# A study of vorticity formation in high energy nuclear collisions

## Abstract

We present a quantitative study of vorticity formation in peripheral ultrarelativistic heavy-ion collisions at \({\sqrt{s}_\mathrm{NN}}= 200\) GeV by using the ECHO-QGP numerical code, implementing relativistic dissipative hydrodynamics in the causal Israel–Stewart framework in \(3+1\) dimensions with an initial Bjorken flow profile. We consider different definitions of vorticity which are relevant in relativistic hydrodynamics. After demonstrating the excellent capabilities of our code, which proves to be able to reproduce Gubser flow up to 8 fm/*c*, we show that, with the initial conditions needed to reproduce the measured directed flow in peripheral collisions corresponding to an average impact parameter \(b=11.6\) fm and with the Bjorken flow profile for a viscous Quark Gluon Plasma with \(\eta /s=0.1\) fixed, a vorticity of the order of some \(10^{-2}\,c\)/fm can develop at freeze-out. The ensuing polarization of \(\Lambda \) baryons does not exceed 1.4 % at midrapidity. We show that the amount of developed directed flow is sensitive to both the initial angular momentum of the plasma and its viscosity.

## 1 Introduction

The hydrodynamical model has by now become a paradigm for the study of the QCD plasma formed in nuclear collisions at ultrarelativistic energies. There has been a considerable advance in hydrodynamics modeling and calculations of these collisions over the last decade. Numerical simulations in \(2+1\) D [1] and in \(3+1\) D [2, 3, 4, 5, 6, 7] including viscous corrections are becoming the new standard in this field, and existing codes are also able to handle initial state fluctuations.

An interesting issue is the possible formation of vorticity in peripheral collisions [8, 9, 10]. Indeed, the presence of vorticity may provide information as regards the (mean) initial state of the hydrodynamical evolution which cannot be achieved otherwise, and it is related to the onset of peculiar physics in the plasma at high temperature, such as the chiral vortical effect [11]. Furthermore, it has been shown that vorticity gives rise to polarization of particles in the final state, so that e.g. \(\Lambda \) baryon polarization – if measurable – can be used to detect it [12, 13]. Finally, as we will show, a numerical calculation of vorticity can be used to make stringent tests of numerical codes, as the *T*-vorticity (see Sect. 2 for the definition) is expected to vanish throughout under special initial conditions in the ideal case.

Lately, vorticity has been the subject of investigations in Refs. [9, 10] with peculiar initial conditions in cartesian coordinates, ideal fluid approximation and isochronous freeze-out. Instead, in this work, we calculate different kinds of vorticity with our \(3+1\) D ECHO-QGP^{1} code [3], including dissipative relativistic hydrodynamics in the Israel–Stewart formulation with Bjorken initial conditions for the flow (i.e. with \(u^x=u^y=u^\eta =0\)), henceforth denoted BIC. It should be pointed out from the very beginning that the purpose of this work is to make a general assessment of vorticity at top RHIC energy and not to provide a precision fit to all the available data. Therefore, our calculations do not take into account effects such as viscous corrections to the particle distribution at the freeze-out and initial state fluctuations, that is, we use smooth initial conditions obtained averaging over many events.

### 1.1 Notations

In this paper we use the natural units, with \(\hbar =c=K=1\). The Minkowskian metric tensor is \(\mathrm{diag}(1,-1,-1,-1)\); for the Levi-Civita symbol we use the convention \(\epsilon ^{0123}=1\).

We will use the relativistic notation with repeated indices assumed to be summed over, however, contractions of indices will be sometimes denoted with dots, e.g. \( u \cdot T \cdot u \equiv u_\mu T^{\mu \nu } u_\nu \). The covariant derivative is denoted \(d_\mu \) (hence \(d_\lambda g_{\mu \nu }=0\)), the exterior derivative by \(\mathbf{d}\), whereas \(\partial _\mu \) is the ordinary derivative.

## 2 Vorticities in relativistic hydrodynamics

Unlike in classical hydrodynamics, where vorticity is the curl of the velocity field \(\mathbf{v}\), several vorticities can be defined in relativistic hydrodynamics which can be useful in different applications (see also the review [14]).

