Effect of bulk viscosity on interferometry correlations in ultrarelativistic heavy-ion collisions

TL;DR

This study investigates how temperature-dependent bulk viscosity influences the evolution and interferometry measurements in ultrarelativistic heavy-ion collisions, revealing subtle effects on system lifetime and correlation radii.

Contribution

It introduces a detailed 3+1D hydrodynamic model incorporating temperature-dependent bulk viscosity to analyze its impact on collision dynamics and interferometry observables.

Findings

Increased bulk viscosity slightly extends system lifetime.

Shape of freeze-out hypersurface is altered, with outer layers lasting longer.

Reduction in R_out/R_side ratio improves agreement with experimental data.

Abstract

A temperature dependent bulk viscosity coefficient is used in 3+1D hydrodynamic simulations. I study the effect of the increase of bulk viscosity around the critical temperature on the system dynamics in central Pb+Pb collisions at $\sqrt{s_{NN}}=2760$~GeV. With increasing bulk viscosity the life-time of the system increases slightly. Also the shape of the freeze-out hypersurface changes, the outer layers of the fireball live longer. This effect causes a small reduction of the ratio of two interferometry radii $R_{out}/R_{side}$, improving the agreement with experimental data.

Equations

$$\zeta /s=\frac{{\zeta }_H}{s}\frac{1}{1+\exp \left(\left(T-T_H\right)/\Delta T_1\right)}+c\frac{{\zeta }_{peak}}{s}\left(T\right)$$

$$\frac{{\zeta }_{peak}}{s}\left(T\right)=\{\begin{array}{ll}{l}_1\exp \left(\frac{T/T_H-1}{s_1}\right)+l_2\exp \left(\frac{T/T_H-1}{s_2}\right)& \\ \text{for}T<T_a& \\ A_0+A_1\frac{T}{T_H}+A_2{\left(\frac{T}{T_H}\right)}^2& \\ \text{for}{T}_a<T<T_b& \\ l_3\exp \left(\frac{1-T/T_H}{s_3}\right)+l_4\exp \left(\frac{1-T/T_H}{s_4}\right)& \\ \text{for}T>T_b& \end{array}$$

$$C(q)=1+\lambda \exp \left(-R_{out}^2{q}_{out}^2-R_{side}^2{q}_{side}^2-R_{long}^2{q}_{long}^2\right).$$

$$s(x,y)\propto (1-\alpha ){\rho }_W(x,,y)+2\alpha {\rho }_{bin}(x,y),$$

Full text

relativistic heavy-ion collisions, interferometry correlations, hydrodynamic expansion, bulk viscosity

I Introduction

The interferometry correlations are a sensitive measure of the space-time evolution of the system formed in heavy-ion collisions. The interferometry, or Handburry-Brown Twiss (HBT) radii, extracted from the same-sign $\pi$-$\pi$ correlation function give an estimate of the size of the region where the pion pair is emitted Lisa:2005dd ; Wiedemann:1999qn . From the studies of Au-Au collisions at BNL Relativistic Heavy Ion Collider energies it has been found that hydrodynamic simulations describe fairly well the experimental data on HBT correlations Broniowski:2008vp ; Pratt:2008qv . Similar agreement is seen for Pb+Pb collisions at the CERN Large Hadron Collider.

From the analysis of the three dimensional $\pi$-$\pi$ correlation function three radii parameters can be extracted Bertsch:1989vn ; Pratt:1986cc , $R_{out}$, $R_{side}$, and $R_{long}$. A key assumption required to reach agreement with the data is the use of a hard equation of state for the matter created in the collision. A softening of the equation of state leads to an increase of the $R_{out}/R_{side}$ ratio Rischke:1996em ; Bozek:2009ty , that would spoil the agreement with the experimental data. The measurement of the $R_{out}/R_{side}$ value close to one suggests a rapid expansion of the fireball. Besides a hard equation of state, other effects that enable transverse expansion lead to a reduction of $R_{out}/R_{side}$. The factors that influence the transverse expansion are shear viscosity, preequilibrium flow or steeper gradients in the initial entropy distribution Broniowski:2008vp ; Pratt:2008qv ; Bozek:2010er ; Karpenko:2012yf . Using a hard equation of state and realistic assumptions for the initial density profile and the preequilibrium flow the observed ratio $R_{out}/R_{side}$ can be reproduced in hydrodynamic models to within $10\%$.

