Constraints on AdS/CFT Gravity Dual Models of Heavy Ion Collisions
aa r X i v : . [ h e p - ph ] J un Constraints on AdS/CFT Gravity Dual Models of Heavy Ion Collisions
Jorge Noronha , Miklos Gyulassy , and Giorgio Torrieri Department of Physics, Columbia University, 538 West 120 th Street, New York, NY 10027, USA Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany (Dated: October 31, 2018)We show that the five-fold constraints due to (1) the observed nuclear modification of heavyquark jets measured via non-photonic electrons R eAA ( p T ∼ v ( p T ∼ dN π /dy , (4) the latticeQCD entropy density deficiency, S/S SB , of strongly coupled Quark-Gluon Plasmas (sQGP), and(5) a causal requirement are analytically correlated in a class of gauge/string gravity dual models ofsQGP dynamics. Current RHIC/BNL and lattice QCD data are found to be remarkably compatiblewith these models if the t’Hooft and Gauss-Bonnet coupling parameters lie in the range λ ≈ − < λ GB < .
09. In addition, the observed five-fold correlation appears to favor color glasscondensate over Glauber initial conditions within current systematic errors.
PACS numbers: 25.75.-q, 11.25.Tq, 13.87.-a
The combined observations of the quenching of hard(high transverse momentum or high quark mass jets) pro-cesses and the nearly “perfect fluid” elliptic flow of soft(low momentum transfer) hadrons produced in Au+Aucollisions at √ s = 200 AGeV at the Relativistic HeavyIon Collider (RHIC) [1] have been interpreted as provid-ing evidence for the formation of a new form of stronglyinteracting quark-gluon plasma (sQGP) [2]. Further-more, bulk multiplicity (entropy) production systemat-ics have been interpreted as evidence for gluon satura-tion of the initial conditions as predicted by the ColorGlass Condensate (CGC) model [3, 4]. However, it hasbeen a challenge to find a single consistent theoreticalframework that can explain simultaneously both soft andhard phenomena. These phenomena include 1) the nu-clear modification of high transverse momenta, p T > p T < α s = g Y M / π → . − . λ = g Y M N c ≫ analytic con-nections between a wide variety of thermodynamic andnonequilibrium dynamic variables at strong coupling thatwere not yet realized with traditional gauge theory tech-niques.In this Letter, we focus on predicted analytic con-nections between three fundamental properties of thesQGP: (1) its equation of state (entropy=S), (2) its long wavelength transport coefficients (viscosity= η ) , and(3) coupling between long wavelength near-equilibrium“soft medium” properties and short-wavelength non-equilibrated “hard probes” (energy loss per unit length, dE/dx ). We show that these three properties togetherwith global entropy and causality restrictions can providevaluable phenomenological constraints of higher dimen-sional gravity dual models of sQGP in heavy ion colli-sions. We consider here the constraints imposed by cur-rent RHIC and LQCD data on a class of gravity dualmodels that include quadratic as well as quartic cur-vature corrections to the classical Einstein-Hilbert ac-tion for the effective 5 dimensional dual gravity action.Therefore, we implement and extend the suggestion madein [16] to include perturbatively both the lowest order O ( λ GB ∼ /N c ) Gauss-Bonnet R and O (1 /λ / ) R curvature corrections to the three properties above.Our analysis is based on the following remarkably sim-ple algebraic expressions relating these three fundamen-tal sQGP properties: S/S SB = 34 (cid:18) λ GB + 158 ζ (3) λ / (cid:19) , (1) η/s = 14 π (cid:18) − λ GB + 15 ζ (3) λ / (cid:19) , (2) τ − Q = µ Q (cid:18) λ GB + 1516 ζ (3) λ / (cid:19) (3)where s = S/V is the entropy density. The heavyquark jet relaxation rate, 1 /τ Q , is controlled by µ Q = √ λ πT / M Q for a heavy quark with mass M Q in aplasma of temperature T . The relaxation time is relatedto the energy loss per unit length through τ Q ( λ, λ GB ) = − / ( d log p/dt ) = − / ( d log E/dx ), where p = M Q γv and v = p/E .We note that the R correction O (1 /λ / ) to the heavyquark jet energy loss is a new result [25] reported in thisLetter and it is needed for a consistent additive perturba- Glauber CGC <Λ< È Λ GB È £ (cid:144) S SB = ± STAR20 - % PHENIXSTARmin bias v H p T = L R AA e H p T = . G e V L FIG. 1: (Color online) The gravity dual model correlationof the nuclear modification factor , R eAA , of hard ( p T = 5 . v , of soft ( p T = 1 GeV, 20-60%centrality) pions assuming Glauber (left) and CGC (right)initial participant distributions and dN π /dy = 1000 are com-pared. The green region corresponds to the range of entropyfunction deficiencies, S/S SB , consistent with infinite volumeextrapolations of lattice QCD data [34]. The red (blue) el-lipse indicates PHENIX (STAR) 0-10% centrality R eAA and20-60% v centrality Au+Au 200 AGeV data from RHIC [35]-[38]. The dashed blue curves show fixed S/S SB = 0 . S/S SB = 0 . S/S SB = 0 .
