular momentum parameter (at least for impact parameterswhich are not very large).To see how this works in detail, consider the metric of an asymptotically AdS planardilatonic Euclidean Reissner-Nordström black hole with magnetic charge parameter P ∗ and mass parameter M ∗ : it takes the form [76] g E (AdSdilP ∗ RN ) = U ( r )d t + d r U ( r ) + [ f ( r )] (cid:2) d ψ + d ζ (cid:3) , (8)where the coordinates are as above; here U ( r ) = − πM ∗ r (cid:20) − (1 + α ) P ∗ M ∗ r (cid:21) − α α + r L (cid:20) − (1 + α ) P ∗ M ∗ r (cid:21) α α , (9)and f ( r ) = r (cid:18) − (1 + α ) P ∗ M ∗ r (cid:19) α α . (10)As before, M ∗ and P ∗ are related to the mass and magnetic charge per unit event hori-zon area: if r = r h at the event horizon, then the mass per unit event horizon area is M ∗ / ( ℓ P f ( r h ) ) , while P ∗ / ( ℓ P f ( r h ) ) is the magnetic charge per unit event horizon area.The dilaton itself, ϕ , is described by a potential of the form V ( ϕ ) = − πℓ P L α ) (cid:2) α (cid:0) α − (cid:1) e − ϕ/α + (cid:0) − α (cid:1) e αϕ + 8 α e αϕ − ( ϕ/α ) (cid:3) , (11) whether the required value of the angular momentum parameter is realistic in the case of collisions withmaximal magnetic fields. We found in [63] (see also [71]) that this is definitely not a problem in thecase of the RHIC plasmas, but that it could be in the LHC case. Since the phenomenon in which weare interested here appears already even at relatively low values of the magnetic field, we will not take astand on this issue here. The coupling takes the form e − αϕ F , where F is the field strength two-form, ϕ is the dilaton, and α is a coupling constant. g E (AdSP ∗ RN ) ;note that, since the dilaton itself is given here by e αϕ ( r ) = (cid:18) − (1 + α ) P ∗ M ∗ r (cid:19) α α , (12)we see that V ( ϕ ) → − / (8 πℓ P L ) , the AdS energy density, when either α → or r → ∞ .That is, the asymptotic AdS energy density is embedded in V ( ϕ ) . (The kinetic term forthe dilaton is the standard one.)The electromagnetic field two-form in this case takes the form F = P ∗ ℓ P (cid:18) − (1 + α ) P ∗ M ∗ r (cid:19) α α d ψ ∧ d ζ . (13)This can be used to define a magnetic field at infinity given by B ∞ = P ∗ L . (14)The function U ( r ) vanishes at r = r h , which is therefore to be found by solving − πM ∗ r h (cid:20) − (1 + α ) P ∗ M ∗ r h (cid:21) − α α + r h L (cid:20) − (1 + α ) P ∗ M ∗ r h (cid:21) α α = 0 , (15)and the Hawking temperature, which determines the temperature at infinity, is πT ∞ = 8 πM ∗ r h (cid:18) − (1 + α ) P ∗ M ∗ r h (cid:19) − α α − π (1 − α ) P ∗ r h (cid:18) − (1 + α ) P ∗ M ∗ r h (cid:19) − α α + 2 r h L (cid:18) − (1 + α ) P ∗ M ∗ r h (cid:19) α α + α P ∗ M ∗ L (cid:18) − (1 + α ) P ∗ M ∗ r h (cid:19) α − α . (16)Finally, for this bulk geometry, the function S E discussed earlier is given [78] [71], upto a positive constant factor, by S E (AdSdilP ∗ RN )( r ) = r L (cid:20) − (1 + α ) P ∗ M ∗ r (cid:21) α α − πM ∗ L r (cid:18) − (1 + α ) P ∗ M ∗ r (cid:19) − α α − L Z rr h s (cid:20) − (1 + α ) P ∗ M ∗ s (cid:21) α α d s. (17)Given values of B ∞ , T ∞ , L , and α , one can solve the three equations (14), (15), and(16) for r h , P ∗ , and M ∗ , and then equation (17) specifies S E (AdSdilP ∗ RN )( r ) completely;one can then check directly (by simply graphing it) whether this function is strictly non-negative as r increases from r h to infinity. In fact, if α is chosen too large, this system of simultaneous equations turns out to have no realsolutions. This interesting phenomenon will not be an issue in this work, where we always choose α tobe as small as possible consistent with the non-negativity of S E .
