HHydrodynamics formalism with Spin dynamics ∗ Rajeev Singh
Institute of Nuclear Physics Polish Academy of Sciences,PL-31342 Krak´ow, PolandWe review the key steps of the relativistic fluid dynamics formalism withspin degrees of freedom initiated recently. We obtain equations of motionof the expansion of the system from the underlying definitions of quantumkinetic theory for the equilibrium phase space distribution functions. Weinvestigate the dynamics of spin polarization of the system in the Bjorkenhydrodynamical background.PACS numbers: 24.70.+s, 25.75.Ld, 25.75.-q
1. Introduction
Spin polarization experimental measurements of Λ hyperons recentlytaken by the STAR Collaboration [1–4] have created a huge interest in thespin polarization studies and in studies correlating between the vorticityand particle spin polarization in relativistic heavy-ion collisions [5–41]; forreviews see [42–46]. Thermal-based models [47–50] which precisely explainthe global polarization of particles, does able to explain differential resultscorrectly [4], these models assume the condition that particle spin polariza-tion emitted at freeze-out hypersurface is defined by the thermodynamicalquantity which is named as thermal vorticity [6, 51], not considering thefact that it may evolve independently during the expansion of the fluid. Inthis article, we follow scheme proposed in Refs. [52–57], and analyze suchpossibility of spin polarization evolution using relativistic hydrodynamicsframework with spin.
2. Distribution functions in equilibrium
If we know phase space distribution function for the system’s equilibriumstate, then it is possible to derive relativistic hydrodynamics from the under-lying kinetic theory definitions [58]. Following ideas developed by Becattini ∗ Presented at Criticality in QCD and the Hadron Resonance Gas, July 29-31, 2020. (1) a r X i v : . [ nu c l - t h ] S e p proceedings˙Rajeev printed on September 16, 2020 et al . [6], we take into consideration the following distribution functions forthe relativistic systems of spin / massive particles (and antiparticles) inthe local equilibrium state. f + rs ( x, p ) = ¯ u r ( p ) X + u s ( p ) , f − rs ( x, p ) = − ¯ v s ( p ) X − v r ( p ) , (1)where x and p is the space-time position coordinate and the four-vector mo-mentum, respectively, with u r ( p ) and v r ( p ) being the Dirac bispinors ( r, s =1 , u r ( p ) u s ( p ) = δ rs and ¯ v r ( p ) v s ( p ) = − δ rs , here δ rs is the Dirac delta function and, X ± have the following form in terms of relativistic Boltzmann distributions X ± = exp (cid:20) ± ξ ( x ) − β µ ( x ) p µ ± ω µν ( x )Σ µν (cid:21) , where β µ ≡ U µ /T and ξ ≡ µ/T , here T , µ and U µ is the temperature,baryon chemical potential and four-vector velocity, respectively, and ω µν isthe second rank asymmetric tensor known as spin polarization tensor withΣ µν ≡ i [ γ µ , γ ν ] being the spin operator.With the help of the expressions from Ref. [59] and Eqs. (1) we canobtain the Wigner functions in equilibrium as follows W ± eq ( x, k ) = e ± ξ m (cid:90) dP e − β · p δ (4) ( k ∓ p ) (2) × (cid:20) m ( m ± /p ) cosh( ζ ) ± sinh( ζ )2 ζ ω µν ( /p ± m )Σ µν ( /p ± m ) (cid:21) , where k being the off mass-shell particle four-momentum, dP = d p/ ((2 π ) E p )is the invariant measure with E p = (cid:112) m + p denoting the on-shell energyof the particle, and ζ = √ √ ω µν ω µν in temrs of spin polarization tensor.Wigner function can also be expanded using Clifford-algebra expansion (2) W ± eq ( x, k ) = 14 (cid:2) F ± eq ( x, k ) + iγ P ± eq ( x, k ) + γ µ V ± eq ,µ ( x, k )+ γ γ µ A ± eq ,µ ( x, k ) + Σ µν S ± eq ,µν ( x, k ) (cid:3) , where X ∈ {F , P , V µ , A µ , S νµ } are the coefficient functions of the Wignerfunction, which can obtained from the trace of W ± eq ( x, k ) multiplying firstby: { , − iγ , γ µ , γ µ γ , µν } .