9, respectively in the range λ = 5 , . . . , λ GB = 0 .
09 is indicated bysolid red. The gray ellipse shows preliminary v for minimumbias data with estimated non-flow effects subtracted [39]. tive application of this class of AdS/CFT models to heavyion reactions. We consider here only heavy quark observ-ables because gravity dual string drag models [13] applyonly for heavy M Q > √ λT jets. The heavy quark jetdrag is then modeled as a trailing string moving in a blackbrane background according to the classical Nambu-Gotoaction (with dilaton neglected) A NG = − πα ′ R d σ √− g where g = det g ab = G µν ∂ a X µ ∂ b X ν is the inducedworldsheet metric, σ a = ( τ, σ ) are the internal world-sheet coordinates, G µν ( X ) is the background metric, and X µ = X µ ( τ, σ ) is the embedding of the string in space-time. The trailing string ansatz (where τ = t, σ = u and X µ ( t, u ) = ( t, x + vt + ξ ( u ) , , , u )) describes theasymptotic behavior of a string attached to a movingheavy quark (the string endpoint) with velocity v in the x direction and located at a fixed AdS radial coordinate u m ≫ u h [13]. The black brane horizon coordinate u h ∝ T α ′ is is determined by G ( u h ) = 0. Using the ansatzabove and the string’s classical equations of motion, onecan show that the drag force dp/dt = − Cv/ (2 πα ′ ), where C is a constant determined by the negativity condition that g ( u ) = G uu (cid:0) G + v G xx (cid:1) (cid:18) C v G G xx (cid:19) − < u h ≤ u ≤ u m . However, both the numerator anddenominator in Eq. (4) change their sign simultaneouslyat a certain u ∗ [13] given by the root of the equation G ( u ∗ ) + v G xx ( u ∗ ) = 0. This fixes C = G xx ( u ∗ )and dp/dt = − v G xx ( u ∗ ) / (2 πα ′ ). Neglecting higher-order derivative corrections in N = 4 SYM one finds u ∗ = u h √ γ , where γ = 1 / √ − v . The condition that u ∗ ≤ u m leads to a maximum “speed limit” for the heavyquark jet to be consistent with this trailing string ansatzgiven by γ max ≤ u m /u h [14].Using the metric derived in [10] to O ( α ′ ), one cancompute the effects of quartic corrections on the dragforce [15] and determine u ∗ perturbatively to O ( λ − / )as [26] u ∗ = u h √ γ h ζ (3) v λ / (cid:16) γ − γ (cid:17)i and thedrag force dpdt = −√ λ T π vγ (cid:20) ζ (3) λ / (cid:18) − γ + 6724 γ (cid:19)(cid:21) . (5)The heavy quark mass at T = 0 is M Q = u m / (2 πα ′ )and, to leading order in 1 /λ , u m /u h ≃ M Q λT . Thus, thecorrected u ∗ displayed above defines a new speed limit γ m ≃ M Q λT (cid:2) − (cid:0) πηs − (cid:1)(cid:3) , after neglecting terms of O (1 /γ, /N c ). Note that γ m and dp/dt decrease withincreasing η/s .For our applications, we consider the range λ ∼ − | λ GB | < . N c = 3. In this parame-ter range the 1 /λ / and λ GB ∼ /N c corrections arecomparable. We neglect known (but formally) higher or-der terms [12] O ( √ λ/N c ) in this first attempt to testpredicted dynamical correlations between hard and softphenomena in high energy A+A collisions.The small Gauss-Bonnet parameter λ GB = ( c − a ) / c is related to the central charges c and a (related to theconformal anomaly in curved spacetime) of the dual CFTas noted in Eq. (2.14) of Ref. [16]. Varying λ GB providesa parametric way to explore deformations of the original N = 4 SU ( N c ) SYM theory. Interest in Gauss-Bonnetdeformations were heightened when Kats and Petrov [21]argued that for N = 2 Sp ( N c ), λ GB = 1 / N c , the KSSviscosity bound on η/s ≥ / π was violated by 17%for N c = 3. As further shown in [16], a large classof other effective CFTs are now known to lead to sim-ilar λ GB ∝ /N c effects. However, the analysis of Refs.[18]-[20] revealed that λ GB deformations are limited byrequirements of causality and positive energy flow to anarrow range − / < λ GB < / τ Q into the observed nuclear mod-ification of single non-photonic electrons, R eAA ( p T =5 . (cid:144) S SB ΠΗ (cid:144) s = H a L . . v = . R AA e . .