10e found in [63] that, for the “maximal” RHIC data used above ( eB = 10 × m π , T ≈ MeV), then S E for the (much more complex) geometry used there is forced tobe positive everywhere when even a very small amount of shearing angular momentumis added to the bulk black hole. In the same way, with these values for B ∞ and T ∞ , thepositivity of S E for the dilatonic black hole geometry can be restored with a dilaton thatis very weakly coupled : one only needs to take α = 0 . , as can be seen in Figure 1.Figure 1: S E (AdSdilP ∗ RN )( r ) , T ∞ ≈ MeV, µ B = 0 , eB ∞ ≈ × m π , α = 0 . .On the other hand, for maximal LHC data, far larger values of the angular momentumwere required to ensure that S E is never negative in that case; similarly, we found in [71]that, for these data, a dilaton coupling at least as large as α ≈ . is needed. Thus, thedilaton coupling does mimic the effect of angular momentum for our purposes .We are now in a position to give a self-consistent holographic computation of thekinematic viscosities of the plasmas produced, in collisions of varying centrality, at the See [71] for the precise meaning of “weak coupling”. Essentially it means that α is much smaller thanits maximal possible value (for these data), which is around 0.720. See the preceding footnote. The precise reason for the fact that the dilaton tends to restore consistency in this way is notcompletely known. As mentioned earlier, the bulk interpretation of the inconsistency involves uncontrolledpair-production of branes. The location of a brane in the bulk can be represented [59] by a scalar field atinfinity, which (because of the form of the conformal coupling term) behaves well if the scalar curvatureat infinity is positive, but it misbehaves if the scalar curvature at infinity is negative, and sometimes ifit vanishes, as it does here. The dilaton turns on a source and a vacuum expectation value for the dualoperator (in fact as a multi-trace operator, as in [79]). In some way that remains to be understood indetail, this has the effect (if the coupling is strong enough) of pushing the scalar field to behave morelike the positive case. Understanding this in detail would require a delicate analysis of high-order termsin the action given in [59].
4. The Kinematic Viscosity as a Function of the Magnetic Field
We now wish to compute the kinematic viscosity, using the assumed holographic dualityof the above AdS planar dilatonic Reissner-Nordström black hole with a field theory thatmodels the QGP when it is subjected to a non-zero magnetic field.We saw above that the mass (that is, the energy) of the black hole per unit eventhorizon area is given by M ∗ / ( ℓ P f ( r h ) ) , where f ( r ) is given by equation (10). The entropyper unit horizon area is (for Einstein gravity, as assumed here, in natural units) / ℓ P .Assuming uniformity in the y direction on the boundary, the ratio of the field theoryentropy density to its energy density is then given holographically by sρ = f ( r h ) M ∗ ; (18)notice that the bulk quantity on the right has no dependence on ℓ P . Combining this withequations (1) and (10), we have finally, since ν = η/ρ , ν = r h (cid:16) − (1+ α ) P ∗ M ∗ r h (cid:17) α α πM ∗ . (19)The right hand side of this relation involves only bulk parameters; but, as we sawabove, holography (in the form of the equations (14), (15), and (16)) allows us to regard r h , P ∗ , and M ∗ as nothing but (very complicated) functions of the known (that is, fixedby considerations of boundary physics) parameters B ∞ , T ∞ , L , and α . (As usual, equalityholds here strictly only at arbitrarily large coupling: in general, one should interpret theright side of (19) as a lower bound on the kinematic viscosity, implying an upper boundon the Reynolds number.)Clearly, neither ν nor its dimensionless version Re enjoy the universality of the holo-graphic prediction for η s ; but we are now, with equation (19), at least in a position tocompute how they vary with the physical boundary parameters.Our objective here is to study the effect of the magnetic fields on the kinematic vis-cosity. To that end, we first compute the quantity ν , which is the kinematic viscositythat one expects for the plasma if magnetic fields are neglected. (Equivalently, this isthe holographically predicted value for ν in the case of central or near-central collisions.)We then compute ν itself, including the effect of various non-zero values for the magneticfield B ∞ . The ratio ν/ν then quantifies the effect we wish to study.Before we proceed, we should briefly discuss the phenomenological model usually em-ployed to estimate the magnetic field B arising in peripheral heavy ion collisions, as afunction of the impact parameter b . In [46] it is found, using the HIJING