3. Kinetic and hydrodynamical equations
The kinetic equation to be followed by Wigner function is( γ µ K µ − m ) W ( x, k ) = C [ W ( x, k )] , (3) roceedings˙Rajeev printed on September 16, 2020 with K µ = k µ + i ¯ h ∂ µ . For global equilibrium state, the Wigner functionfollows exactly Eq. (3) with the collision term C [ W ( x, k )] = 0. The widelyused method of treating Eq. (3) is the semi-classical expansion method ofthe coefficient functions of the Wigner function X = X (0) + ¯ h X (1) + ¯ h X (2) + · · · . Up to the first order (i.e. next-to-leading order) in ¯ h the treatment of Eq. (3)gives the following kinetic equations to be followed by the two independentcoefficients which are: F eq and A ν eq , k µ ∂ µ F eq ( x, k ) = 0 , k µ ∂ µ A ν eq ( x, k ) = 0 , k ν A ν eq ( x, k ) = 0 . (4)In the case of global equilibrium Eqs. (4) are satisfied exactly which in-turn yields the conditions that β µ is a Killing vector, whereas, ξ and ω µν are constant, but ω µν does not necessarily be equal to thermal vorticity (cid:36) µν = − ( ∂ µ β ν − ∂ ν β µ ) = const. But in the case of local equilibriumEqs. (4) are not exactly followed, here we follow [60] and by permitting β , ξ and ω dependence on x , we need to have only certain moments inmomentum space of the kinetic equations (4) which are satisfied, whichlead to conservation laws for charge, energy-linear momentum and spin [56] ∂ µ N µ = 0 , (5) ∂ µ T µν GLW = 0 , (6) ∂ λ S λ,αβ GLW = 0 , (7)here the baryon current, the energy-momentum and the spin tensors arebased on the forms by the de Groot - van Leeuwen - van Weert (GLW) [59] N α = nU α , (8) T αβ GLW = ( ε + P ) U α U β − P g αβ , (9) S α,βγ GLW = cosh( ξ ) (cid:104) n (0) U α ω βγ + A (0) U α U δ U [ β ω γ ] δ (10)+ B (0) (cid:16) U [ β ∆ αδ ω γ ] δ + U α ∆ δ [ β ω γ ] δ + U δ ∆ α [ β ω γ ] δ (cid:17)(cid:105) , (11)where ∆ αβ = g αβ − U α U β is the spatial projection operator which is or-thogonal to the hydrodynamic flow 4-vector U .For the case of polarization tensor in the leading order, the baryon numberdensity, the energy density and, the pressure are expressed respectively as n = sinh( ξ ) n (0) ( T ) , (12) ε = cosh( ξ ) ε (0) ( T ) , (13) P = cosh( ξ ) P (0) ( T ) , (14) proceedings˙Rajeev printed on September 16, 2020 where for spin-less and neutral massive Boltzmann particles, thermodynam-ical properties are defined by [61] n (0) ( T ) = 2 T π ˆ m K ( ˆ m ) , (15) ε (0) ( T ) = 2 T π ˆ m (cid:104) K ( ˆ m ) + ˆ mK ( ˆ m ) (cid:105) , (16) P (0) ( T ) = T n (0) ( T ) . (17)Here, K ( ˆ m ) and K ( ˆ m ) are modified Bessel functions of 1st and 2nd kindrespectively. The thermodynamical quantities B (0) and A (0) are expressedas B (0) = − m s (0) ( T ) , A (0) = − B (0) + 2 n (0) ( T ) (18)with entropy density s (0) = (cid:0) ε (0) + P (0) (cid:1) /T and ˆ m = m/T .
