Critical Exponents and Transport Properties Near the QCD Critical Endpoint from the Statistical Bootstrap Model

Critical exponents and transport properties near the QCD critical endpoint from the statistical bootstrap model

Guruprasad Kadam∗

Department of Physics, Shivaji University, Kolhapur, Maharashtra - 416 004, India

Hiranmaya Mishra†

Theory Division, Physical Research Laboratory, Navarangpura, Ahmedabad - 380 009, India

Marco Panero‡

Department of Physics, University of Turin and INFN, Turin, Via Pietro Giuria 1, I-10125 Turin, Italy (Dated: November 5, 2020) We present an estimate of the behaviour of the shear and bulk viscosity coefficients when the QCD critical point is approached from the hadronic side, describing hadronic matter within the statistical bootstrap model (SBM) of strong interactions. The boostrap model shows critical behavior near the quark-hadron transition temperature if the parameter characterizing the degeneracy of Hagedorn states is properly chosen. We calculate the critical exponents and amplitudes of relevant thermodynamic quantities near the QCD critical point and combine them with an Ansatz for the shear and bulk viscosity coefficients to derive the behavior of these coefficients near the critical point. The shear viscosity to entropy density ratio is found to decrease when the temperature is increased, and to approach the Kovtun- Son-Starinets bound 1/(4π) faster near the critical point, while the bulk viscosity coefficient is found to rise very rapidly.

PACS numbers: 12.38.Mh, 12.39.-x, 11.30.Rd, 11.30.Er arXiv:2011.02171v1 [hep-ph] 4 Nov 2020