Monte Carloevent generator (subsequently updated to HIJING++ [80]), that B rises, almost linearly,from zero (for exactly central collisions) up to a maximum value (which is determinedby the collision energy: it is much larger at the LHC than at the RHIC). This maxi-mum occurs at a value of b which is so large that one can question whether a plasmais actually formed, to any significant degree, beyond that point. As it is known that12he temperature of the plasma varies much more slowly with the collision energy thanthe magnetic field [46], it is a reasonable approximation, for moderate impact parametercollisions, to assume that the temperature is independent of the impact parameter: thisis how we will analyse how the kinematic viscosity varies with the magnetic field in ourholographic model. This allows us to separate the effects of magnetic fields from thoseof varying temperatures, for example when we move up from RHIC collision energies toLHC energies. Once we have understood these effects, we will be in a position to discusscollisions with large impact parameters.Let us begin with the RHIC data. Taking T ∞ ≈ MeV in this case, we allow eB ∞ to range in steps from zero up to around 10 m π [44–48]. For each choice of eB ∞ , we havechosen a value of α close to the minimal value capable of ensuring that S E , evaluatedusing equation (17), remains non-negative everywhere on its domain. (In fact, we findthat our results vary slowly with α , so the precise choice is not important.)As explained above, we denote the value of ν (at the stated temperature) for B ∞ = α = 0 by ν . Using equations (14), (15), and (16), one finds that in fact ν = 3 / (8 πT ∞ ) ,so that in this case the predicted kinematic viscosity of the RHIC plasma satisfies (thenotation fc again corresponding to the finite coupling case) ν fc0 > ν ≈ . × − eV − . (20)With the approximate values for u and δ used in [15] ( u ≈ . , δ ≈ fm), this yieldsa bound for the Reynolds number given by Re fc0 < Re ≈ . , or let us say Re fc0 < ≈ . This is in very good agreement with the experimentally determined lower bound on Re ( QGP ) for the RHIC plasmas, around 8.5 (the inequality (4) given above). In short, a combination of experimental and holographic techniques puts the Reynoldsnumber of the RHIC plasmas somewhere between 8.5 and 20, for near-central collisions.
For peripheral collisions, we allow B ∞ to be non-zero. For sufficiently small values ofthe field, it is consistent to take α to be negligible, and in this case equations (15), (16),and (19) simplify to such an extent that an analytic discussion becomes possible. In fact,a straightforward manipulation shows that one has, in this case, r h L − πT ∞ r h − πB ∞ L = 0 (21)and ν = 4 r h L − πT ∞ . (22)Notice that r h is obtained by solving a quartic, with B ∞ in the (negative) constantterm. If we fix the other parameters and increase B ∞ , then, the quartic drops lower andthe positive root, that is, r h , shifts to the right. From equation (22) we see at once thatthe effect of this is to cause ν to decrease. Thus we see that, at least for relatively smallmagnetic fields, an increase in the field always causes ν to decrease, all else being equal.Evidently this is a basic prediction of the holographic model.For larger magnetic fields the pattern turns out to be the same, though in this case α can no longer be neglected and a numerical analysis of equations (15), (16), and (19)becomes necessary. We state the results for ν by comparing it with ν (and for Re bycomparing it with Re ; note that this ratio does not depend on u or δ ). The results areshown in the table and (for ν/ν ) in Figure 2.13 B ∞ /m π α ν/ν Re/Re B ∞ . Thus, for example, if we take eB ∞ = 4 m π , we find that ν dropsto about half of its value when the magnetic field is neglected, so the Reynolds numberroughly doubles . To put it another way: holography suggests that the flow of the plasmais not best described by a single Reynolds number, but rather by a range of values whichdepend on the centrality of the corresponding collision . In the case of RHIC collisions,perhaps the best way to state the case is that there is an upper bound on Re fc , rangingfrom about 20 for very central collisions, up to around 60 for a small subset of collisions(those at high impact parameter such that a plasma is nevertheless actually formed).Next we turn to the LHC plasmas. Taking T ∞ ≈ MeV in this case, we allow eB ∞ to range from zero up to around 70 m π [46, 50]. With the same notation as above, we find Strictly speaking, this is not the case, since we are dealing with inequalities when finite coupling istaken into account: that is, we now have Re fc < ≈ instead of Re fc0 < ≈ . But it is natural to interpretthis as jump in the predicted value of Re fc .