32 0 . - - Λ G B Λ (cid:144) S SB ΠΗ (cid:144) s = H b L . v = . v = . R AA e . .
32 0 . - - Λ Λ G B FIG. 2: (Color online) Five fold phenomenological constraintsin the t’Hooft and Gauss-Bonnet parameter space ( λ, λ GB ).(1) The green region with black dashed contours is from latticeQCD constraints on 0 . < S/S SB < .
9. (2) The cyan regionis determined from noncentral elliptic flow v ( p T = 1 , − . ± .
01 [38]. Blue dashed contours correspondto fixed 4 πη/s = 1 , . ,
2. The inversion λ GB ( λ, v ( η/s )) isbased on minimum bias viscous hydro results of [31] assumingCGC initial eccentricities scaled by a factor 1.1 in panel (a)and unscaled 1.0 in panel (b). The entropy is constrainedby the (3) dN π /dy = 1000 pion rapidity density in centralcollisions. (4) The gray region and contours are determinedfrom central R eAA ( p T = 5 . . ± .
07 data [35].(5) The horizontal red line constraint is the causality upperbound for λ maxGB = 0 .
09 [19, 20]. The yellow trapezoidal regionwith the purple boundary is the intersection of the five foldconstraint bands. Note the red circled five fold conjunctionarea in panel (a) ( λ ≈ , λ GB ≈ .
08) that is absent in theunscaled panel (b). in Eq. (3) to compute the path length, L , dependentheavy quark fractional energy loss ǫ ( L ). The heavyquark jet nuclear modification factor is then R AA = h (1 − ǫ ) n Q i L , where n Q ( p T ) is the flavor dependent spec-tral index n Q + 1 = − dd ln p T ln (cid:16) dσ Q dydp T (cid:17) obtained fromFONLL production cross sections [28] as used in [27].The path length average of the nuclear modification atimpact parameter b is computed using a Woods-Saxonnuclear density profile with Glauber profiles T A ( ~x ⊥ ) with σ NN = 42 mb. For 0-10% centrality triggered databoth Glauber and CGC geometries lead to similar nu-merical results [29]. The distribution of initial hard jetproduction points at a given ~x ⊥ and azimuthal direc-tion φ is taken to be proportional to the binary par-ton collision density, T AA ( ~x ⊥ , b ). We assume a longi- tudinally expanding local (participant) parton density ρ ( ~x ⊥ , b ) = χρ part ( ~x ⊥ , b ) /τ , where χ ≡ ( dN π /dy ) /N part and ρ part is the Glauber participant nucleon profile den-sity. However, we evaluate (3) with a reduced tempera-ture T CF T = 0 . S/S SB ) / T QCD to take into accountthe smaller number of degrees of freedom in a strongly-coupled QCD plasma, which is similar to the prescriptiongiven in [30]. We compute the heavy quark modificationfactor R QAA via R QAA ( p T , b ) = Z π dφ Z d ~x ⊥ T AA ( ~x ⊥ , b )2 π N bin ( b ) × exp (cid:20) − n Q ( p T ) Z τ f τ dττ c ( ~x ⊥ + τ ˆ e ( φ ) , φ ) (cid:21) (6)where N bin is the number of binary collisions. Here, τ = 1 fm/c is the assumed plasma equilibration timeand τ f is determined from T ( ~ℓ, τ f ) = T f = 140 MeV,i.e, the time at which the local temperature falls belowa freeze-out temperature taken from [31]. Systematic er-rors associated with the freeze-out condition will be dis-cussed elsewhere.In order to compute v (1 GeV) for the C (20-60%) cen-trality class, we employ a linear fit to the numerical vis-cous hydrodynamic results of Luzum and Romatschke[31]. The dependence of v on viscosity for both Glauber[32] and CGC [4] initial transverse profiles can be well fitwith v ( p T , η/s, C ) = a ( p T ) ǫ ( C ) (1 − b η/s ) (7)where ǫ ( C ) = h y − x i C / h x + y i C is the average initialelliptic geometric eccentricity for the centrality class C .To rescale the minimum bias viscous hydro results of Ref.[31] to the considered 20-60% centrality class we use thefactor ǫ Glaub (20 − /ǫ Glaub (0 − . / .