4. Bjorken expansion set-up
Since the spin polarization tensor ω µν is a 2nd rank asymmetric tensor,so in analogy to the Faraday electromagnetic field strength tensor, it canbe written into electric-like ( κ ) and magnetic-like ( ω ) components ω µν = κ µ U ν − κ ν U µ + (cid:15) µναβ U α ω β , (19)where κ and ω are 4-vectors, orthogonal to fluid flow vector U µ . For longi-tudinal boost-invariant and transversely homogeneous systems [62, 63], onecan write the following basis vectors U α = 1 τ ( t, , , z ) = (cosh( η ) , , , sinh( η )) ,X α = (0 , , , ,Y α = (0 , , , ,Z α = 1 τ ( z, , , t ) = (sinh( η ) , , , cosh( η )) , (20)where longitudinal proper time is defined as τ = √ t − z and, the space-time rapidity is defined as η = ln(( t + z ) / ( t − z )). The normalizationconditions satisfied by the basis vectors (20) are U · U = 1 ,X · X = Y · Y = Z · Z = − ,X · U = Y · U = Z · U = 0 , (21) X · Y = Y · Z = Z · X = 0 . roceedings˙Rajeev printed on September 16, 2020 Using the fact that κ and ω are orthogonal to U µ and Eqs. (21), κ µ and ω µ can be written as κ α = C κX ( τ ) X α + C κY ( τ ) Y α + C κZ ( τ ) Z α ,ω α = C ωX ( τ ) X α + C ωY ( τ ) Y α + C ωZ ( τ ) Z α , (22)where one can notice that the scalar functions depend only on proper time( τ ).Putting Eqs. (22) in Eq. (7) and then using the projection method, weproject the resulting tensor on different combination of basis vectors U α X β , U α Y β , U α Z β , Y α Z β , X α Z β and X α Y β , we get the six equations of motionsas diag ( L , L , L , P , P , P ) ˙ C = diag ( Q , Q , Q , R , R , R ) C , (23)where C = ( C κX , C κY , C κZ , C ωX , C ωY , C ωZ ), ˙( . . . ) ≡ U · ∂ = ∂ τ and L ( τ ) = A − A − A , P ( τ ) = A , Q ( τ ) = − (cid:20) ˙ L + 1 τ (cid:18) L + 12 A (cid:19)(cid:21) , Q ( τ ) = − (cid:18) ˙ L + L τ (cid:19) , R ( τ ) = − (cid:20) ˙ P + 1 τ (cid:18) P − A (cid:19)(cid:21) , R ( τ ) = − (cid:18) ˙ P + P τ (cid:19) . with A = cosh( ξ ) (cid:0) n (0) − B (0) (cid:1) , A = cosh( ξ ) (cid:0) A (0) − B (0) (cid:1) , A = cosh( ξ ) B (0) , Eqs. (23) implies that the C functions evolve independently of each other forthe case of Bjorken flow and, C κX and C κY (similarly C ωX and C ωY ) followsthe same form of evolution equations due to the rotational invariance.Charge current conservation (5) for Bjorken type flow is expressed as dndτ + nτ = 0 (24) proceedings˙Rajeev printed on September 16, 2020 whereas the energy and linear momentum conservation law (6) (after pro-jecting on U ) yields dεdτ + ( ε + P ) τ = 0 . (25)
5. Particle spin polarization at freeze-out
To calculate the mean spin polarization per particle, the following for-mula is used [56] (cid:104) π µ (cid:105) = E p d Π µ ( p ) d p /E p d N ( p ) d p , (26)with E p d Π µ ( p ) d p being the total value of the Pauli-Luba´nski vector (after in-tegrating over the freeze-out hypersuface, ∆Σ λ ), E p d Π µ ( p ) d p = − cosh( ξ )(2 π ) m (cid:90) ∆Σ λ p λ e − β · p ˜ ω µβ p β , and E p d N ( p ) d p = 4 cosh( ξ )(2 π ) (cid:90) ∆Σ λ p λ e − β · p , is the total momentum density of both particles and antiparticles with four-momentum given as p λ = ( m T cosh y p , p x , p y , m T sinh y p ).After performing the the canonical boost [64] of (26), we obtain the polar-ization vector (cid:104) π (cid:63)µ (cid:105) in the local rest frame of the particle as (cid:104) π (cid:63)µ (cid:105) = − m (cid:16) p x sinh y p b (cid:17) a i + (cid:16) χ p x cosh y p b (cid:17) a j +2 C κZ p y − χC ωX m T (cid:16) p y sinh y p b (cid:17) a i + (cid:16) χ p y cosh y p b (cid:17) a j − C κZ p x − χC ωY m T − (cid:16) m cosh y p + m T b (cid:17) a i − (cid:16) χ m sinh y p b (cid:17) a j , (27)with a i = χ ( C κX p y − C κY p x ) + 2 C ωZ m T , a j = C ωX p x + C ωY p y , b = m T cosh y p + m , and χ = ( K ( ˆ m T ) + K ( ˆ m T )) /K ( ˆ m T ) and ˆ m T = m T /T . roceedings˙Rajeev printed on September 16, 2020 Fig. 1. Dependence of the temperature re-scaled by its initial value (solid blackline) and the ratio of baryon chemical potential over temperature re-scaled by theinitial ratio (dotted blue line) on the proper-time.
