Universal Properties of Bulk Viscosity Near the QCD Phase Transition Frithjof Karsch Brookhaven National Laboratory

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2008 Universal properties of bulk viscosity near the QCD phase transition Frithjof Karsch Brookhaven National Laboratory

Dmitri Kharzeev Brookhaven National Laboratory

Kirill Tuchin Iowa State University, [email protected]

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Abstract We extract the bulk viscosity of hot quark-gluon matter in the presence of light quarks from the recent lattice data on the QCD equation of state. For that purpose we extend the sum rule analysis by including the contribution of light quarks. We also discuss the universal properties of bulk viscosity in the vicinity of a second-order phase transition, as it might occur in the chiral limit of QCD at fixed strange quark mass and most likely does occur in two-flavor QCD. We point out that a chiral transition in the O (4) universality class at zero baryon density as well as the transition at the chiral critical point which belongs to the Z (2) universality class both lead to the critical behavior of bulk viscosity. In particular, the latter universality class implies the divergence of the bulk viscosity, which may be used as a signature of the critical point. We discuss the physical picture behind the dramatic increase of bulk viscosity seen in our analysis, and devise possible experimental tests of related phenomena.

Keywords RIKEN BNL Research Center

Disciplines Astrophysics and Astronomy | Physics

Comments This article is from Physics Letters B 663 (2008): 217, doi: 10.1016/j.physletb.2008.01.080. Posted with permission.

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This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/physastro_pubs/135 Physics Letters B 663 (2008) 217–221

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Physics Letters B

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Universal properties of bulk viscosity near the QCD phase transition ∗ Frithjof Karsch a, Dmitri Kharzeev a, , Kirill Tuchin b,c a Department of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000, USA b Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA c RIKEN BNL Research Center, Upton, NY 11973-5000, USA article info abstract

Article history: We extract the bulk viscosity of hot quark–gluon matter in the presence of light quarks from the recent Received 15 November 2007 lattice data on the QCD equation of state. For that purpose we extend the sum rule analysis by including Accepted 2 January 2008 the contribution of light quarks. We also discuss the universal properties of bulk viscosity in the vicinity Available online 8 April 2008 of a second-order phase transition, as it might occur in the chiral limit of QCD at ﬁxed strange quark Editor: J.-P. Blaizot mass and most likely does occur in two-ﬂavor QCD. We point out that a chiral transition in the O (4) universality class at zero baryon density as well as the transition at the chiral critical point which belongs to the Z(2) universality class both lead to the critical behavior of bulk viscosity. In particular, the latter universality class implies the divergence of the bulk viscosity, which may be used as a signature of the critical point. We discuss the physical picture behind the dramatic increase of bulk viscosity seen in our analysis, and devise possible experimental tests of related phenomena. © 2008 Elsevier B.V. Open access under CC BY license.