14n this case that ν fc0 > ν ≈ . × − eV − ; (23)with the same value of uδ as before, this yields Re fc0 < ≈ . Thus, holography leads usto expect larger values of the Reynolds number for central or near-central LHC collisionsthan for their RHIC counterparts (though the value is still “small” in the sense we havediscussed) The results for non-zero magnetic fields are shown in the table and in Figure 3. eB ∞ /m π α ν/ν Re/Re
10 0.04 0.4446 2.24920 0.15 0.2800 3.57130 0.18 0.2137 4.67940 0.24 0.1758 5.68850 0.27 0.1513 6.60960 0.28 0.1344 7.44070 0.29 0.1220 8.197Figure 3: Relative Kinematic Viscosity vs. Magnetic Field, LHC Data.As in the RHIC case, the effect of including the magnetic field is to reduce the relativevalue of the kinematic viscosity, that is, to increase the Reynolds number, which againbecomes centrality-dependent: the upper bound is around 26 for central or near-central If the kinematic viscosity of the LHC plasmas approaches . × − eV − , which is around 0.023cSt, this would be among the lowest kinematic viscosities of all liquids (possibly rivalled only by verydense metals with very high boiling points, such as rhenium, near their critical points). • In this work, we have focused on the kinematic viscosity, because of its relation withthe most important dimensionless quantity in hydrodynamics, the Reynolds number. Ifone is interested in the dynamic viscosity, it can of course be reconstructed from ν simplyby multiplying by the energy density. For example, a plasma resulting from a near-central collision at the RHIC facility typically [16] has an energy density, at the relevanttime , of around ≈ GeV/fm ≈ . × eV , so with the above lower bound on ν ≈ . × − eV − we find that our model predicts a lower bound of around . × eV for η in this case; this is, in fact, below, but not very far below, the upper bound on the dynamic viscosity of the actual RHIC plasma given in [14], ≈ . × eV . For the LHC, where the lower bound on ν is ≈ . × − eV − but the energydensity is around 2.3 times larger [81], the predicted lower bound on η is higher , around . × eV . Notice that the LHC plasmas are predicted to have a lower kinematicviscosity, but a higher dynamic viscosity, than their RHIC counterparts: this behaviourwith increased temperature is to be expected for dense relativistic liquids. • The surprisingly good agreement of Re with observational/phenomenological ex-pectations — that is, the prediction that it should be “small” — is partly, but only partly,due to the success of the KSS bound: for we have used (1) to derive (19). However, as wehave stressed, there is a “non-universal” contribution to the right side of equation (19),and this contribution might well have deviated strongly from order unity: indeed, weknow that it does so in the presence of a strong magnetic field. In other words, equation(18) is also necessary for our prediction of the Reynolds number to be satisfactory. (Notethat, for example, M ∗ is typically of order hundreds of thousands to millions (of fm) withthese data; so it is not clear that the factor on the right side of (18) will be of orderunity.) To put it yet another way: holography produces good results not only for theentropy-normalized dynamic viscosity, but also for the kinematic and dynamic viscositiesthemselves. • At fixed temperature, the relative kinematic viscosity (that is, ν/ν ) always de-creases with increasing magnetic field. At fixed magnetic field, it always increases withtemperature. • In order to avoid confusion, we have (in both the RHIC and LHC cases) keptthe temperature fixed as the magnetic field is increased. As explained earlier, this isa reasonable approximation for collisions with moderate impact parameters, but it isnot accurate for highly peripheral collisions. For these, the temperature is lower thanin the nearly central case; unfortunately, the numerical extent of