Fig. 2. Dependence of scalar functions C κX (solid black line), C κZ (dashed-dottedblue line), C ωX (dotted red line) and C ωZ (dashed green line) on the proper-time.
6. Results
Here we show the solutions of the differential equations (23), (24), and(25). System is initialized at the initial proper time τ = 1 fm with initialtemperature and the initial baryon chemical potential as T = T ( τ ) = 150MeV and µ = µ ( τ ) = 800 MeV, respectively. Here the system is as-sumed to be formed with Λ particles having mass m = 1116 MeV. In Fig. 1,proper-time dependence of temperature and baryon chemical potential isdepicted, where the temperature decreases with proper-time, whereas theratio of baryon chemical potential and temperature increases with proper-time. From Fig. 2, proper time dependence of the C functions can be knowndescribing the spin polarization evolution of the system.Using the information of the thermodynamic parameters and C coefficients proceedings˙Rajeev printed on September 16, 2020 - - - - - - - - - - - - - - - - - - - - - - - Fig. 3. Different components of the mean polarization of Λ particles in the restframe of the particle obtained with the initial values µ = 800 MeV, T = 155 MeV, C κ, = (0 , ,
0) and C ω, = (0 , . ,
0) for y p = 0. evolution, we can calculate the different components of the mean polariza-tion vector in the rest frame of the particles (cid:104) π (cid:63)µ (cid:105) at freeze-out, see Fig. 3.We note that (cid:104) π (cid:63)y (cid:105) is negative reflecting the system’s initial spin polarization.Because of the Bjorken symmetry which we have assumed in our calcula-tions in this article, the longitudinal component ( (cid:104) π (cid:63)z (cid:105) ) of the mean polar-ization vector is vanishing which is not in agreement with the quadrupolestructure of the longitudinal component of the spin polarization seen in theexperiment. But we note here that (cid:104) π (cid:63)x (cid:105) shows quadrupole structure. Wehowever see and note that the Bjorken set-up is very simple to address themeasurements done by the experiment.
7. Summary
We briefly presented the key ingredients of relativistic perfect-fluid hy-drodynamics with spin framework initiated recently. From the definitionsof kinetic theory for the equilibrium phase space distribution functions inthe local equilibrium we obtained the equations of motions for the expan-sion of the the system. For the case of Bjorken type of flow we investigatedthe system’s spin polarization dynamics, which in turn show that the scalarfunctions describing the dynamics of the spin polarization evolve indepen-dently of each other. These results are used to obtain the particle spinpolarization at the freeze-out hypersurface. We however note that withinthe current simple set-up of Bjorken symmetry experimental measurementscannot be addressed properly.
Acknowledgments
This research is supported in part by the Polish National Science CenterGrants No. 2016/23/B/ST2/00717 and No. 2018/30/E/ST2/00432. roceedings˙Rajeev printed on September 16, 2020 REFERENCES [1] L. Adamczyk et al. [STAR Collaboration],
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