1. Introduction in terms of the general theory of critical phenomena? Can it be used in the experimental studies to isolate the signatures of phase Recently, it was pointed out that bulk viscosity in SU(3) gluo- transitions in heavy ion collisions? In this Letter we address these dynamics rises dramatically in the vicinity of the deconfinement questions by generalizing the previous analysis to the case of QCD phase transition [1]. The analysis was based on an exact sum rule with two light quarks and a strange quark, which is of direct rele- derived from low-energy theorems of broken scale invariance, the vance to heavy ion experiments. We will use as an input the very high-statistics lattice data on the equation of state [2], and an recent high statistics lattice data on the equation of state of QCD ansatz for the spectral density of the correlation function of the with almost physical quark masses from the RIKEN-BNL-Columbia- energy–momentum tensor. The rapid increase of bulk viscosity has Bielefeld Collaboration [5]. been associated with the fast growth of the thermal expectation The Letter is organized as follows. In Section 2 we review the definition of bulk viscosity and the formalism relating it to the value of the trace of the energy–momentum tensor θT = E − 3P , where E is the energy density and P is the pressure. Very re- correlation function of the trace of the energy–momentum tensor cently, bulk viscosity in SU(3) gluodynamics has been measured based on Kubo’s formula. In Section 3 we discuss the low-energy on the lattice by the direct analysis of the correlation function of theorems based on QCD renormalization group which constrain the trace of the energy–momentum tensor [3] (see also [4]). The the zero-momentum correlation functions involving the trace of numerical results of the study [3] agree very well with the pre- the energy–momentum tensor. In this section we generalize the + vious analysis [1]. Since the methods used in Refs. [1,3], and the sum rule of Ref. [1] to full QCD with 2 1 flavors. In Section 4 we latticeobservablesusedasaninput,arequitedifferent,thisagree- briefly review the relevant lattice thermodynamics results, intro- ment indicates that the dramatic growth (by about three orders of duce and motivate our ansatz for the spectral density, and use the magnitude!) of bulk viscosity in the vicinity of the deconfinement sum rule to extract the bulk viscosity near the critical temperature. phase transition is indeed a prominent feature of SU(3) gluody- We find that the behavior of bulk viscosity in full QCD is qualita- namics. tively similar to the case of SU(3) gluodynamics. We then proceed Is this behavior specific for the first-order phase transition in to the discussion of uncertainties associated with our method, and SU(3) gluodynamics? What is the “microscopic” dynamics respon- estimate the error bars which should be associated with our result. sible for the growth of bulk viscosity? Can this growth be classified In Section 5 we classify the critical behavior responsible for the rapid growth of bulk viscosity near the phase transition. We argue that at zero baryon density, this behavior belongs to the O (4) uni- * Corresponding author. versality class. The bulk viscosity in this case does not diverge at E-mail address: [email protected] (D. Kharzeev). T = Tc , but has a cusp. On the other hand, in the vicinity of the

0370-2693 © 2008 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2008.01.080 218 F. Karsch et al. / Physics Letters B 663 (2008) 217–221 chiral critical point at finite baryon density, the critical behavior Using the Kramers–Kronig relation the retarded Green’s function is in the Z(2) universality class. This means that the bulk viscos- can be represented as = ity should diverge at T Tc . This behavior should manifest itself ∞ ∞ in heavy ion collisions through the decrease of average transverse 1 Im G R (u, p) ρ(u, p) G R (ω, p) = du = du. (7) momentum of produced particles, accompanied by the increase in π u − ω − iε ω − u + iε total multiplicity. This is due to both the increase in entropy as- −∞ −∞ sociated with a large bulk viscosity, and the associated quenching The retarded Green’s function G R (ω, p) of a bosonic excitation of the transverse hydrodynamical expansion of the system (“ra- is related to the Euclidean Green’s function G E (ω, p) by analytic dial flow”). Finally, we summarize our findings and discuss the continuation qualitative physical picture which arises from our analysis. In particular, we argue that the rapid growth of bulk viscosity favors the G E (ω, p) =−G R (iω, p), ω > 0. (8) scenario of “soft statistical hadronization” in which the expand- =− − ing system hadronizes at the phase transition by producing a large Using (7) and the fact that ρ(ω, p) ρ( ω, p) we recover number of color screening soft partons. ∞ ρ(u, 0) G ≡ lim G E (ω, 0) = 2 du. (9) 2. Kubo’s formula for bulk viscosity ω→0 u 0