this effect is not wellunderstood. However, as stated above, the relative kinematic viscosity decreases as thetemperature is lowered (for a fixed magnetic field), so ν/ν is smaller in the peripheralcase, for a given value of the magnetic field. That is, taking this effect into account willonly accentuate the tendency of the magnetic field to reduce ν/ν . (In fact, however, theeffect of this is rather small.) • Taking these observations into account, one can arrive at a rough idea as to how ν/ν The definition of the energy density in heavy ion collisions is a subtle matter: see the discussion ofthis issue in [16]. − m π on the horizontal axis in Figure 3 corresponds to a range [46]of impact parameters b ≈ − fm. We then proceed as in the construction of Figure 3,specifying lower temperatures associated with each step up in the magnetic field. In doingso, we are guided by remarks in [82,83], to the effect that temperatures in peripheral LHCcollisions approximate those in their central RHIC counterparts. We assume, for the sakeof definiteness, that this begins to happen at around b = 10 fm. We then (somewhatarbitrarily) interpolate linearly. The results are shown in the table ( α being chosen inthe same manner as before) and in Figure 4; they are “schematic” in the sense that theyare based on simplified assumptions regarding the way the temperature varies with theimpact parameter. b (fm) T ∞ (MeV) ν/ν Re/Re b = 0 fm to around b = 3 fm, but then declines much more slowly for largerimpact parameters. 17 . Conclusion Much of the work on the holographic description of the QGP has (rightly) focused onthe entropy-normalized dynamic viscosity, to the point where η s is often described as “theviscosity” of the QGP. Here we have argued that other measures of viscosity, notably thekinematic viscosity (or the Reynolds number) may usefully supplement the role of η s , bybeing capable of responding to strong variations in physical conditions from collision tocollision: in particular, ν and Re can and do respond to the powerful magnetic fieldswhich arise in peripheral collisions.We found in fact that a very simple holographic model imposes an upper bound on Re , the Reynolds number for the QGP produced by a nearly central collision, which isin remarkably good agreement with the lower bound deduced from observations. On theother hand, the same model predicts that Re/Re , where Re is the Reynolds number fora generic (peripheral) collision, varies surprisingly strongly with the magnetic field andtherefore with the impact parameter b .As always, we must bear in mind that holography is still far from being a precisiontool; indeed one reason for pursuing studies such as the present one is to determine howseriously holographic predictions regarding the behaviour of the QGP should be taken. Inthe present case, there is reason to doubt the validity of the precise numerical predictionsof the model: the reader may prefer to interpret our results as simply implying thatholography indicates that there might be a significant difference between the Reynoldsnumber of the plasma produced by a nearly central collision and that of its counterpartsappearing in peripheral collisions.Nevertheless, even at this qualitative level, one might think that the rather dramaticpredicted effect of magnetic fields on the Reynolds number or kinematic viscosity wouldbe easily detected in the data from experimental studies focusing on centrality dependence[84, 85]. But this may not be the case. For while our (partly conjectural) Figure 4 doesshow a strong variation of ν/ν with b , most of the variation occurs in the region of verysmall b , the graph being otherwise rather flat; depending on the resolution, this couldmean that the effect will appear to be independent of centrality. (On the other hand, itshould be sensitive to global parameters, such as the impact energy.)Furthermore, the extreme magnetic fields we have been considering, and of courseconsequently their influence on ν/ν and Re , are thought to be strongly time-dependent,and it is not yet clear whether they persist for a sufficiently long time to have a significantobservable effect. It will be interesting to see whether variations of the Reynolds numbercan be detected in future studies, either directly or perhaps indirectly; for example, ininvestigations of paramagnetic squeezing [42, 50]. Acknowledgements
