Probing vorticity in heavy ion collision with dilepton production

We study the effect of vorticity present in heavy ion collisions (HICs) on the temperature evolution of hot quark-gluon plasma in the presence of spin-vorticity coupling. The initial global rotation entails a nontrivial dependence of the longitudinal flow velocity on the transverse coordinates and also develops a transverse velocity component that depends upon longitudinal coordinate. Both of these velocities leads to a 2+1 dimensional expansion of the fireball. It is observed that with the finite vorticity and spin-polarization fireball cools faster in comparison with the case without vorticity. Furthermore, we also discuss the consequence of this on thermal dilepton production.


Introduction
Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) have made the strongly interacting deconfined state of matter i.e., Quark Gluon Plasma (QGP), accessible in laboratory by colliding two heavy nuclei at sufficiently high energy. The colliding nuclei have a very large initial angular momentum(L), L ∝ b √ s, with b as impact parameter, for example; at RHIC energies (Au-Au collision, √ s = 200 GeV) L ∼ 10 5 and at LHC energies (Pb-Pb collision, √ s = 2.76 TeV) L ∼ 10 7 , for impact parameter b = 5 fm respectively [1]. After the collision, most of the angular momentum is carried away by the spectators and a significant fraction of it is retained in the interaction region that could further be transferred to QGP. This can bring about a non-vanishing local vorticity which is perpendicular to the reaction plane [2,3]. The average vorticity in the heavy ion collision (HIC) experiment has been estimated to be of order 10 21 s −1 [4].
There have been efforts to study the vorticity evolution and its dynamics in the HIC experiments. One important observation of these studies is that the event averaged vorticity decreases with increase in collision energy [4]. Several other models such as A Multi-Phase Transport (AMPT) model [5,6], hydrodynamics [7,8], AMPT model including chiral kinetic equation [9] have been used to study the collective rotational motion of QGP. In Ref. [10], it has been shown that for large Reynolds number Re = uLη −1 ; which for η s = 1 4π , flow velocity u = (0.1 − 1), fluid length scale L = 5 fm and temperature T = 300 MeV at RHIC varies between 10 − 100; of QGP the time evolution of vorticity gets diluted in the plasma and is significant up to freeze-out time. However for smaller Reynolds number the vorticity is damped and it is less significant at freeze-out time. We restrict ourselves in the large Reynolds number , so we ignore viscosity in the present work. In the presence of a non-vanishing vorticity the longitudinal velocity develops a dependence on transverse coordinate and vice versa, it is expected that with the finite vorticity hydrodynamic expansion of the thermalized plasma will be very different from the usual 1D Bjorken flow . Ultimately, the hydrodynamic expansion in this case is in 2+1 dimension.
The manifestation of vorticity is in the polarization of secondary produced particles [11][12][13][14], for example for hyperons, at RHIC energies the polarization is reported in [15]. However, in the present analysis, we shall focus on another effect of vorticity that induces a spin-vorticity coupling in the rotating fluid. Such a coupling term even modifies the thermodynamic relation and also responsible for Barnett effect [16][17][18]. The modified thermodynamic relation suggest the quantum nature of the fluid (spin vorticity coupling) which we would like to investigate through its consequences on the thermal dilepton production. Quark and anti-quark global polarization that arises from spinorbit coupling [12] lead to observable effects such as, emission of circularly polarized photons [19] and spin alignment of vector meson [20]. Another effect of rotation of matter produced in HIC experiments on two-particle correlation function has been proposed in Ref. [21] by employing differential Hanbury Brown and Twiss (HBT) analysis. In QGP it may also be possible to have parity odd effects associated with chiral vortical effect(CVE) leading to charge separation and induced currents [6,22] and chiral vortical wave (CVW). The later one lead to the elliptical flow splitting of baryon/antibaryon [23,24] At present, both theoretical as well as experimental efforts are in progress to suggest and detect possible signals for such large vorticity in HIC experiments. Phenomenologically, it is also important to quantify the rotational motion of the QGP in these collisions. In the present analysis we mainly focus on the effect of vorticity and spin polarization as discussed for example in Ref. [18,25] on the hydrodynamic evolution of QGP subsequent to a off central HIC. In this regard, we attempt to investigate the temperature evolution of thermalized hot QGP. We show that in presence of spin-vorticity coupling the rate of cooling of fireball is increased which can lead to an early hadronization. Furthermore, we investigate its consequences on the thermal dilepton production from plasma. As anticipated from effect of vorticity on temperature, dilepton yield is suppressed and the suppression is more for larger initial vorticity.
Our paper is organized as follows, in section(2) we calculate the vorticity evolution by employing the hydrodynamic analysis. In section(3) we have studied the temperature evolution of the fireball and discuss the effect of a local vorticity on critical temperature which is followed by the thermal dilepton production from a vortical thermalized plasma in section (4). Finally in section(5) we summarize and conclude our results.