According to Kubo’s formula the shear η and the bulk ζ vis- In the next section we use the low-energy theorems to relate G to cosities are related to the correlation function of the stress tensor the energy density and the pressure of hot hadronic matter. θij(x), i, j = 1, 2, 3, as 3. Low-energy theorems at finite T 2 η(ω) δilδkm + δimδkl − δikδlm + ζ(ω)δikδlm 3 In a conformally invariant theory the trace of the energy– ∞ momentum tensor vanishes. Therefore, Eq. (4) implies that the 1 3 i(ωt−kr) bulk viscosity is a measure for the violation of conformal invari- = lim d x dt e θik(t, r), θlm(0) . (1) ω k→0 ance. QCD is a conformally invariant theory at the classical level. 0 However, quantum fluctuations generate the scale anomaly which manifests itself in the non-conservation of the dilatational current Contracting the i,k and l,m indices gives the bulk viscosity as a s [6,7]: static limit of the correlation function of the trace of the stress μ tensor g μ μ ¯ β( ) 2 ∂ sμ = θμ = mqqq + F ≡ θF + θG , (10) ∞ 2g 1 1 = 3 iωt ζ lim dt d re θii(x), θkk(0) . (2) 2 ≡ aμν a ¯ 9 ω→0 ω where we adopted the shorthand notation F F Fμν and q, q 0 are three-vectors in the quark flavor space (i.e., summation over flavors is assumed). β(g) is the QCD β-function, which governs This formula can be written in terms of Lorentz-invariant operators the behavior of the running coupling if we recall that the ensemble average of the commutator of the Hamiltonian H with any operator O in equilibrium is given by dg(μ) = μ β(g). (11) ∂O dμ d3x θ (x), O = [H, O] = i = 0. (3) 00 eq =− 3 2 eq ∂t eq At the leading order β bg /(4π) . Although the scale symmetry of the QCD Lagrangian is broken Thus we can write (2) as by quantum vacuum fluctuations, there is a remaining symmetry ∞ imposed by the requirement of the renormalization group invari- 1 1 3 iωt μ μ ance of observable quantities [8]. This symmetry manifests itself ζ = lim dt d re θμ (x), θμ (0) . (4) 9 ω→0 ω in a chain of low-energy theorems (LET) for the correlation func- 0 2 tions of the operator θG (x) = (β/2g)F (x). These low-energy theo- In the following we are going to calculate the bulk viscosity us- rems entirely determine the dynamics of the effective low-energy ing the low-energy theorems involving the Green’s functions of the theory. This effective theory has an elegant geometrical interpreta- trace of energy–momentum tensor θ(x). Therefore, it is convenient tion [9]; in particular, gluodynamics can be represented as a clas- to re-write Kubo’s formula (4) using the retarded Green’s function sical theory formulated on a curved (conformally flat) space–time as background [10]. At finite temperature, the breaking of scale invariance by quantum fluctuations results in θ = E − 3P = 0clearly ∞ observed on the lattice for SU(3) gluodynamics [2]; the presence of 1 1 3 iωt R 1 1 R ζ = lim dt d re iG (x) = lim iG (ω, 0) quarks [11] including the physical case of two light and a strange 9 ω→0 ω 9 ω→0 ω 0 quark [5,12,13], or considering large Nc [14] does not change this conclusion. 1 1 R =− lim Im G (ω, 0). (5) The LET of Refs. [8,9] were generalized to the case of finite tem- 9 ω→0 ω perature in [15,16] (lattice formulation has been discussed very The last equation follows from the fact that due to P-invariance, recently in Ref. [17]). The lowest in the chain of relations reads the function Im G R (ω, 0) is odd in ω while Re G R (ω, 0) is even in (at zero baryon chemical potential): ω. Note that the non-vanishing ζ implies the existence of a mass- 4 ∂ less excitation in the spectral density ρ. d x T θG (x), O(0) = T − d OT , (12) Let us define the spectral density ∂ T 1 where d is a canonical dimension of the operator O.Inparticu- ρ(ω, p) =− Im G R (ω, p). (6) π lar, F. Karsch et al. / Physics Letters B 663 (2008) 217–221 219 4 ∂ where s is the entropy density and c is the speed of sound. In de- d x T θG (x), θG (0) = T − 4 θG T , (13a) s ∂ T riving (19) we used basic thermodynamic relations to relate temperature derivatives of energy density and pressure to the entropy 4 ∂ d x T θG (x), θF (0) = T − 3 θF T , (13b) = = E ∂ T density, s ∂ P/∂ T , specific heat, cv ∂ /∂ T and the velocity of 2 = E = sound, cs ∂ P/∂ s/cv .Eq.