∗ [email protected] † [email protected] ‡ [email protected] 2

I. INTRODUCTION

The phase diagram of strongly interacting matter has attracted a lot of attention in the past couple of decades [1–3]. At low temperature T and low baryon chemical potential µB, quantum chromodynamics (QCD) matter consists of colorless hadrons, while at high temperature and baryon density the degrees of freedom are colored quarks and gluons. Lattice QCD simulations indicate that, at µB ∼ 0, the transition from the hadronic phase to the quark-gluon plasma phase (QGP) is actually an analytic cross over [4–9]. At finite baryon densities µB/T & 1, however, reliable lattice QCD results are not available, since these calculations are affected by the sign problem [10–13]. In this regime of the phase diagram one has to resort to effective models of QCD, such as the Nambu- Jona-Lasinio model [14, 15], the quark-meson-coupling model [16] etc. Finally, at low temperature and sufficiently large baryon density many phenomenological models predict the transition between the hadronic phase to the deconfined phase to be of first order [17–25]: see also the discussion in ref. [26]. Apart from the effective QCD models mentioned above, a simple model to describe the hadronic phase of QCD is the hadron resonance gas model (HRG). This model is based on the S-matrix formulation of statistical mechanics [27]. At low density, as it turns out, the thermodynamics of can be approximately modeled in terms of a non-interacting gas of hadrons and resonances [28– 30]. The predictions of this model have been compared with lattice QCD simulations, finding good agreement for temperatures up to T ∼ 150 MeV except for some discrepancies in the trace anomaly [31–33]. Later studies found that the agreement can be improved, if the continuum density of states is included in the mass spectrum of the HRG [34–37]. Remarkably, analogous results have been obtained also in lattice simulations of SU(N) Yang-Mills theories without dynamical quarks [38, 39], and even in three spacetime dimensions [40]. A description of the density of hadron states in terms of a continuous distribution is the basis of the statistical bootstrap model (SBM) [41, 42], which attracted a lot of attention in the particle physics community in the pre-QCD era. The mass spectrum of abundant formation of heavy resonances and higher angular momentum states can be consistently described by a self-similar structure of hadrons through the bootstrap condition. These high-mass resonances have an interesting effect on the strong interaction thermodynamics: in the thermodynamic system dominated by exponentially rising resonance states there is a finite limiting temperature TH, called Hagedorn temperature. The existence of this limiting temperature indicates that the hadron resonance gas cannot exist at physical temperatures T > TH, and suggests that strongly interacting matter should then enter a different phase. The bootstrap condition of the SBM requires the density of a states to be of the form ρ(m) ∼ m exp(m/TH)[43–45], where a is a constant. Interestingly, the string model (or dual resonance model) of strong interactions [46] also predicts this type of density of states. The a constant plays an important role in determining the thermodynamics of the SBM near the Hagedorn temperature. In fact, for the choice a = −4 both the energy density and the entropy density remain finite near TH and one expects a phase transition to take place [23, 43, 47], so that TH can be interpreted as a critical temperature, Tc. A particularly interesting point in the QCD phase diagram is the conjectured critical end point (CEP). It should be remarked that, so far, the existence of the CEP has neither been proven theoretically, nor has it been observed experimentally. However, its existence is strongly suggested by the aforementioned model calculations investigating the phase diagram region at low temperatures and baryon densities larger than that of nuclear matter, which predict a first-order transition line separating the hadronic phase from a deconfined phase: since that line is known not to extend all the way to the µB = 0 axis (where the transition is actually a crossover), it should end at a CEP, 3 where the transition should be a continuous one [48]. A lot of theoretical investigation has been carried out, and is still going on, to locate the CEP and predict possible experimental signatures, see refs. [49–52] for reviews. On the experimental side, an entire experimental program, namely the Beam Energy Scan (BES) program, has been devoted at the Super Proton Synchrotron (SPS) and at the Relativistic Heavy Ion Collider (RHIC) to search for the CEP [53, 54]. In particular, as suggested in ref. [48], the existence and the location of the CEP could be revealed by the obser- vation of a suppression in temperature and chemical potential fluctuations on an event-by-event basis, and by large fluctuations in the multiplicity of low-energy pions. A very important feature of the critical point is the emergence of a universal critical mode. As the system approaches the critical point, this mode rises very rapidly with some power of the correlation length ξ, which eventually diverges at the critical point. For instance, the variance, skewness and kurtosis of the non-Gaussian fluctuation of the critical mode grow as ξ2, ξ9/2 and ξ7, respectively [55–57]. In the experimental search of the critical endpoint, these critical fluctuations can be accessed by measuring event-by-event fluctuations of particle multiplicities [53]. While “static” critical phenomena have been extensively studied theoretically, an avenue that has been been explored less is the one of “dynamical” critical phenomena. As it turns out, critical singularities can also occur in quantities encoding the dynamical properties of the medium, like transport coefficients. Away from the critical point, the dynamic properties of a system can be characterized by hydrodynamics, which provides an effective description of the fluid in the low- frequency, long-wavelength limit. Hydrodynamics describes fluctuations of conserved quantities at equilibrium and any additional slow variable that occurs due to the existence of a spontaneously broken symmetry. In the hydrodynamic effective theory the dynamical critical fluctuations are described by coupling the order-parameter field with the conserved momentum density. In this model, which is called the H model in the classification of dynamical critical phenomena [58] by Hohenberg and Halperin, the transport coefficients depend on the correlation length as

1 3 η ∼ ξ 19 , κ ∼ ξ, D ∼ , ζ ∼ ξ . (1) B ξ This behavior suggests that the transport coeffients would affect the bulk hydrodynamic evolution of the matter created in heavy-ion collisions near the QCD critical point [59–66]. It is worth empasizing that, while lattice calculations remain the tool of choice for theoretical first-principle studies of various quantities relevant for strong-interaction matter, their applicabil- ity in studies of transport coefficients in the proximity of the QCD critical endpoint is severely limited, for a two-fold reason. One one hand, as we remarked above, the existence of the sign problem poses a formidable barrier to lattice simulations at finite baryon-number density—one, which might even be impossible to overcome with classical computers, if it is related to fundamental computational-complexity issues [67]. On the other hand, even at zero baryon-number density, the lattice determination of transport coefficients of QCD matter involves its own difficulties, due to the fact that lattice QCD calculations are done in a Euclidean spacetime, and typically the extraction of quantities involved in real-time dynamics requires a Wick rotation back to Minkowski signature, with the reconstruction of a continuous spectral function from a finite set of Euclidean data, which is an ill-posed numerical problem [68]. Despite some recent progress (see, e.g., refs. [69–74]), a general solution to this type of problems is still unknown. For these reasons, phenomenological models remain a useful theoretical tool to get some insight into the physics near the QCD critical endpoint. In this work, we extract critical exponents [43] and amplitudes of thermodynamic quantities relevant near the critical point within the statistical bootstrap model. We then derive the singularities characterizing shear and bulk viscosity coeffi- 4 cients, starting from an Ansatz for viscosity coefficients [75] that is suitable for a hydrodynamic system with conserved baryon charge. We then estimate viscosity coefficients near the critical point from the hadronic side using the critical exponents of this model. We organize the paper as follows. In sectionII we extract critical exponents and amplitudes of thermodynamic quantities relevant near the critical point within the statistical bootstrap model. In sectionIII we derive the singuliarities of shear and bulk viscosity near the critical point. In sectionIV we discuss our results. Finally, in sectionV we summarize our findings and conclude.