The author is extremely grateful for the hospitality of the HECAP Section of the AbdusSalam ICTP, where this work was initiated.18 eferences [1] Makoto Natsuume, AdS/CFT Duality User Guide, Lect.Notes Phys. 903 (2015),arXiv:1409.3575 [hep-th][2] Veronika E. Hubeny, The AdS/CFT Correspondence, Class.Quant.Grav. 32 (2015)124010, arXiv:1501.00007 [gr-qc][3] Youngman Kim, Ik Jae Shin, Takuya Tsukioka, Holographic QCD: Past, Present,and Future, Prog. Part. Nucl. Phys. 68 (2013) 55, arXiv:1205.4852 [hep-ph][4] Oliver DeWolfe, Steven S. Gubser, Christopher Rosen, Derek Teaney, Heavy Ionsand String Theory, Prog. Part .Nucl. Phys. 75 (2014) 86, arXiv:1304.7794 [hep-th][5] Romuald A. Janik, AdS/CFT and Applications, PoS EPS-HEP 2013 (2013) 141,arXiv:1311.3966 [hep-ph][6] Paul M. Chesler, Wilke van der Schee, Early thermalization, hydrodynamics andenergy loss in AdS/CFT, Int.J.Mod.Phys. E24 (2015) 10, 1530011, arXiv:1501.04952[nucl-th][7] P.Kovtun, D.T.Son, A.O.Starinets, Viscosity in Strongly Interacting QuantumField Theories from Black Hole Physics, Phys.Rev.Lett. 94 (2005) 111601,arXiv:hep-th/0405231[8] D. T. Son, A. O. Starinets, Viscosity, Black Holes, and Quantum Field Theory,Ann.Rev.Nucl.Part.Sci.57:95-118,2007, arXiv:0704.0240 [hep-th][9] Sera Cremonini, The Shear Viscosity to Entropy Ratio: A Status Report, Mod. Phys.Lett.B, Volume: 25 (2011) 1867, arXiv:1108.0677 [hep-th][10] Huichao Song, QGP viscosity at RHIC and the LHC - a 2012 status report, Nucl.Phys. A 904-905, 114c (2013), arXiv:1210.5778 [nucl-th][11] Jinfeng Liao, Volker Koch, On the Fluidity and Super-Criticality of the QCD matterat RHIC, Phys.Rev. C81 (2010) 014902, arXiv:0909.3105 [hep-ph][12] Volker Koch, Adam Bzdak, Jinfeng Liao, Facets of the QCD Phase-Diagram, EPJWeb Conf. 13 (2011) 02001, arXiv:1101.4245 [nucl-th][13] L F Kozin, Steve C Hansen,
Mercury Handbook : Chemistry, Applications and En-vironmental Impact , RSC Publishing, 2013[14] Thomas Schaefer, Derek Teaney, Nearly Perfect Fluidity: From Cold Atomic Gasesto Hot Quark Gluon Plasmas, Rept.Prog.Phys.72:126001,2009, arXiv:0904.3107 [hep-ph][15] L. P. Csernai, D. D. Strottman, Cs. Anderlik, Kelvin-Helmholz instability in highenergy heavy ion collisions, Phys.Rev.C85.054901(2012), arXiv:1112.4287 [nucl-th]1916] B.B. Back et al., The PHOBOS perspective on discoveries at RHIC,Nucl.Phys.A757:28-101,2005, arXiv:nucl-ex/0410022[17] V. Skokov, A.Yu. Illarionov, V. Toneev, Estimate of the magnetic field strength inheavy-ion collisions, Int.J.Mod.Phys. A24 (2009) 5925, arXiv:0907.1396 [nucl-th][18] Kirill Tuchin, Particle production in strong electromagnetic fields in relativisticheavy-ion collisions, Adv.High Energy Phys. 2013 (2013) 490495, arXiv:1301.0099[hep-ph][19] Xu-Guang Huang, Electromagnetic fields and anomalous transports in heavy-ion col-lisions — A pedagogical review, Rept.Prog.Phys. 79 (2016) 076302, arXiv:1509.04073[nucl-th][20]