(20) is the main result of our Letter. where θG and θF are the contributions to the trace of energy– momentum tensor from the gauge field and the quarks corre- 4. Bulk viscosity from the lattice data spondingly defined in (10). Using (10) and (13a), (13b) we obtain 4.1. Spectral density d4x T θ(x), θ(0) In order to extract the bulk viscosity ζ from (20) we need to make an ansatz for the spectral density ρ.Athighfrequency,the ∂ ∂ spectral density should be described by perturbation theory; how- = T − 4 θG T + 2 T − 3 θF T ∂ T ∂ T ever, the perturbative (divergent) contribution has been subtracted in the definition of the quantities on the r.h.s. of the sum rule 4 + d x T θF (x), θF (0) (19), and so we should not include the high frequency perturba- ∼ 2 4 tive continuum ρ(u) αs u on the l.h.s. as well. Indeed, when ∂ ∂ the frequency ω is much larger than the temperature of the sys- ≈ T − 4 θT + T − 2 θF T , (14) ∂ T ∂ T tem T , ω T the spectral density ρ(ω) should be temperature- independent. Therefore the subtraction of the expectation value where in the last line we neglected the T θ (x), θ (0) correla- F F of the trace of the energy–momentum tensor performed on the tion function which is proportional to m2. We will see in the next lattice should remove the high frequency perturbative part of the section that the quark mass m can indeed be treated as a small spectral density. parameter.1 Note that the subtraction of the zero-temperature expectation To relate the thermal expectation value of θG T to the quantity 4 ∗ value θ0 (16) removes the ultra-violet (UV) singularity 1/a (a is (E − 3P) computed on the lattice, we should keep in mind that ∗ the lattice spacing) in (15), so that the resulting quantity (E − 3P) ∗ (E − 3P) =θT −θ0, (15) is UV finite. Such a subtraction implies the removal of local contact terms in the renormalized correlation function of the trace of the i.e., the zero-temperature expectation value of the trace of the energy–momentum tensor. energy–momentum tensor In the small frequency region, we will assume the following ansatz θ0 =−4| v | (16) 2 ρ(ω, 0) 9ζ ω0 has to be subtracted; it is related to the vacuum energy density = , (21) 2 + 2 ω π ω0 ω v < 0. Analogously, ∗ which satisfies (5) and (6). Substituting (21) in (19) we arrive at θF T =mqq¯ +mqq¯ 0. (17) 1 ∂ ∗ 9ω ζ = Ts − 3 − 4(E − 3P) + T − 2 mqq¯ Since the lattice data we are going to use correspond to al- 0 2 cs ∂ T most physical quark masses, we can use the PCAC relations to + 16| |+6 M2 f 2 + M2 f 2 . (22) express the zero-temperature vacuum expectation value mqq¯ 0 v π π K K of the scalar quark operator through the pion and kaon masses The parameter ω0 = ω0(T ) is a scale at which the perturbation Mπ , M K and decay constants fπ , f K : theory becomes valid. On dimensional grounds, we expect it to be proportional to the temperature, ω ∼ T . We estimate it as the mqq¯ =−M2 f 2 − M2 f 2 (18) 0 0 π π K K scale at which the lattice calculations of the running coupling [18] Eq. (9) implies that the l.h.s. of (14) is just the constant G.Us- coincide with the perturbative expression at a given temperature ing (13a), (13b) and (15), (17), (18) in the r.h.s. of (14) we derive (see discussion below). the following sum rule We note that the expression for bulk viscosity (22) consists of a thermal part that can be determined through lattice calculations, ∞ ρ(u, 0) and a vacuum contribution, which we fixed using PCAC relations 2 du for the quark condensates and using the gluon condensate value u | |1/4 = 250 MeV. With this the vacuum part is given numerically 0 v by ∂ ∗ ∂ ∗ = T − 4 (E − 3P) +θ0 + T − 2 mqq¯ +θF 0 ∂ T ∂ T 3 4 4 16| v | 1 + · 1.6 (560 MeV) (3Tc ) . (23) 1 8 = Ts − 3 − 4(E − 3P) 2 E − 4 cs Using the lattice results for trace anomaly, ( 3P)/T ,the fermionic contribution to it as well as the square of the velocity of ∂ ∗ + T − 2 mqq¯ + 16| v |−6mqq¯ 0 (19) sound presented in Ref. [5] we can determine the bulk viscosity. To ∂ T be specific we use the data set obtained from simulations on lat- 1 ∂ ∗ tices with temporal extent N = 6 [5]. These are the lattice results = Ts − 3 − 4(E − 3P) + T − 2 mqq¯ τ 2 cs ∂ T closest to the continuum limit. Our results for the bulk viscosity in units of the entropy density are shown in Fig. 1. We note that + 16| |+6 M2 f 2 + M2 f 2 , (20) v π π K K the contribution from the fermion sector of the trace anomaly is at least a factor 3 smaller than the gluonic contribution. Once lattice calculations with physical quark masses are performed we expect 1 Note that the heavy quark terms decouple in the low-energy matrix elements of the energy–momentum tensor canceling against the heavy quark part of the beta- this contribution to become even less important. In Fig. 1(right) we function in the θG operator [8]. show results for ζ/s using 500 MeV ω0 1500 MeV. 220 F. Karsch et al. / Physics Letters B 663 (2008) 217–221