Throughout the paper, we work in natural units (~ = c = kB = 1).

II. STATISTICAL BOOTSTRAP MODEL: CRITICALITY AND CRITICAL POINT EXPONENTS

A. Critical exponents

The analysis of critical phenomena is based on the assumption that, in the T → Tc limit, any relevant thermodynamic quantity can be separated into a regular part and a singular part. The singular part may be divergent or it can have a divergent derivative. It is further assumed that the singular part of all the relevant thermodynamic quantities is proportional to some power of t, where t = (T − Tc)/Tc. These powers, called critical exponents, characterize the nature of singularity at the critical point. The critical exponents, α, β, γ and ν are deﬁned through following power laws [76] (in the limit t → 0−):

−α CV = C− |t| (2) nB β 1 − = N− |t| (3) nB,c −γ kT = K− |t| (4) −ν ξ = Ξ− |t| (5) where CV , nB,c, kT and ξ respectively denote the speciﬁc heat, the critical baryon density, the isothermal compressibility and the correlation length , while C−, N−, K− and Ξ− are the corresponding amplitudes from the hadronic side (T < Tc). Note that the second line in the equations above is an equation of state, relating baryon density nB and pressure p near the critical point.

B. Hagedorn resonance gas model

We follow Ref. [43] to extract the amplitudes and critical exponents within the statistical bootstrap model. Consider an ideal gas of pions and all possible resonance states as non-interacting constituents. The partition function of this system can then be written as [77]

N N Z 3 X (V zB) Y d pi Z(T, V, z ) = dm ρ (m )e−Ei/T (6) B N! (2π)3 i i i N 1

µ /T where zB = e B is the fugacity and ρ(m) is the hadron spectrum included in the HRG model. It is customary to include all the hadrons and resonances that have been detected experimentally P and take ρ(m) = j δ(m − mj) with an upper cut-oﬀ Λ. Such discrete mass spectrum leads to the physical hadron resonance gas model. In the physical HRG, if gi is the degeneracy of the 5 i-th hadronic species, then for spin degrees of freedom the degeneracy factors turns out to be 2 gi ∼ mi [47]. Thus, one sees that the spin multiplicity already can result in an unbounded increase in resonances. The upshot of the m2 dependence of resonance degeneracy is that the partition function of the physical resonance gas and all of its higher-order derivatives remain ﬁnite at Tc. Thus, the required degeneracy structure is absent in the physical resonance gas and hence it does not show critical behavior. It turns out that the degeneracy structure required to show critical behavior is present in the Hagedorn density of states. Consider a density of states of the form

ρ(m) = gπ · δ(m − mπ) + ρH(m) (7) where gπ (which we take to be equal to its physical value 3) is the pion degeneracy, mπ is the pion mass and ρH is the continuum density of states. The logarithm of the partition function (6) with the density of states in eq. (7) is written as

ln Z(T, V, zB) = ln Zπ(T,V ) + ln ZH(T, V, zB) (8) where " !# Z d3p pp2 + m2 ln Z (T,V ) = −g ln 1 − exp π (9) π π (2π)3 T and

ln ZH(T, V, zB) = V zBΦB(T ) (10) with Z ∞ Z d3p −Ei/T ΦB(T ) = dmρH(m) 3 e (11) M (2π) where M = 2mπ is the two-pion threshold. Performing the momentum integration, one obtains Z ∞ T 2 ΦB(T ) = 2 dm m ρH(m)K2(m/T ) (12) 2π M where Kn(z) is a modiﬁed Bessel function of the second kind of order n. For large real values of its argument, one has r π K (z) ' e−z 1 + O(z−1) . (13) n 2z Hence, when m/T 1 one gets