### 2.1 The kinematical vorticity

*u*is the four-velocity field. This tensor includes both the acceleration

*A*and the relativistic extension of the angular velocity pseudo-vector \(\omega _\mu \) in the usual decomposition of an antisymmetric tensor field into a polar and pseudo-vector fields:

*u*) projector:

### 2.2 The *T*-vorticity

*T*-vorticity as the exterior derivative of a vector field (1-form)

*Tu*, that is \(\Omega = \mathbf {d} (Tu)\). Indeed, Eq. (6) implies – through the Cartan identity – that the Lie derivative of \(\Omega \) along the vector field

*u*vanishes, that is,

*T*

*u*and \(\mathbf{d}\mathbf{d} = 0\). Equation (7) states that the

*T*-vorticity is conserved along the flow and, thus, if it vanishes at an initial time it will remain so at all times. This can be made more apparent by expanding the Lie derivative definition in components:

*T*-vorticity has the same property as the classical vorticity for an ideal barotropic fluid, such as the Kelvin circulation theorem, so the integral of \(\Omega \) over a surface enclosed by a circuit comoving with the fluid will be a constant.

*T*-vorticity and kinematical vorticity by expanding the definition (5):

### 2.3 The thermal vorticity

*u*, that is a hydrodynamical frame, is introduced, but it can also be taken as a primordial quantity to define a velocity through \(u \equiv \beta /\sqrt{\beta ^2}\) [16]. The thermal vorticity features two important properties: it is adimensional in natural units (in cartesian coordinates) and it is the actual constant vorticity at the global equilibrium with rotation [17] for a relativistic system, where \(\beta \) is a Killing vector field whose expression in Minkowski space–time is \(\beta _\mu = b_\mu + {\varpi }_{\mu \nu } x^\nu \),

*b*and \({\varpi }\) being constant. In this case the magnitude of thermal vorticity is – with the natural constants restored – simply \(\hbar \omega /k_BT\) where \(\omega \) is a constant angular velocity. In general (replacing \(\omega \) with the classical vorticity defined as the curl of a proper velocity field) it can be readily realized that the adimensional thermal vorticity is a tiny number for most hydrodynamical systems, though it can be significant for the plasma formed in relativistic nuclear collisions.

*T*-vorticity and thermal vorticity:

## 3 High energy nuclear collisions

In nuclear collisions at very large energy, the QCD plasma is an almost uncharged fluid. Therefore, according to previous section’s arguments, in the ideal fluid approximation, if the transversely projected vorticity tensor \(\omega ^\Delta \) initially vanishes, so will the transverse projection \(\Omega ^\Delta \) and \({\varpi }^\Delta \) and the kinematical and thermal vorticities will be given by the formulas (9) and (14), respectively. Indeed, the *T*-vorticity \(\Omega \) will vanish throughout because also its longitudinal projection-vanishes according to Eq. (6). This is precisely what happens for the usually assumed BIC for the flow at \(\tau _0\), that is \(u^x=u^y=u^\eta =0\), where one has \(\omega ^\Delta =0\) at the beginning as it can be readily realized from the definition (1). On the other hand, for a viscous uncharged fluid, transverse vorticities can develop even if they are zero at the beginning.

It should be noted, though, that even if the space–space components (\(x,y,\eta \) indices) of the kinematical vorticity tensor vanish at the initial Bjorken time \(\tau _0\), they can develop at later times even for an ideal fluid if the spatial parts of the acceleration and velocity fields are not parallel, according to Eq. (9). The equation makes it clear that the onset of spatial components of the vorticity is indeed a relativistic effect as, with the proper dimensions, it goes like (\(\mathbf{a} \times \mathbf{v})/c^2\).

In the full longitudinally boost invariant Bjorken picture, that is, \(u^\eta = 0\) throughout the fluid evolution, in the ideal case, as \(\omega ^\Delta =0\), the only allowed components of the kinematical vorticity are \(\omega ^{\tau x}, \omega ^{\tau y}\) and \(\omega ^{xy}\) from the first Eq. (2). The \(\omega ^{xy}\) component, at \(\eta =0\), because of the reflection symmetry (see Fig. 1) in both the *x* and the *y* axes, can be different from zero but it ought to change sign by moving clockwise (or counterclockwise) to the neighboring quadrant of the *xy* plane; for central collisions it simply vanishes.

*A*the mass number of the colliding nuclei, and

*x*axis and the two nuclei have initial momentum along the

*z*axis (whence the reaction plane is the

*xz*plane) and their momenta are directed so as to make the initial total angular momentum oriented along the negative

*y*axis (see Fig. 1). The wounded nucleons weight function \(W_\mathrm{N}\) is then defined:

*collision hardness*parameter, which can vary between 0 and 1.