Fig. 1. Bulk viscosity in the high temperature phase of QCD versus temperature in units of the transition temperature. Part (a) of the figure shows various contributions to 9ω0(T )ζ/Ts given in (22). In part (b) of the figure we show the bulk viscosity in units of the entropy density for ω0 = 0.5, 1, 1.5 GeV (top to bottom) which reflects the uncertainty in the determination of this scale parameter.

4.2. Uncertainties of the method Table 1 Critical exponents of 3-d O (4) [21] using β and δ as input and Z(2) [22] symmetric Our analysis relies on lattice results for the equation of state. spin models Recent calculations with O(a2) improved actions yield quite con- Model α β γ δ sistent results [5,13], however it should be noted that they are still O (4) −0.21 0.38 1.47 4.82 not extrapolated to the continuum limit and the light quark masses Z(2) 0.11 0.33 1.24 4.79 used in the calculation are about a factor two larger than the physical values. Since the sum rule we use is exact (up to the terms Here b is an arbitrary scale factor. It is expected that the chiral quadratic in light quark masses), the main source of uncertainty phase transition in QCD can be described by an effective, three- is the ansatz for the spectral density. We have made the sim- dimensional theory for the chiral order parameter, which in the plest possible assumption about its shape consistent with general case of two-flavor QCD would amount to an O (4) symmetric spin physical requirements—in particular, the spectral function must be model. linear for small frequencies. We also expect that the spectral den- The scaling behavior of the specific heat is controlled by the sity entering our sum rule should vanish above a certain frequency critical exponents α = (2yt − 1)/yt . Exponents relevant for a dis- ω T at which the spectral density becomes perturbative and 0 cussion of other susceptibilities as well as the quark mass de- temperature-independent. pendence of these quantities are β = (1 − y )/yt , γ = yt /y and We have estimated this frequency using lattice results on the h h δ = yh/(1 − yh). Their numerical values for Z(2) and O (4) sym- temperature dependence of the running coupling [18,19]; ω is 0 metric spin models in three dimensions are given in Table 1. taken to be the inverse distance at which this running coupling be- Eq. (24) can be used to extract scaling laws for various quan- comes approximately temperature-independent and runs according tities, valid in the vicinity of the critical point. We are at present to the zero temperature β-functions. Choosing this scale involves only interested in the scaling behavior of the specific heat as func- some uncertainty. Moreover, we have to admit that there is a siz- tion of temperature for vanishing external field able uncertainty in our ansatz. A different functional form of the −α spectral density would change the numerical value of extracted cv (t) = ct + const, (25) bulk viscosity. Fortunately, the temperature dependence of bulk which is obtained from Eq. (24) after taking two derivatives with − viscosity is not sensitive to this uncertainty, and so our main con- respect to t and choosing b = t 1/yt ; a constant c can be both pos- clusion about the rapid growth of this quantity near the phase itive and negative. In the O (4) universality class the exponent α is transition is robust. Moreover, we note that the spectral density negative. The specific heat thus will not diverge but will only have extracted from the analyses of correlation functions on the lattice a cusp. From the relations given by Eq. (19) we conclude that sim- [3,20] is quite similar to our ansatz. ilarly the bulk viscosity will not diverge but will have a maximum at Tc and the velocity of sound will not vanish but will only attain 5. Universality a minimum.