3/2 Z ∞ T 3/2 −m/T ΦB(T ) = dm m ρH(m)e . (14) 2π M All the thermodynamic functions can be readily obtained from the partition function in eq. (10) once the mass spectrum ρ(m) is speciﬁed. In the statistical bootstrap model (see refs. [78, 79] for reviews), hadronic matter at high temperature is dominated by formation of resonances whose number grows exponentially. Self-consistency requires a mass spectrum of the form [41, 42]

a bm ρH(m) = A m e (15) 6 where A, a, and b are constant parameters. In particular, the parameter A provides the normal- ization of the resonance contributions relative to that of the pions. The parameter a speciﬁes the nature of the degeneracy of high-mass resonances, and also determines the critical behavior of hadronic matter. One possible solution of the bootstrap condition was derived in ref. [80], yielding a ' 3. Finally, the parameter b turns out to be the inverse of the Hagedorn temperature at which thermodynamic functions show singular behavior. The partition function with the continuous spectrum (15) reads

3/2 Z ∞ T a+3/2 −(b−1/T )m ln ZH(T, V, zB) = AV zB dm m e (16) 2π M In the next section we shall calculate the critical exponents by taking appropriate derivatives of the partition function (16) and then taking the T → 1/b limit.

C. Critical exponents in the statistical bootstrap model

With the change of variable w = m(1/T − b), we get

3/2 Z ∞ T −(a+5/2) a+3/2 −w ln ZH(T, V, zB) = AV zB (1/T − b) dw w e 2π M(1/T −b) T 3/2 5 = AV z (1/T − b)−(a+5/2)Γ a + ,M(1/T − b) , (17) B 2π 2 having expressed the integral in terms of the upper incomplete Γ function. The energy density can then be written as T 2 ∂ ln Z = H (18) V ∂T and for T → 1/b one ﬁnds that

 Az 7 −(a+ 7 ) B Γ a + (1/T − b) 2 , for a > −7/2  (2πb)3/2 2 AzB ' − 3/2 ln[M(1/T − b)], for a = −7/2 . (19)  (2πb)  constant, for a < −7/2

Hence for a < −7/2 the energy density remains ﬁnite (and approaches some critical value c) as

T → TH, implying that the system cannot exist in this state for > c and suggesting that a phase transition must take place. The speciﬁc heat at constant volume can then be written as

2 T 2 ∂2 C = + ln Z (20) V T V ∂T 2 H and for T → 1/b one gets

2  Ab zB Γ a + 9 (1/T − b)−(a+9/2), for a > −9/2  (2πb)3/2 2  2 C ' Ab zB . (21) V − 3/2 ln[M(1/T − b)], for a = −9/2  (2πb)  constant, for a < −9/2 7

Comparing eq. (21) with eq. (2) we deduce the amplitude C− as

2 ( Ab zB Γ a + 9 , for a −9/2 (2πb)3/2 2 > C− = 2 (22) Ab zB , for a < −9/2 (2πb)3/2 while the critical exponent α reads a + 9 , for a −9/2 α = 2 > . (23) 0, for a < −9/2

The baryon number density nB can be evaluated as 3/2 zB ∂ T −(a+5/2) 5 nB = ln ZH = AzB (1/T − b) Γ a + ,M(1/T − b) , (24) V ∂zB 2π 2 hence for T close to 1/b we get the critical density as

 AzB Γ a + 5 (1/T − b)−(a+5/2), for a > −5/2  (2πb)3/2 2 n ' − AzB ln[M(1/T − b)], for a = −5/2 . (25) B,c (2πb)3/2  constant, for a < −5/2 The inverse of the isothermal compressibility is deﬁned as −1 ∂p kT = −V (26) ∂V T and for a non-interacting resonance gas it takes the following, very simple form:

−1 kT = nBT. (27) For temperatures close to 1/b, one obtains,

 AzB Γ a + 5 (1/T − b)−(a+5/2), for a > −5/2  b(2πb)3/2 2 k−1 ' − AzB ln[M(1/T − b)], for a = −5/2 , (28) T b(2πb)3/2  constant, for a < −5/2 from which it is straightforward to deduce the amplitude K−  −1  AzB Γ a + 5 , for a > −5/2  b(2πb)3/2 2 K− = −1 . (29)  AzB , for a < −5/2  b(2πb)3/2 and the critical exponent γ as − a + 5 ; for a > −5/2 γ = 2 . (30) 0; for a < −5/2 We note that a continuous density of states with a < −7/2 makes the energy and entropy densities finite , while all higher-order derivatives diverge near TH. In our analysis of transport coefficients we shall consider the a = −4 case [23, 47, 81] which leads to normal behaviour of the hadronic system near the boundaries of the quark-hadron phase transition line, since it does not allow the energy density to become infinite even for pointlike particles. 8

III. TRANSPORT COEFFICIENTS NEAR THE CRITICAL POINT

Approaching the critical point, the thermodynamic quantities relevant for the computation of transport coefficients are: energy density (), baryon number density (nB), specific heats (CV 2 and Cp), isothermal compressibility (kT ), speed of sound (Cs ) and correlation length (ξ). A set of Ansätze for the transport coefficients near the critical point can be written in terms of thermodynamic quantities as [75] η T Cp = 2 Fη , s Csξ s CV ζ hCsξ Cp = Fζ . (31) s s CV Near the critical point, the correlation length ξ is the only relevant length scale. Further, longitudi- nal perturbations can be assumed to be those of the non-equilibrium modes near Tc. A particularly simple form of the functions Fη,ζ , namely Fη(Cp/CV ) = fη × (Cp/CV ) and Fζ = fζ × (Cp/CV ), can be obtained from a perturbative treatment of conventional fluids. Here, fη and fζ are non-universal dimensionless constants and depend on the microscopic length scale of the system. Substituting the singular part of the thermodynamic quantities from eqs. (2) into eqs. (31) we get, as t → 0−

(i.e T → Tc from the hadronic side) s 3 γ−α −1 η fηK−λc Tc hc C−|t| −γ+2ν+α/2 = 2 1 + 2 |t| (32) s − ξ−sc C− Tcλc K− s 3 3 γ−α −3 ζ fζ K−ξ−λc Tc hc C−|t| −γ−ν+3α/2 = 3 1 + 2 |t| (33) s − sc C− Tcλc K−

Here, hc and sc respectively denote the enthalpy and entropy densities at Tc, both of which are

finite when one sets a = −4 in the Hagedorn density of states, while λc = (∂p/∂T )V at T = Tc. Finally, we leave fη and fζ as free parameters. Alternatively, they could be fixed by imposing some constraint on the viscosity coefficients near Tc. For instance, gauge-gravity duality principle constrain (rather a lower bound) the ratio η/s to the value 1/(4π). Similar constrain can be imposed on ζ/s as well.

IV. RESULTS AND DISCUSSION

3 ρ Tc(GeV) A1(GeV ) A2 m0(GeV) 3 ρ1 0.160 15 Tc - 0.5 3/2 ρ2 0.180 - 0.63 (GeV ) -

TABLE I. Parameters of continuum density of states [35, 47].

Before discussing the behavior of viscosity coeﬃcients near Tc, it is instructive to point out a few remarks about the thermodynamics of the Hagedorn resonance gas model. Fig.1 shows the equilibrium thermodynamic quantities, namely pressure (p), energy density () and entropy density (s) in units of T 4 (for p and ) and T 3 (for s). We have also plotted the trace of the 9

2 6

- 4 bootstrap model with density of states ρ (m) = A m exp (bm) 5.5 1.8 1 1 bootstrap model with ρ1 density of states 2 2 - 5/4 bootstrap model with ρ density of states bootstrap model with density of states ρ2 (m) = A2 (m +m0 ) exp (bm) 5 2 1.6 pion-gas contribution pion-gas contribution 4.5 1.4 4