281 =1 .
128 from Ref. [33]. Our fit to the rescaled numericalresults of [31] give b ≈ . a ( p T = 1) ǫ (20 − ≈ .
14 (0 . p T ∼ v (4) and PHENIX v ( BBC ) data andnon-flow effects [39] that complicate the interpretationof minimum bias data in [31] are reduced.We find (see Figs. 1 and 2) that a (reasonable) com-bination of model parameters ( λ and λ GB ) can accountfor the correlation between the reported R eAA [35, 36]and v [38] taking into account the LQCD constrainton the equations of state deficiency S/S SB [34] (in therange T ∼ − T c ). Our results suggest that withincurrent systematic errors a small positive quadratic cur-vature correction with 0 < λ GB < . λ GB ( λ, v ( η/s )) inver-sion. We find that a 10% reduction of the viscous hydro-dynamic v ( η/s ) (as for minimum bias centrality) virtu-ally eliminates the yellow overlap region and could falsifythe AdS/CFT description based on Eqs. (1-3). Greatcare is called for at this time to avoid premature con-clusions. Systematic studies on viscous hydrodynamicdependence on centrality cuts and improved experimen-tal control over non-flow corrections will be needed be-fore definitive conclusions could be reached. The goodnews demonstrated by comparing Fig. 2a and 2b is thatif improved theoretical and experimental control (bet-ter than 10%) over elliptic flow systematics and the nu-clear modification of heavy quark jet observables can bereached, then rather strong experimental constraints onthe AdS/CFT gravity dual model parameters could beachieved. We close by emphasizing [27] that future com-parison of the nuclear modification of identified bottomand charm quark jets at RHIC and LHC combined withthe fivefold (hard/soft) constraints considered in this Let-ter will provide especially stringent tests of AdS/CFTgravity dual phenomenology applied to high energy heavyion reactions.We thank A. Dumitru, S. Gubser, W. Horowitz,A. Poszkanzer, B. Cole, and W. Zajc for useful com-ments. J.N. and M.G. acknowledge support from US-DOE Nuclear Science Grant No. DE-FG02-93ER40764.G.T. acknowledges support from the Helmholtz In-ternational Center for FAIR within the framework ofthe LOEWE program (Landesoffensive zur EntwicklungWissenschaftlich- ¨Okonomischer Exzellenz) launched bythe State of Hesse. [1] I. Arsene et al. [BRAHMS Collaboration], Nucl. Phys.A , 1 (2005); B. B. Back et al. , [PHOBOS Collab-oration] Nucl. Phys. A , 28 (2005); J. Adams et al. [STAR Collaboration], Nucl. Phys. A , 102 (2005);K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A , 184 (2005).[2] M. Gyulassy and L. McLerran, Nucl. Phys. A , 30(2005); E. V. Shuryak, Nucl. Phys. A , 64 (2005).[3] D. Kharzeev, E. Levin and M. Nardi, Phys. Rev. C ,054903 (2005); L. D. McLerran and R. Venugopalan,Phys. Rev. D , 2233 (1994).[4] A. Dumitru, E. Molnar, and Y. Nara, Phys. Rev. C ,024910 (2007).[5] P. Danielewicz and M. Gyulassy, Phys. Rev. D , 53(1985).[6] D. Molnar and M. Gyulassy, Nucl. Phys. A , 495(2002) [Erratum-ibid. A , 893 (2002)].