This parametrization, and especially the chosen forms of the functions \(f_\pm \), are certainly not unique as a given angular momentum can be imparted to the plasma in infinitely many ways. Nevertheless, as has been mentioned, it proved to reproduce correctly the directed flow in a \(3+1\) D hydrodynamical calculation of peripheral Au–Au collisions at high energy [18], thus we took it as a good starting point. A variation of this initial condition will be briefly discussed in Sect. 7. Besides, the parametrization (20) essentially respects the causality constraint that the plasma cannot extend beyond \(\eta = y_\mathrm{beam}\). Indeed, at \({\sqrt{s}_\mathrm{NN}}=200\) GeV \(y_\mathrm{beam}\simeq 5.36\) while the 3\(\sigma \) point in the gaussian profile in Eq. (22) lies at \(\eta = \eta _\mathrm{flat}/2 + 3 \sigma _\eta \simeq 4.4\).

Parameters defining the initial configuration of the fluid in Bjorken coordinates. The last two parameter values have been fixed for the last physical run

Parameter | Value |
---|---|

\(\sqrt{s_{\mathrm{NN}}}\) | \(200\,\text {GeV}\) |

\(\alpha \) | 0. |

\(\epsilon _0\) | \(30\,\text {GeV}/\text {fm}^\text {3}\) |

\(\sigma _{\mathrm{in}}\) | \(40\,\text {mb}\) |

\(\tau _0\) | \(0.6\,\text {fm}/{c}\) |

\(\eta _{\mathrm{flat}}\) | 1 |

\(\sigma _{\eta }\) | 1.3 |

\(T_{\mathrm{fo}}\) | \(130\, \text {MeV}\) |

| \(11.57\text {fm}\) |

\(\eta _m\) ideal | 3.36 |

\(\eta _m\) viscous | 2.0 |

\(\eta /s\) | 0.1 |

We have run the ECHO-QGP code in both the ideal and the viscous modes with the parameters reported in Table 1 and the equation of state reported in Ref. [20]. The impact parameter value \(b=11.57\) was chosen as, in the optical Glauber model, it corresponds to the mean value of the 40–80 % centrality class (\(9.49 < b < 13.42\) fm [21]) used by the STAR experiment for the directed flow measurement in Ref. [22]. The initial flow velocities \(u^x,u^y,u^\eta \) were set to zero, according to BIC. The freeze-out hypersurface – isothermal at \(T_{\mathrm{fo}}=130\) MeV – is determined with the methods described in Refs. [3, 23].

## 4 Qualification of the ECHO-QGP code

To show that our code is well suited to model the evolution of the matter produced in heavy-ion collisions and hence to carry out our study on the development of vorticity in such an environment, we have performed two calculations, referring to an ideal and viscous scenario, respectively, providing a very stringent numerical test.

*T*-vorticity by \(1/T^2\) in order to have an adimensional number. Since the

*T*-vorticity has always been determined at the isothermal freeze-out, in order to get its actual magnitude, one just needs to multiply it by \(T_{\mathrm{fo}}^2\).

### 4.1 *T*-vorticity for an ideal fluid

Since the fluid is assumed to be uncharged and the initial *T*-vorticity \(\Omega \) is vanishing with the BIC, it should be vanishing throughout, according to the discussion in Sect. 2. However, the discretization of the hydrodynamical equations entails a numerical error, thus the smallness of \(\Omega \) in an ideal run is a gauge of the quality of the computing method. In Fig. 2 we show the mean of the absolute values of the six independent Bjorken components at the freeze-out hypersurface, of the *T*-vorticity divided by \(T^2\) to make it adimensional, as a function of the grid resolution (the boundaries in \(x,y,\eta \) being fixed).^{2} As is expected, the normalized *T*-vorticity decreases as the resolution improves.

### 4.2 Gubser flow

*R*refers to the rapidity coordinate. It is then convenient to perform a coordinate transformation (

*q*is an arbitrary parameter setting an energy scale for the solution once one goes back to physical dimensionful coordinates)

*T*and the components \(\pi ^{xx}\), \(\pi ^{xy}\), and \(\pi ^{\eta \eta }\) of the viscous stress tensor, respectively, at different times. The initial energy density profile was taken from the exact Gubser solution at the time \(\tau =1\) fm/

*c*.

### 4.3 *T*-vorticity for a viscous fluid

Unlike for an ideal uncharged fluid, *T*-vorticity can be generated in a viscous uncharged fluid even if it is initially vanishing. Thus, the *T*-vorticity can be used as a tool to estimate the numerical viscosity of the code in the ideal mode by extrapolating the viscous runs.

A comment is in order here. In general, in addition to standard truncation errors due to finite-difference interpolations, all shock-capturing upwind schemes are known to introduce numerical approximations that behave roughly as a dissipative effects, especially in the simplified solution to the Riemann problems at cell interfaces [30]. It is therefore important to check whether the code is not introducing, for a given resolution, numerical errors which are larger than the effects induced by the physics. We refer to the global numerical errors generically as numerical viscosity.