1.2 3.5 4 4 T T 1 3 / / ε p 0.8 2.5 2 0.6 1.5 0.4 1

0.2 0.5

0 0 50 75 100 125 150 175 50 75 100 125 150 175 T [MeV] T [MeV] 4 4

bootstrap model with ρ density of states bootstrap model with ρ density of states 3.5 1 3.5 1 bootstrap model with ρ2 density of states bootstrap model with ρ2 density of states pion-gas contribution pion-gas contribution 3 3

2.5 2.5 4 3 T 2 T 2 / / s ∆

1.5 1.5

1 1

0.5 0.5

0 0 50 75 100 125 150 175 50 75 100 125 150 175 T [MeV] T [MeV]

FIG. 1. Equilibrium thermodynamics quantities for diﬀerent types of continuous resonance spectrum distributions, as a function of the temperature T , in MeV. The solid blue curves correspond to ρ1(m) = −4 2 2 −5/4 A1 m exp(bm), whereas the dashed red curves are obtained for ρ2(m) = A2(m + m0) exp(bm). In addition, we also plot the curves representing the contribution due to an ideal pion gas (dotted green curves).

energy-momentum tensor in units of the fourth power of the temperature, ( − 3p)/T 4. The blue −4 curve corresponds to a continuous density of states of the form ρ1(m) = A1 m exp(bm), whereas 2 2 −5/4 the red one is obtained for a spectral density ρ2(m) = A2(m + m0) exp(bm). Finally, the green curve shows the contribution from the ideal pion gas. We note that the resonances become significant around T ' 80 MeV, leading to a rapid increase in the energy density, as shown in fig.1. Furthermore, in the case of ρ1 both the energy and the entropy densities remain finite for

T ≤ Tc. This reflects the fact that, for a = −4, the second derivative of the partition function is 3 4 divergent, but the first is not. In fact, setting A1 = 15Tc corresponds to c/Tc ' 4 [47]. In the case of ρ2, both and s diverge at T = Tc: in this case, the model predicts that reaching the limiting Hagedorn temperature requires an infinite energy density. Fig.2 shows the predictions for the shear and bulk viscosities near the QCD critical point based on eqs. (32) and (33). We take the correlation length amplitude to be ξ− = 1 fm and the estimate for the critical point location to be (Tc, µc) = (160 MeV, 220 MeV) [82]. For a = −4 one can easily derive the critical exponents and amplitudes needed for the estimate of the viscosity coefficients near Tc. The critical exponents are not independent but are constrained by scaling laws. In particular, the exponents α and ν are related by the Josephson scaling law νd = 2 − α, 10

4 3.5

bootstrap model with fη=0.5 bootstrap model with fζ=0.85 3.5 1 / (4π), from P. Kovtun et al., Phys. Rev. Lett. 94 (2005) 111601 3 A. Monnai et al., Phys. Rev. C 95 (2017) 034902 G. S. Denicol et al., Phys. Rev. C 88 (2013) 064901 N. G. Antoniou et al., Phys. Rev. C 96 (2017) 055207 3 D. Fernandez-Fraile et al., Eur. Phys. J. C 62 (2009) 37 N. G. Antoniou et al., Phys. Rev. C 96 (2017) 055207, with fη=0.0325 2.5 N. G. Antoniou et al., Phys. Rev. C 96 (2017) 055207, with f =0.0082 2.5 η 2 s s / 2 / ζ η 1.5 1.5

1 1

0.5 0.5

0 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -2 -1.5 -1 -0.5 0 0.5 ( T - Tc ) / Tc ( T - Tc ) / Tc

FIG. 2. The left-hand-side panel shows a comparison of the shear viscosity to entropy density ratio with the predictions from other models [75, 83–85], whereas the right-hand-side panel shows the predictions of the statistical bootstrap model for the bulk viscosity to entropy density ratio in comparison with other works [59, 60, 75]. Our results correspond to fη = 0.5 and fζ = 0.85.