A sizable value of the bulk viscosity may slow down the expansion of the fireball. It is expected that around the QCD critical temperature the value of bulk viscosity rises Karsch:2007jc . This peak in the temperature dependence of bulk viscosity has been implemented in a number of hydrodynamic simulations Denicol:2009am ; Song:2009rh ; Ryu:2015vwa ; Bernhard:2016tnd . The conclusion from these studies is that bulk viscosity of matter influences the transverse expansion of the fireball. Bulk viscosity leads to a reduction of the effective pressure.

This rises the question, whether introducing a significant bulk viscosity around the critical temperature would not invalidate the good description of the interferometry data. In an expanding system bulk viscosity reduces the effective pressure and acts in a similar way as a soft point in the equation of state. In this paper I investigate this question in explicit hydrodynamic simulations for central Pb-Pb collisions. Contrary to naive expectations, I find that the presence of a large bulk viscosity peak around the critical temperature does not lead to an increase of the $R_{out}/R_{side}$ ratio. In fact one finds a small decrease in that ratio for simulations with bulk viscosity, driving it closer to the experimental value.

II Model and calculations

I perform calculations in a $3+1$-dimensional viscous hydrodynamic model Schenke:2010rr . The shear viscosity to entropy density ration is $\eta/s=0.08$. The bulk viscosity to entropy ratio is temperature dependent

[TABLE]

where

[TABLE]

and $T_{H}=180$MeV, $T_{a}=179.1$MeV, $T_{b}=189$MeV, $\Delta T_{H}=4$MeV, $A_{0}=-13.45$, $A_{1}=27.55$, $A_{2}=-13.77$, $l_{1}=0.9$, $l_{2}=0.26$, $l_{3}=0.9$, $l_{4}=0.256$, $s_{1}=0.0025$, $s_{2}=0.022$, $s_{3}=0.025$, $s_{4}=0.13$. The parametrization of the bulk viscosity peak is similar as in Denicol:2009am . In the following I study two different cases $c=0$, $1$ (Fig. 1). The first case corresponds to no bulk viscosity peak, with bulk viscosity appearing only at low temperatures. Such a small bulk viscosity appears naturally in a mixture of massive particles with nonequilibrium distributions. The parametrization with $c=1$ includes a bulk viscosity peak around the critical temperature as expected from lattice QCD results Karsch:2007jc . The height of the bulk viscosity peak is also estimated from fits to particle spectra and azimuthal harmonic flow coefficients Bernhard:2016tnd . In the following I denote the two scenarios by the value of bulk viscosity at $T_{H}$, that is $\zeta/s=0.02,\ 0.35$. An increase in the bulk viscosity implies larger dissipative correction in the hydrodynamic evolution. The corresponding entropy production in the evolution is compensated by a reduction of the initial entropy for calculation with larger bulk viscosity to obtain the same final particle multiplicity at central rapidity.

The results presented further correspond to the freeze-out temperature of $150$ MeV. The Cooper-Frye formula for the emission of hadrons from the freeze-out hypersurface includes nonequilibrium corrections from both shear tensor and bulk viscosity. Bulk viscosity corrections are included using the relaxation time approximation Bozek:2009dw . I have checked that varying the freeze-out temperature down to $140$ MeV does not change the conclusions.

From pion pairs a Bose-Einstein symmetrized correlation function is constructed Chojnacki:2011hb and binned in bins of average transverse momentum of the pion pair. The three dimensional correlation function in relative pair momentum $q$ is fitted with a Gaussian formula Bertsch:1989vn ; Pratt:1986cc

[TABLE]

The three components of $q$ are defined along the beam axis ($q_{long}$), along the pair transverse momentum ($q_{out}$), and transverse to those ($q_{side}$). The interferometry analysis is summarized by the three HBT radii $R_{i}$ as functions of the average pair momentum $k_{T}$.

III Smooth initial conditions

To present a simple example I calculate the evolution of the fireball with smooth initial density density. The entropy density is obtained from the optical Glauber model. The density in the transverse plane is proportional to a combination of densities of participant nucleons $\rho_{w}(x,y)$ and binary collisions $\rho_{bin}(x,y)$

[TABLE]

with $\alpha=0.15$ Bozek:2011ua . The normalization of the initial density is adjusted in order to reproduce the charged particle multiplicity. The calculation is performed for zero impact parameter for illustration.