We have thus calculated the *T*-vorticity for different physical viscosities (in fact \(\eta /s\) ratios), in order to provide an upper bound for the numerical viscosity of ECHO-QGP in the ideal mode. The mean value of the *T*-vorticity is shown in Fig. 5 and its extrapolation to zero occurs when \(|\eta /s| \lesssim 0.002\) which is a very satisfactory value, comparable with the one obtained in Ref. [4]. The good performance is due to the use of high-order reconstruction methods that are able to compensate for the highly diffusive two-wave Riemann solver employed [3].

## 5 Directed flow, angular momentum, and thermal vorticity

*y*axis with negative value and can be written as

*b*(see Fig. 10) (see also Ref. [33]).

At \(b=11.57\) fm the above angular momentum is about \(3.58 \times 10^3\) in \(\hbar \) units. This means that, with the current parametrization of the initial conditions, for that impact parameter about 89 % of the angular momentum is retained by the hydrodynamical plasma, while the rest is possibly taken away by the corona particles.

*y*coordinate \(y=0\), is shown in Fig. 13 where it can be seen that it attains a top (negative) value of about 0.05 corresponding to a kinematical vorticity, at the freeze-out temperature of 130 MeV, of about 0.033

*c*/fm \(\simeq 10^{22} \mathrm{s}^{-1}\). In this respect, the Quark Gluon Plasma would be the fluid with the highest vorticity ever made in a terrestrial laboratory. However, the mean value of this component at the same value of \(\eta /s = 0.1\) is of the order of \(5.4 \times 10^{-3}\), that is about ten times less than its peak value, as shown in Fig. 12. This mean thermal vorticity is consistently lower than the one estimated in Ref. [13] (about 0.05) with the model described in Refs. [9, 10], implying an initial non-vanishing transverse kinematical and thermal vorticity \({\varpi }^\Delta \). This reflects in a quite low value of the polarization of \(\Lambda \) baryons, as will be shown in the next section.

## 6 Polarization

*y*component is predominantly negative, oriented along the initial angular momentum vector and a magnitude of the order of 0.1 %. Indeed, the main contribution to the polarization stems from the longitudinal component \(\Pi _0^z\), with a maximum and minimum along the bisector \(|p^x|=|p^y|\).

The obtained polarization values are – as expected – consistently smaller than those estimated in Ref. [13] (of the order of several percent with a top value of 8–9 %) with the already mentioned initial conditions used in Refs. [9, 10]. This is a consequence of the much lower value of the implied thermal vorticity, as discussed in the previous section. Also, the \(\Pi _0^y\) pattern is remarkably different, with different location of maxima and minima.

## 7 Conclusions, discussion and outlook

To summarize, we have calculated the vorticities developed in peripheral (\(b=11.6\) fm) nuclear collisions at \({\sqrt{s}_\mathrm{NN}}=200\) GeV (\(b=11.6\) fm) with the most commonly used initial conditions in the Bjorken hydrodynamical scheme, by using the code ECHO-QGP implementing second-order, causal, relativistic dissipative hydrodynamics. An extensive testing of the high accuracy and very low numerical diffusion properties of the code has been carried out, followed by long-time simulations (up to \(\tau =8\) fm / *c*) of the so-called viscous Gubser flow, a stringent test of numerical implementations of Israel–Stewart theory in Bjorken coordinates.

We have found that the magnitude of the \(1/\tau \) \(x-\eta \) component of the thermal vorticity at freeze-out can be as large as \(5 \times 10^{-2}\), and yet its mean value is not large enough to produce a polarization of \(\Lambda \) hyperons much larger than 1 %, which is a consistently lower estimate in comparison with other recent calculations based on different initial conditions. We have found that the magnitude of directed flow, at this energy, has an interestingly sizeable dependence on both the shear viscosity and the longitudinal energy density profile asymmetry parameter \(\eta _m\) which in turn governs the amount of initial angular momentum retained by the plasma.

We plan to extend this kind of calculation to different centralities, different energies and with initial state fluctuations in order to determine the possibly best conditions for vorticity formation in relativistic nuclear collisions.

## Footnotes

- 1.
The code is publicly available at the web site http://theory.fi.infn.it/echoqgp.

- 2.
It should be pointed out that, throughout this work, by mean values of the vorticities we mean simple averages of the Bjorken components (possibly rescaled by \(1/\tau \)) over the freeze-out hypersurface without geometrical cell weighting. Therefore, the plotted mean values have no physical meaning and they should be taken as descriptive numbers which are related to the global features of vorticity components at the freeze-out.

## Notes

### Acknowledgments

We are grateful to L. Csernai and Y. Karpenko for very useful comments and suggestions.

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