DOI: 10.22661/AAPPSBL.2019.29.6.61
Vorticity and Dileptons from Heavy Ion Collision
HIRANMAYA MISHRA^{1,}*
^{1}THEORY DIVISION, PHYSICAL RESEARCH LABORATORY, NAVRANGPURA, AHMEDABAD 380 009, INDIA
^{* }Email address: hm@prl.res.in
The study of vorticity in heavy ion collisions has been a topic of intense theoretical investigation, particularly, after the observation of Λ polarization by STAR collaboration. Vorticity in heavy ion collisions can also affect dilepton production in quark gluon plasma (QGP). In a recent work by the author alongwith his colleagues (Phys. Rev. D100, 056018, 2019), it was shown that the presence of vorticity the QGP matter causes it to cool faster and further leads to a suppression of dilepton production from such a system.
INTRODUCTION
Non central heavy ion collisions (HIC) have opened up many interesting as well as theoretically challanging areas in the field of quark gluon plasma. A strong magnetic field is produced due to electric current produced by the spectator protons constituting the colliding ions. The non central HICs that have large angular momentum also generate vorticity of the QGP fluid. In the limit of massless (chiral) quarks this can lead to novel transport phenomena related to chiral effects like the chiral magnetic effect and chiral vortical effect where a charge current is induced in the direction of the magnetic field and vorticity respectively.
One of the recent intriguing phenomenona occuring in this rotating fluid is that the medium can be polarised leading to polarization of the emitted hadrons. Indeed, the study of polarization of Λhyperon leads an estimate of the vorticity of QGP as ω ~ (9 ± 1) × 10^{21}s^{1}. This makes the matter produced in heavy ion collision a perfect vortical fluid. Apart from the polariszation of the produced hadrons, another manifestation of vorticity is the polarization of quarks and antiquarks due to spin orbit coupling leading to observable effects like emmission of circularly polarised photons and spin alignment of vector mesons. In a recent work, the effect of spin vorticity coupling on QGP dynamics and thermal dilepton production was investigated.
EVOLUTION OF VORTICITY IN HIC
Let us note that in the presence of nonvanshing vorticity, the longitudinal velocity of the fluid develops a dependence on the transverse coordinates. Therefore, it is expected that with finite vorticity hydrodynamic expansion will be different from the usual 1dimensional bjorken flow. This makes the hydrodynamic expansion in presence of vorticity to be in 2+1 dimensions. The relativistic Euler equation for an ideal fluid can be written as
 (1)

where ϵ and P are the energy density and pressure respectively. Here u^{μ} = 𝛾(1, v) is the flow four velocity, with, 𝛾^{1} = _{} and ∇^{µα}g^{µα} u^{µ}u^{α} is the projector perpendicular to the direction of the velocity. In the rest frame, u^{µ}∂µ and ∇^{µν} become the temporal and spatial derivatives respectively. We make the approximation that the velocities are small and the velocities are subsonic so that the vorticity is given by ω = ∇ × v, where v(r) is the fluid velocity. Specifically, let us take zdirection as the beam direction, xaxis as the impact parameter axis so that vorticity is established in the ydirection. We can decompose the flow velocity into an irrotational part v_{0} and the other being v_{r} so that v = v_{0} + v_{r}. The rotational velocity is v_{r} is defined in terms of vorticity as _{}v_{r} = ω × r. The longitudinal part of the velocity v, v_{0z} = z /t while the transverse components v_{0T} are given by the solutions of the equation
 (2)

which can be derived from the Euler equation Eq.(1) in the limit of small magnitude of velocity and we have used the equation of state and have considered the lowest order in the departure from the equllibrium [1]. In the above, s, P, ϵ are entropy, pressure and energy densities respectively. One can take the initial entropy density distribution as, Ref.[2],
 (3)

where σ_{x} and σ_{y} are transverse distribution root mean square widths [2]. Solving Eq.(2) by using the entropy density defined in Eq.(3), transverse velocities are
 (4)

 (5)

Taking curl of the spatial part of the Euler equation for the velocity, one has the evolution equation for the vorticity as
 (6)

dt Substituting in the above equation for v_{0}, one can obtain the following solution for ω as
 (7)

where ω_{0}(r_{0},t_{0}) is the initial vorticity at position r_{0} and time t_{0}. The factor represents the area swapped by the stream line from time t_{0} to t. From the above equation it is clear that vorticity gets diluted due to the expansion. The vorticity is shown in Fig. 1 as a function of time for typical values of initial vorticities (ω_{0} and for t_{0}=0.5 fm, σ_{y}=2.5 fm) consistent with values as reported by STAR at freeze out. Due to the vorticity of the medium, the single particle distribution functions get modified which for spin 1/2 particles are given by [4]
Fig. 1: Variation of vorticity (ω) as a function of time for different values of initial vorticity (ω_{0}) at t_{0} = 0.5 fm. It may be noted here that we have used here natural units velocity of light c = 1 so that length an d time have same units. Thus in these units 1 second ≃ 3 × 10^{23} fermi.
 (8)