Vorticity evolution in HIC
In this section, we discuss the time evolution of vorticity in QGP by employing hydrodynamic analysis. In the relativistic limit there are many quantities related to vorticity such as the thermal vorticity, T vorticity, and the kine- , are defined; for detail see [26]. We start with the relativistic Euler equation which for an ideal fluid can be written as where, and P are energy and pressure density respectively. Here, Using the thermodynamic relation P ( + P ) −1 = c 2 s ln s and second law of thermodynamics (∂ µ s µ = 0), the spatial part of above equation can be writ-ten as where s is entropy density and c s is speed of sound. Here, we work in the non-relativistic limit for simplicity by assuming γ ≈ 1 while we retain the the relativistic equation of state. This can be a reasonable approximation as the value of γ for fluid is very close to its non-relativistic value. Thus the fluid motion is regarded as non-relativistic while microscopic motion is regarded as relativistic [27]. A similar approximation for vorticity has been taken in Ref. [10,28]. In Ref. [28] the authors have studied vorticity development and distribution in both relativistic and non-relativistic limits by implementing numerical simulations and found that in both the limits the average values of vorticity are of similar order, though the relativistic one differ by a factor of 2-3 from its non-relativistic counterpart. It has also been observed that at later time (∼ 4) fm vorticity can be as high as 3 fm −1 depending upon favorable configuration with Kelvin Helmholtz instability (KHI). For a less favorable configuration at the same time vorticity can be as high as 1 fm −1 . In the non-relativistic limit, the vorticity ω is defined as where v(r) is velocity. The non-relativistic definition of vorticity as in Eq. [3] is similar to the angular velocity in the case of rigid body rotator with a global angular velocity about an axis. On the other hand, in a fluid, the vorticity need not be a constant which make it different from a rotating rigid body. Generally, in a non-central collision, the average vorticity is perpendicular to the reaction plane. Assumingẑ axis as the beam direction, x axis as impact parameter axis that makes reaction plane in x − z plane, the average vorticity is alongŷ direction. For flow velocity we adopt the assumptions same as taken by the authors of [10], where the fluid velocity v is decomposed into two parts; one is non-rotational flow ( v 0 ) and other is rotational The rotational flow velocity is defined as Further v 0 has three components, one is longitudinal Bjorken flow part v 0z = z t and the other two components which comes from transverse expansion which are given by the differential equation [29] ∂ where s, P , are entropy, pressure and energy densities. Assuming the initial entropy density as where σ x and σ y are transverse distribution root mean square widths. Solving Eq.(6) by using the entropy density defined in Eq. (7), transverse velocities are The transverse expansion is due to very high initial pressure gradient as noted in Ref. [29] and dominate only in the time scale t σx cs . Taking curl of Eq.(2) and using Eqs.(4), (5), we get Substituting in the above equation for v 0 , one can obtain the following solution for ω as also found in [10] ω( r, t) where ω 0 ( r 0 , t 0 ) is the initial vorticity at position r 0 and time t 0 .

Temperature evolution with finite vorticity
We take the effect of vorticity in the 4-velocity profile which can be defined as u µ = (γ, γv x , 0, γv z ), satisfying the normalization condition u µ u µ = 1. Lorentz factor γ is where v 2 = v 2 x + v 2 z , subsequently we set γ ≈ 1 as explained earlier.
Velocities v z and v x are longitudinal and transverse velocities that can be written as where where and P are energy and pressure densities respectively. To obtain the evolution equation for temperature, we write the projection of energymomentum conservation (∂ µ T µν = 0) on fluid four velocity leading to ∂ µ (( + P )u µ ) = u ν g µν ∂ µ P.
Using the covariant proper time derivative d dτ = u µ ∂ µ and standard thermodynamic relation + P = T s, c 2 s = d dP , Eq.(17) can be rewritten in terms of proper time evolution of temperature as where T is temperature and τ is proper time. While in case of 1-D Bjorken flow the rhs in above equation reduces to − c 2 s τ . Here, it contains the time derivative and the divergences of velocity profiles, that include γ, in terms of vorticity. These time and space derivatives plays very important role for temperature evolution. However in the limit γ ≈ 1 these derivatives becomes less significant and the temperature evolution with finite vorticity is identical to that of in Bjorken's 1-D picture. Now let us consider the case in which the global angular momentum in non-central heavy ion collision generate the spin polarization of the fluid (QGP). This polarization in a rotating fluid is basically induced by the spin-vorticity coupling. The modified thermodynamic relation can be written as [18] + p = T s + ωw (19) where w = 4 sinh (ζ)n 0 with n 0 as number density of particles obeying Maxwell-Boltzmann distribution and ζ = ω 0 T , which vanishes for non-vortical fluids. In above equation it is clear that spin term can lead to change in entropy density of the system. For entropy to be conserved one must have ∂ µ (wu µ ) = 0. The evolution equation, modified by Eq. [19] becomes also note that here we have τ = t. The spin-vorticity coupling term in Eq. (20) effect the evolution of temperature which turned out to be very different from that of Eq. [18], where spin polarization is not considered. Using the velocity  profiles as defined in Eqs. (8) and (9), we solve temperature evolution Eq. (20) numerically. For this purpose, we take c 2 s = 1 3 , τ 0 = 0.5 fm, T 0 = 300 MeV. For a non-central collision of b = 7 fm, the rms widths are σ x = 2 fm, σ y = 2.6 fm. We take the vorticity profile as given in Eq. (11), where ω 0 is a free parameter. Since the rotational motion should be such that the velocity should not exceed the speed of light so we consider ω 0 values such that the condition ωR < 1 is always satisfied during the lifetime of fireball. Figure(1) shows the behavior of temperature (T ) vs time(τ ). The red curve is 1D Bjorken flow which we have reproduced in the limit of zero vorticity. In the presence of spin-vorticity coupling the cooling of the fireball is faster than that of without vorticity due to which the hadronization time is also reduced as shown in table (1). With increase in vorticity critical time (by which system reaches at critical temperature) decreases. Note that there is no contribution of vorticity in term ∂ µ u µ as divergence of of vorticity is zero so it does not play any role in reducing the temperature. Thus, the only term that is responsible for the faster cooling is spin-vorticity coupling that can be seen clearly from Eq.(20).