where d is the number of space dimensions [76]. In the left-hand side panel of fig.2, the solid blue curve shows the shear viscosity to entropy density ratio within the statistical bootstrap model, with density of states specified by ρ1. In the figure, we also compare the critical solution for η/s with those obtained from various other models: in particular, the. dashed black curve corresponds to the conjectured universal lower bound 1/(4π) for this ratio, that was derived in ref. [83], while the brown curve describes the result that one would obtain for a pion gas [84], and the magenta curve shows the result that can be derived assuming the medium to be described in terms of a hadronic mixture [85] at low temperature and density. Finally, the red curve corresponds to the same solution for the viscosity coefficients as in this work, but with the critical exponents of three- dimensional Ising model and the amplitudes constrained by universality arguments [75]. We note that the critical behavior of the statistical bootstrap model leads to a linear decrease in η/s as a function of the temperature, and that at low temperatures the estimated magnitude of η/s is in agreement with that of a pion gas, or of the hadron gas mixture. Near Tc there is a mild violation of the bound conjectured in ref. [83]. Such violation has also been noted for one of the solutions discussed in ref. [75], shown by the dashed red curve in fig.2.

The right-hand side panel of fig.2 shows the bulk viscosity to entropy density ratio, in which one notes that the statistical bootstrap model predicts a rapid increase in the bulk viscosity as a function of the temperature. In fact, near Tc, our results are in remarkable agreement with those from ref. [59, 60] (shown by the dashed green curve), where the bulk viscosity has been estimated under the assumption of a QCD critical point belonging to the dynamical universality class of the so-called H model [58]. A large bulk viscosity should manifest itself in heavy-ion collisions through the decrease of the average transverse momentum of final-state hadrons. Moreover, due to the increase in entropy associated with the dissipation through large bulk viscosity, this effect should be accompanied by an increase in total multiplicity for final-state hadrons. The large bulk viscosity near the critical point would play a particularly important role in the elliptic flow measurement of the matter produced in the BES program. 11