[7] A. Majumder, B. Muller and X. N. Wang, Phys. Rev.Lett. , 192301 (2007).[8] O. Fochler, Z. Xu and C. Greiner, arXiv:0806.1169 [hep-ph].[9] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998);E. Witten, Adv. Theor. Math. Phys. , 253 (1998);S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B , 105 (1998).[10] S. S. Gubser, I. R. Klebanov and A. A. Tseytlin, Nucl.Phys. B , 202 (1998).[11] A. Buchel, J. T. Liu and A. O. Starinets, Nucl. Phys.B , 56 (2005); A. Buchel, Nucl. Phys. B , 166(2008).[12] A. Buchel, Nucl. Phys. B , 166 (2008); R. C. My-ers, M. F. Paulos and A. Sinha, Phys. Rev. D ,041901 (2009); A. Buchel, R. C. Myers, M. F. Paulosand A. Sinha, Phys. Lett. B , 364 (2008).[13] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz andL. G. Yaffe, JHEP , 013 (2006); S. S. Gubser, Phys.Rev. D , 126005 (2006); J. Casalderrey-Solana andD. Teaney, Phys. Rev. D , 085012 (2006).[14] J. Casalderrey-Solana and D. Teaney, JHEP , 039(2007); S. S. Gubser, Nucl. Phys. B , 175 (2008).[15] J. F. Vazquez-Poritz, arXiv:0803.2890 [hep-th].[16] A. Buchel, R. C. Myers and A. Sinha, JHEP , 084(2009).[17] A. Karch and E. Katz, JHEP , 043 (2002).[18] M. Brigante, H. Liu, R. C. Myers, S. Shenker andS. Yaida, Phys. Rev. D , 126006 (2008).[19] M. Brigante, H. Liu, R. C. Myers, S. Shenker andS. Yaida, Phys. Rev. Lett. , 191601 (2008).[20] D. M. Hofman and J. Maldacena, JHEP , 012(2008).[21] Y. Kats and P. Petrov, arXiv:0712.0743 [hep-th].[22] P. K. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev.Lett. , 111601 (2005).[23] R. G. Cai, Phys. Rev. D , 084014 (2002).[24] K. B. Fadafan, JHEP , 051 (2008).[25] J. Noronha et al, to be published.[26] The drag force in this case had been computed numeri-cally in Ref. [15] but the analytical expression in Eq. (5)is, as far as we know, a new result.[27] W. A. Horowitz and M. Gyulassy, Phys. Lett. B , 320(2008).[28] M. Cacciari, P. Nason, R. Vogt, Phys. Rev. Lett. ,122001 (2005); M. L. Mangano, P. Nason, G. Ridolfi,Nucl. Phys. B , 295 (1992).[29] A. Adil, H. J. Drescher, A. Dumitru, A. Hayashigaki andY. Nara, Phys. Rev. C , 044905 (2006).[30] S. S. Gubser, Phys. Rev. D , 126003 (2007)[31] M. Luzum and P. Romatschke, Phys. Rev. C , 034915(2008); Erratum-ibid. , 039903 (E) (2009).[32] P. F. Kolb, U. W. Heinz, P. Huovinen, K. J. Eskola, andK. Tuominen, Nucl. Phys. A , 197 (2001).[33] K. Adcox et al. [PHENIX Collaboration], Phys. Rev.Lett. , 212301 (2002); D. d’Enterria, private commu-nication.[34] A. Bazavov et al. , arXiv:0903.4379 [hep-lat]; M. Cheng et al. , Phys. Rev. D , 014511 (2008); C. Bernard etal. , Phys. Rev. D , 094505 (2007); Y. Aoki, Z. Fodor,S. D. Katz and K. K. Szabo, JHEP , 089 (2006).[35] S. S. Adler et al. [PHENIX], Phys. Rev. Lett. , 032301(2006); A. Adare et al. [PHENIX], Phys. Rev. Lett. ,172301 (2007).[36] B. I. Abelev et al. [STAR], Phys. Rev. Lett. , 192301(2007).[37] A. Afanasiev et al. [PHENIX Collaboration],arXiv:0905.1070 [nucl-ex].[38] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C77