 (9)

where u_{r}(p) and v_{s}(p) are bispinors for particle and antiparticle respectively. Further, X^{±}, appearing in the above equations for the single particle distribution function is defined as the product of the Boltzmann distribution function and the matrices M^{±} and can be written as [35]
 (10)

with M^{±} = exp[±_{}ω_{µν} ∑^{µν}], β_{µ} = βu^{µ}, and ω_{µν} being the spin polarization tensor, ∑_{µν} = _{}[𝛾_{µ},𝛾_{ν}] is the spin operator in terms of Dirac matrices. The polarization tensor ω_{µν}, can have a tensor decomposition as done in Ref. [3]
 (11)

Here, k_{µ} and ω_{µ} are orthogonal to the flow velocity u_{µ}. The space like component of ω_{µ} is ω = ∇ × v. Choosing ω^{µ} = (0, 0, ω, 0) i.e., rotation in the xz plane, the matrices M^{±} can be written as
 (12)

where _{}. The thermodynamic relation in the presence of vorticity is given as
 (13)

where, Ω, the vorticity is given by _{}. Further, w is the spin density given by
 (14)

with n_{0} = T^{3}/𝜋^{2} being the number density in the massless limit.
Vorticity also affects the temperature evolution during the expansion. The longitudinal velocity will have a dependence on the transverse coordinates making the hydrodynamic expansion to be in 2+1 dimensions. If we take the vorticity in the ydirection, the flow velocity in ydirection v_{y} = 0 so that v_{z} and v_{x} are respectively the longitudinal and transverse components and can be written in terms of vorticity as v_{x} = _{}ωz and v_{z} = _{}ωx. For small velocities, the Euler equation can be reduced to the temperature evolution equation as
 (15)

where, one uses the thermodynamic relation eq.(13).
Fig. 2: Variation of temperature with τ for various values of ω_{0} with τ_{0} =0.5 fm and T_{0} = 300 MeV. Red curve corresponds to usual 1D Bjorken flow. With increase in vorticity the system cools faster.
Fig. 3: Left: Fractional change in dilepton production as a function of invariant mass for different values of ω_{0}. Right: Fractional change in dilepton production as a function of transverse momentum for different values of ω_{0}, M =0.4 GeV.
Figure(2) shows the behavior of temperature (T) vs time(τ). The red curve is the 1D Bjorken flow which we have reproduced in the limit of zero vorticity. As may be observed from Fig.[2], the spinvorticity coupling term leads to faster cooling of the fireball as compared to the case with ω = 0. The modification of the distribution functions due to spin vorticity coupling as well as the temperature evolution in the presence of vorticity affect the dilepton production from QGP. In Fig.(3), the fractional change in dilepton production due to vorticity is shown. Here 𝓡 denotes corresponding production rates. The left figure is variation with the invariant mass and the right one is with the transverse momentum (p_{T}). In the left figure, it is clear that suppression is maximum at an invariant mass around 1 GeV. For an initial vorticity ω_{0} = 0.4 fm^{1} maximum suppression is 15% and for ω_{0} = 0.7 fm^{1} the suppression can be as large as 28% as compared to the case of zero vorticity. On the other hand for p_{T} variation (right figure) suppression is more around transverse momentum p_{T} = 1  1.5 GeV. For the same values of initial vortices ω_{0} = 0.4 fm^{1} and ω_{0} = 0.7 fm^{1} the maximum suppression is about 15% and 28% respectively. For p_{T} less than 1 GeV the suppression is small.
SUMMARY
The present study analyzes the role of spinpolarization and vorticity on the evolution of the QGP created in relativistic heavyion collisions. Because of the initial vorticity, one needs to modify the Bjorken flow describing the initial stages of hydrodynamic evolution. Inclusion of vorticity leads to a 2+1 dimensional hydrodynamic expansion of the system. We find that in the absence of spinpolarization, vorticity alone cannot significantly influence the temperature evolution of the QGP. This situation changes when the effect of spinvorticity coupling is incorporated in the thermodynamic relation given in Eq.(13). The expanding plasma cools at a much faster rate in comparison with the case without the spinpolarization. This can lead to early hadronization of the system. Further, it is found that the production rates for dileptons from QGP are suppressed due to the faster cooling of the system. These results are also useful in testing the presence of a vorticity induced term in the thermodynamic relation i.e., Eq.(13). If such a term is present it is shown that its effect can be studied via the production of thermal dilepton pairs.
References
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