Dilepton production
We consider the thermal dilepton production only from QGP in which the dominant channel is annihilation of quark and its anti-quark i.e., qq → γ * → ll. We use kinetic theory and assume that two particle scattering is not affected by rotation. In the limit of massless quarks, dilepton production per unit space-time volume is written as [30] dN where p 1,2 , E 1,2 is quark anti-quark momenta and energy.
is distribution function of quark and its anti-quark and σ(M ) is cross-section of thermal dilepton production. Note that in distribution function the effect of rotation is only in temperature T which we have evaluated by employing hydrodynamics in previous section. In Born approximation, σ(M ) = f 16πα 2 q 2 f Nc 3M 2 g 2 , which for N f = 2 and N c = 3 reduces to [31] For our study, dilepton rate per unit momentum is more useful which is given as In the limit M T , Fermi-Dirac distribution function is replaced by Maxwell-Boltzmann distribution function, Eq.(23) reduces to Notice that Eq.(24) is in the rest frame of the system, for an expanding plasma p 0 is replace by u · p, where p µ is particle's four velocity that can be written as where, η s is space time rapidity, p T is particle's transverse momentum, φ is azimuthal angle and m T = p 2 T + M 2 with M as dilepton's invariant mass write more. Fluid four velocity u µ can be written as u µ = (cosh η s cosh ξ, sinh ξ, 0, sinh η s cosh ξ), (26) where, sinh ξ is related to the transverse velocity and η s is space-time rapidity. The space-time volume element can be written as where φ is azimuthal angle and R = 1.2(A) 1 3 is the nucleus radius used for collision (for Gold, A=197).Dilepton production rate as a function of photon invariant mass (M ), transverse momentum p T and particle rapidity y can be written as [32] dN τ 0 and τ are initial thermalization time and the time by which the temperature of the system reaches at its critical value. Here we have taken τ 0 = 0.5 fm and critical temperature T c = 150 MeV. τ for different values of ω 0 are shown in table 1. Further, we take η max = 5.3 and the particle rapidity y = 0. Fig(2) shows the variation dilepton production rate with invariant mass (left) and with transverse momentum (right). As anticipated from the cooling rate of the plasma the dilepton production rate is suppressed in presence of vorticity. The red curve is the production rate without vorticity which is consistent with the results shown in Ref [33]. In Fig.(3), the fractional change in dilepton production due to vorticity is shown, here R denotes corresponding production rate as defined in Eqs. (28) and (29). Left figure is variation with invariant mass and right one is with transverse momentum (p T ). In the left figure it is clear that suppression is maximum at invariant mass around 1 GeV. For an initial vorticity ω 0 = 0.3 fm −1 maximum suppression is 21% and for ω 0 = 0.6 fm −1 the suppression can be as large as 37% as compared to the case of zero vorticity. On the other hand for For p T less than 1 GeV the suppression is small.

Summary and Conclusion
In this work we have studied the effect of vorticity in the temperature evolution of QGP created in HICs in presence of spin-vorticity coupling. Because of vorticity, the longitudinal velocity depends on transverse coordinates and vorticity induced transverse velocity depends on longitudinal coordinate which leads to 2+1 dimensional expansion of the system. We found that for sufficient large global rotation temperature of the hot plasma can fall very fast, this leads to an early hadronization of the system e.g., T 0 = 300 MeV, critical temperature T c = 150 MeV the hadronization time is almost reduced by a factor of half. We studied the consequence of rotation on the temperature evolution of the system and its effect on the thermal dilepton production. We demonstrated that the production rate is suppressed in presence of vorticity. This is a first attempt to include vorticity in the study of thermal evolution and thermal dilepton production. The analytic solutions used here for velocity profile may not be valid at late time evolution. Within this limit we predict suppression of dilepton production and early hadronization as a consequence of vorticity.