The bulk viscosity peak reduces the transverse flow at the edge of the fireball. It causes a retardation of the expansion at the edge. The resulting freeze-out hypersurface is slightly modified (Fig. 2). The effect on the freeze-out hypersurface can be described as a change from a scenario with earlier freeze-out in the outer layers (“burning log scenario”) to a freeze-out at constant proper time. The total life-time of the system is similar in the two scenarios.

Both calculations, with and without a bulk viscosity peak, reproduce the HBT radii fairly well (Fig. 3). The differences in the HBT radii between the two scenarios are small, less than 5%. The increase of bulk viscosity around the critical temperature does not modify strongly the dynamics. Note that the $R_{out}/R_{side}$ ratio decreases when the bulk viscosity peak is introduced (Fig. 4). This decrease may be understood as due to the observed change in the shape of the freeze-out surface Kisiel:2006is . The ratio $R_{out}/R_{side}$ is larger when the outside layers of the fireball freeze-out earlier.

IV Event-by-event hydrodynamic simulations

A realistic modeling of the collective expansion in heavy-ion collisions requires hydrodynamic simulations to be performed for an ensemble of fluctuating initial conditions. I use a Glauber Monte Carlo model to generate the initial entropy density Rybczynski:2013yba . The density in the transverse plane is written as a sum of Gaussians of width $0.5$ fm centered at the positions of the participant nucleons. A contribution of binary collisions is added with $\alpha=0.15$. In Fig. 5 are shown the freeze-out hypersurfaces corresponding to the same initial Monte Carlo Glauber event (one particular event with 377 participant nucleons) but evolved with hydrodynamics with or without a peak in the temperature dependence of bulk viscosity. The effect of bulk viscosity is qualitatively similar as for smooth initial conditions. Quantitatively the difference between the two scenarios is larger at the very edge of the fireball, due to larger local gradients of the density in event-by-event simulations. The increase in bulk viscosity hinders the expansion in the outer layers of the fireball, where the initial gradients are the largest. It leads to a freeze-out hypersurface, where inner layers freeze-out earlier.

For each event a freeze-out hypersurface is defined and the pion pair correlation function is constructed. After averaging over events, HBT radii are extracted from the Gaussian formula (Eq. 3). For realistic event-by-event simulations one gets similar conclusions as for the expansion from the smooth initial density discussed in the previous section. The change in the value of the radii is small (Fig. 6). The final effect on the HBT radii is to reduce the $R_{out}/R_{side}$ ratio by up to 3% for the calculation with a bulk viscosity peak (Fig. 7). This improves the description of the data.

V Conclusions

I have tested the effect of a peak in the temperature dependence of bulk viscosity on the size and the life-time of the source in Pb-Pb collisions at $2760$GeV. The increase of bulk viscosity around the critical temperature slows down the expansion of the fireball in the outer layers, where initially the gradients are the largest and the temperature of matter is close to the critical temperature. The relative increase in the life-time of the fireball is small in central Pb-Pb collisions, and limited to the outer layers of the fireball.

The effective softening of the equation of state due to the bulk viscosity peak does not lead to an increase of the $R_{out}/R_{side}$ ratio. This observation is contrary to expectation from earlier studies of the sensitivity of the HBT radii to the equation of state Rischke:1996em ; Bozek:2009ty . Surprisingly, the ratio $R_{out}/R_{side}$ is reduced when the bulk viscosity peak is introduced. Although the expansion is slightly slower with bulk viscosity, the shape of the hypersurface is modified in such a way as to reduce the problematic $R_{out}/R_{side}$ ratio by a few percent, making it closer to the data.

It would be interesting to include the interferometry radii in the data constraining the parameters of the matter created in relativistic heavy-ion collisions using exhaustive Bayesian statistical analyses. Existing analysis have studied a matter either without a bulk viscosity peak or they do not take the HBT radii in the data used to constraint the parameters Novak:2013bqa ; Pratt:2015zsa ; Bernhard:2015hxa ; Bernhard:2016tnd . The present result suggests that the presence of a peak in the temperature dependence of bulk viscosity would be consistent with an enlarged analysis including HBT radii in the data set. Further insight could be gained by studying the effects of bulk viscosity on the HBT radii in small, rapidly expanding systems, as in p+Pb collisions.

Acknowledgements.

Research supported by the Polish Ministry of Science and Higher Education (MNiSW), by the National Science Centre grant DEC-2012/06/A/ST2/00390, as well as by PL-Grid Infrastructure.