Strongly interacting matter in magnetic fields , Eds. D. Kharzeev, K. Landsteiner, A.Schmitt, H.-U. Yee, Lect. Notes Phys. 871, 1 (2013), arXiv:1211.6245[hep-ph][21] Koichi Hattori, Xu-Guang Huang, Novel quantum phenomena induced by strongmagnetic fields in heavy-ion collisions, Nucl.Sci.Tech. 28 (2017) 26, arXiv:1609.00747[nucl-th][22] David Dudal, Diego Rocha Granado, Thomas G. Mertens, Deconfinement andchiral transition in AdS/QCD wall models supplemented with a magnetic field,arXiv:1611.04285 [hep-th][23] Umut Gursoy, Improved Holographic QCD and the Quark-gluon Plasma,arXiv:1612.00899 [hep-th][24] Sera Cremonini, Umut Gursoy, Phillip Szepietowski, On the Temperature Depen-dence of the Shear Viscosity and Holography, JHEP 1208 (2012) 167, arXiv:1206.3581[hep-th][25] Nikita Astrakhantsev, Viktor Braguta, Andrey Kotov, Temperature dependence ofshear viscosity of SU(3)–gluodynamics within lattice simulation, arXiv:1701.02266[hep-lat][26] Fei Gao, Yu-xin Liu, Temperature Effect on Shear and Bulk Viscosities of QCDMatter, arXiv:1702.01420 [hep-ph][27] Gojko Vujanovic, Gabriel S. Denicol, Matthew Luzum, Sangyong Jeon, Charles Gale,Investigating the temperature dependence of the specific shear viscosity of QCD withdilepton radiation, arXiv:1702.02941 [nucl-th][28] You Zhou (for the ALICE Collaboration), Overview of recent azimuthal correlationmeasurements from ALICE, arXiv:1610.08311 [nucl-ex][29] Robert C. Myers, Miguel F. Paulos, Aninda Sinha, Holographic Hydrodynamics witha Chemical Potential, JHEP 0906:006,2009, arXiv:0903.2834[30] Gary T. Horowitz, Introduction to Holographic Superconductors, arXiv:1002.1722[hep-th] 2031] Sean A. Hartnoll, Andrew Lucas, Subir Sachdev, Holographic quantum matter,arXiv:1612.07324 [hep-th][32] Johanna Erdmenger, Patrick Kerner, Hansjörg Zeller, Non-universal shear viscosityfrom Einstein gravity, Phys.Lett.B699:301-304,2011, arXiv:1011.5912 [hep-th][33] Kiminad A. Mamo, Holographic RG flow of the shear viscosity to entropy densityratio in strongly coupled anisotropic plasma, JHEP 1210 (2012) 070, arXiv:1205.1797[hep-th][34] R. Critelli, S. I. Finazzo, M. Zaniboni, J. Noronha, Anisotropic shear viscosity of astrongly coupled non-Abelian plasma from magnetic branes, Phys. Rev. D 90, 066006(2014), arXiv:1406.6019 [hep-th][35] Sachin Jain, Rickmoy Samanta, Sandip P. Trivedi, The Shear Viscosity in AnisotropicPhases, JHEP 1510 (2015) 028, arXiv:1506.01899 [hep-th][36] Marek Rogatko, Karol I. Wysokinski, Viscosity bound for anisotropic superfluids withdark matter sector, arXiv:1612.02593 [hep-th][37] Sean A. Hartnoll, David M. Ramirez, Jorge E. Santos, Entropy production, viscositybounds and bumpy black holes, JHEP 1603 (2016) 170, arXiv:1601.02757[38] Lasma Alberte, Matteo Baggioli, Oriol Pujolas, Viscosity bound violation in holo-graphic solids and the viscoelastic response, JHEP 1607 (2016) 074, arXiv:1601.03384[hep-th][39] Piyabut Burikham, Napat Poovuttikul, Shear viscosity in holography and effectivetheory of transport without translational symmetry, Phys.Rev. D94 (2016) 106001,arXiv:1601.04624 [hep-th][40] Sangyong Jeon, Laurence G. Yaffe, From Quantum Field Theory to Hydrodynamics:Transport Coefficients and Effective Kinetic Theory, Phys.Rev. D53 (1996) 5799-5809, arXiv:hep-ph/9512263[41] Kirill Tuchin, On viscous flow and azimuthal anisotropy of quark-gluon plasma instrong magnetic field, J.Phys. G39 (2012) 025010, arXiv:1108.4394 [nucl-th][42] G. S. Bali, F. Bruckmann, G. Endrodi, A. Schafer, Paramagnetic squeezing of QCDmatter, Phys. Rev. Lett. 112, 042301 (2014), arXiv:1311.2559 [hep-lat][43] JW Schmelzer, ED Zanotto, VM Fokin, Pressure dependence of viscosity, J ChemPhys. 2005 122(7):074511[44] V. Voronyuk, V.D Toneev, W. Cassing, E.L. Bratkovskaya, V.P. Konchakovski,S.A. Voloshin, Electromagnetic field evolution in relativistic heavy-ion collisions,Phys.Rev.C83:054911,2011, arXiv:1103.4239 [nucl-th][45] Adam Bzdak, Vladimir Skokov, Event-by-event fluctuations of magnetic and electricfields in heavy ion collisions, Phys.Lett. B710 (2012) 171, arXiv:1111.1949 [hep-ph]2146] Wei-Tian Deng, Xu-Guang Huang, Event-by-event generation of electromagneticfields in heavy-ion collisions, Phys. Rev. C 85 044907 (2012), arXiv:1201.5108 [nucl-th][47] Victor Roy, Shi Pu, Event-by-event distribution of magnetic field energy over initialfluid energy density in √ s NN =