V. DISCUSSION AND CONCLUSIONS

A. Discussion

In this work we derived the predictions of the statistical bootstrap model for thermodynamic quantities and transport coefficients near the critical endpoint of QCD. As we have seen, a thermodynamic system with exponentially rising density of states shows critical behavior which might capture the correct physics of the critical endpoint of QCD as T → Tc and that, for a density of a the form ρ(m) ∼ m exp (m/Tc) with a < −7/2, the energy and entropy densities remain finite even for point-like hadrons, while all higher-order derivatives diverge near Tc. For a = −4 the energy density remains finite as T → Tc, signalling the existence of a phase transition. In passing, it is worth mentioning that a continuous spectral density of the form required for self-consistency in the statistical bootstrap model, eq. (15), also arises if one models hadrons in terms of confining strings of glue (as was done, for example, in ref. [86]). Next, we studied the critical behavior of shear and bulk viscosities within the statistical bootstrap model. Identifying the thermodynamic quantities whose singular parts would contribute to the viscosity coefficients it is possible to write down Ansätze for the viscosity coefficients valid near the critical point. Using the Ansätze in eq. (31) together with the singular part of relevant thermodynamic quantities in eq. (2), one can obtain the dominating contributions for the viscosity coefficients in eqs. (32) and (33). We found that the statistical bootstrap model predicts the shear viscosity to entropy density ratio η/s to decrease quite rapidly near Tc. We observe that the magnitude of η/s away from the critical point is in good agreement with the predictions of non-critical models, and that there is a mild violation of the η/s ≥ 1/(4π) bound [83] near Tc. It is worth emphasizing that this (slight) violation of the η/s ≥ 1/(4π) bound might be unphysical, i.e. an artifact of the model. In fact, it is also worth remarking that, while the conjecture of the η/s ≥ 1/(4π) bound was first derived in a holographic context [83] (and is expected to be saturated in strongly coupled gauge theories with a known gravity dual [87], such as the N = 4 supersymmetric Yang-Mills theory [88]), its origin is, in fact, much more general, being related to the uncertainty principle of quantum systems. As for the ζ/s ratio, we found it to rise very rapidly rear Tc, in remarkable agreement with refs. [59, 60]. The anomalous behavior of shear and bulk viscosity coefficients near the critical endpoint might be very important for heavy-ion collision experiments. In particular, an enhanced bulk viscosity should manifest itself in heavy ion collisions through a decrease of the average transverse momentum of final state hadrons, and a corresponding increase in entropy. This feature may be particularly important for the experimental search for the critical endpoint of QCD through the BES program. It is interesting to discuss a comparison of our findings with other related works. In particular, a study similar to ours was recently reported in ref. [89], which also predicts the transport properties of hot QCD matter within a hadron resonance model, finding a very low shear viscosity to entropy density ratio near TH, in agreement with our results. In contrast to our work, however, the focus of that article is not on the behavior near criticality (where, as we have discussed in detail above, much information can be derived with purely analytical calculations and general universality arguments), but instead on their numerical determination in a wider range, using the Gießen Boltzmann- Uehling-Uhlenbeck transport model [90] and Monte Carlo calculations. The fact that the numerical approach used in ref. [89] reproduces our analytical results close to criticality is a non-trivial cross- check of the results. We should emphasize again that there is no fundamental proof that the statistical bootstrap model described in this work should necessarily provide a complete description of the fundamental 12 properties at the CEP. Based on very general arguments (including the continuous nature of the transition at the critical endpoint, spacetime dimensionality, and the underlying symmetries— or lack thereof—of the thery), one may instead argue that the universality class of the critical endpoint of QCD should instead be the one of the Ising model in three dimensions (for a review, see ref. [91]). The critical exponents in this model have recently been computed to very high precisions using conformal bootstrap techniques [92, 93], finding α = 0.11008(1), γ = 1.237075(10), etc., which are clearly incompatible with those predicted by the model that we considered here. If the critical endpoint of QCD exists, it may well be that its actual critical exponents are those of the three-dimensional Ising model, and that deviations from the description in terms of the statistical bootstrap model start to occur when one approaches the CEP. In this respect, it would be interesting to study theoretically how these deviations start to manifest themselves when the system is off, but close to, the critical point—perhaps using the analytical tools of conformal perturbation theory [94] (see also ref. [95], for an explicit example of application), as recently proposed in ref. [96]. An interesting issue associated with the description of the critical endpoint of QCD in terms of the three-dimensional Ising model concerns the identification of the lines, in the QCD phase diagram, that describe relevant deformations of the model: what are the directions that correspond to a “thermal” and to a “magnetic” perturbation of the critical point? How do they affect the reliability of the description of the thermodynamics of the hadronic phase in terms of a hadron resonance gas with a spectrum of the form in eq. (7)? The answers to these questions may have important phenomenological implications for the evolution of the medium in energy scans going through or close to the critical endpoint, since they could directly affect the dynamics of hadrons before freeze-out. Finally, it should be noted that, by construction, the statistical bootstrap model does not allow one to describe the approach to the critical endpoint of QCD from “above”, i.e. from the deconfined phase. Perturbative computations show that the η/s ratio is generally large for a weakly coupled quark-gluon plasma [97, 98] (a seemingly counter-intuitive result, which, in fact, reflects the fact that suppression of interactions makes “transverse” propagation of momentum difficult), but it is well known that thermal weak-coupling expansions are affected by non-trivial divergences, which are not present at T = 0 (see, for example, ref. [99] and references therein), and require a sophisticated treatment [100–103].

B. Conclusions

To summarize, in this work we derived the theoretical predictions of the statistical bootstrap model in the vicinity of the critical endpoint of QCD. We showed that, for a suitable choice of its parameters, this model “predicts” the existence of a phase transition, and allows one to derive the associated critical exponents in an analytical way. Moreover, the model also gives predictions for the transport properties near the CEP, which are encoded in the shear and bulk viscosities. In spite of the relative simplicity of the model, these results are qualitatively and quantitatively very similar to those obtained from other models. These ﬁndings may hopefully guide the future experimental identiﬁcation of the CEP in heavy-ion collision experiments and the determination of the physical properties of QCD matter in the proximity of the critical endpoint. 13

ACKNOWLEDGMENTS

Guruprasad Kadam is ﬁnancially supported by the DST-INSPIRE faculty award under Grant No. DST/INSPIRE/04/2017/002293. G.K. also thanks Sangyong Jeon for useful comments.

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