5.1. The case of zero baryon density 5.2. Chiral critical point

Let us discuss here a generic second order phase transition as The situation may be different at the chiral critical point, i.e., it might occur in the chiral limit of QCD at fixed strange quark the second-order phase transition point that might exist in the mass and most likely does occur in two-flavor QCD. The critical QCD phase diagram at non-vanishing chemical potential [23].Ifit behavior of thermodynamic quantities is in general controlled by exists, this critical point belongs to the universality class of the 3-d two external parameters, the reduced temperature and the baryon Ising model, which has a positive specific heat exponent α.Thesit- chemical potential. uation here, however, is a bit more complicated as the energy-like In the vicinity of the critical point the behavior of bulk thermo- and magnetization-like directions of the effective Ising model do dynamic quantities is governed by thermal (yt ) and magnetic (yh) not coincide with the temperature and symmetry breaking (quark critical exponents, which characterize the scaling behavior of the mass) directions of QCD, nor is the latter controlled by the baryon singular part of the free energy density, chemical potential. Derivatives of the partition function with respect to temperature, which give the specific heat, thus will usually T −1 y y f (t, h) ≡− ln Z = b f b t t, b h h . (24) be related to mixed derivatives with respect to the energy-like V F. Karsch et al. / Physics Letters B 663 (2008) 217–221 221 and magnetization-like direction of the effective Ising-Hamiltonian tistical” because the hadronization pattern is unlikely to depend characterizing the universal behavior in the vicinity of the chiral upon the phase space distributions of “pre-existing” partons—the critical point. When approaching this critical point on a generic memory about these distributions is largely erased by the pro- trajectory typical in a heavy ion collision, say for fixed s/nB ,the duced entropy. One may speculate that the association of inherent critical behavior thus will not be controlled by α but by γ /βδ entropy with the hadronization process is analogous to the “black which will give the dominant singular behavior [23].Thisleadsto hole hadronization” scenario, in which confinement is associated −γ /βδ an even more rapid divergence of the specific heat, cv ∼ h with an event horizon for colored partons [25–27]. with γ /βδ 0.8 and where h = A|T − Tc|/Tc + B|μB − μc|/μc In the vicinity of the chiral critical point, the divergence of bulk parametrizes the distance to the critical point. viscosity should manifest itself in heavy ion collisions through the As should be clear from our discussion given in Section 4 this decrease of average transverse momentum of produced particles, also implies that the bulk viscosity will diverge at the critical point accompanied by an increase in total multiplicity. This is due to and the velocity of sound will vanish. Numerical evaluation of bulk both the increase in entropy associated with a large bulk viscosity, viscosity close to chiral critical point would require lattice data at and the associated quenching of the transverse hydrodynamical ex- finite μB . The sum rule in this case would also get modified due to pansion of the system (“radial flow”). the addition of terms containing derivatives in the chemical potential. We do not attempt such a numerical analysis in the present Acknowledgements Letter. However, the arguments for the divergence of bulk viscosity in the vicinity of chiral critical point which we presented above The work of F.K. and D.K. was supported by the US Department are sufficiently general. of Energy under Contract No. DE-AC02-98CH10886. K.T. was supported in part by the US Department of Energy under Grant No. 6. Summary and discussion DE-FG02-87ER40371; he would like to thank RIKEN, BNL, and the US Department of Energy (Contract No. DE-AC02-98CH10886) for Our results for the bulk viscosity obtained by combining low- providing facilities essential for the completion of this work. energy theorems with lattice results for QCD with a physical strange quark mass and almost physical light quark masses, indi- References cate that the behavior is qualitatively similar to the one observed previously for the case of SU(3) gluodynamics—the bulk viscosity [1] D. Kharzeev, K. Tuchin, arXiv: 0705.4280 [hep-ph]. [2] G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lutgemeier, B. Peters- rises dramatically in the vicinity of the phase transition. There- son, Nucl. Phys. B 469 (1996) 419, hep-lat/9602007. fore the increase of bulk viscosity near the phase transition is a [3] H.B. Meyer, arXiv: 0710.3717 [hep-lat]. sufficiently general phenomenon and is not specific only to the [4] S.S.A. 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