Spin polarization issue in heavy-ions collisions Rajeev Singh rajeev.singh@ifj.edu.pl Ph.D. Advisors: Radoslaw Ryblewski (IFJ PAN) and Wojciech Florkowski (IF UJ) Primary References: Phys. Rev. C 99, 044910 (2019) Prog. Part. Nucl. Phys. 108 (2019) 103709 November 16 - 20, 2020 YITP Workshop: Strings and Fields 2020 Online Rajeev Singh (IFJ PAN) Spin Hydrodynamics 1 / 36
Outline: Motivation 1 Spin polarization tensor and perfect fluid hydrodynamics for spin 1/2 2 particles Boost-invariant flow and spin polarization 3 Numerical solutions 4 Physical observable: Spin polarization 5 Results and plots 6 Summary 7 Rajeev Singh (IFJ PAN) Spin Hydrodynamics 2 / 36
Motivation: Rajeev Singh (IFJ PAN) Spin Hydrodynamics 3 / 36
Motivation: Non-central relativistic heavy ion collisions creates global rotation of matter. This may induce spin polarization reminding us of Einstein and De-Haas effect and Barnett effect. Rajeev Singh (IFJ PAN) Spin Hydrodynamics 4 / 36
Einstein-De Haas Effect (1915): Rotation induced by Magnetization Rajeev Singh (IFJ PAN) Spin Hydrodynamics 5 / 36
Barnett Effect (1915): Magnetization induced by Rotation Rajeev Singh (IFJ PAN) Spin Hydrodynamics 6 / 36
Motivation: Non-central relativistic heavy ion collisions creates global rotation of matter. This may induce spin polarization reminding us of Barnett effect and Einstein and de-Haas effect. Emerging particles are expected to be globally polarized with their spins on average pointing along the systems angular momentum. Source: CERN Courier Rajeev Singh (IFJ PAN) Spin Hydrodynamics 7 / 36
Other works: Other theoretical models used for the heavy-ions data interpretation dealt mainly with the spin polarization of particles at freeze-out, where the basic hydrodynamic quantity giving rise to spin polarization is the ‘thermal vorticity’ expressed as ̟ µν = − 1 2 ( ∂ µ β ν − ∂ ν β µ ). F. Becattini et.al. (Annals Phys. 338 (2013)), F. Becattini, L. Csernai, D. J. Wang (PRC 88, 034905), F. Becattini et.al. (PRC 95, 054902), Iu. Karpenko, F. Becattini (EPJC (2017) 77: 213), F. Becattini, Iu. Karpenko(PRL 120, 012302 (2018)) Rajeev Singh (IFJ PAN) Spin Hydrodynamics 8 / 36
Our hydrodynamic framework: Solving the standard perfect-fluid hydrodynamic equations without spin Rajeev Singh (IFJ PAN) Spin Hydrodynamics 9 / 36
Our hydrodynamic framework: Solving the standard perfect-fluid hydrodynamic equations without spin Determination of the spin evolution in the hydrodynamic background Rajeev Singh (IFJ PAN) Spin Hydrodynamics 10 / 36
Our hydrodynamic framework: Solving the standard perfect-fluid hydrodynamic equations without spin Determination of the spin evolution in the hydrodynamic background Determination of the Pauli-Lubanski (PL) vector on the freeze-out hypersurface Rajeev Singh (IFJ PAN) Spin Hydrodynamics 11 / 36
Our hydrodynamic framework: Solving the standard perfect-fluid hydrodynamic equations without spin. Determination of the spin evolution in the hydrodynamic background. Determination of the Pauli-Luba´ nski (PL) vector on the freeze-out hypersurface. Calculation of the spin polarization of particles in their rest frame. The spin polarization obtained is a function of the three-momenta of particles and can be directly compared with the experiment. Rajeev Singh (IFJ PAN) Spin Hydrodynamics 12 / 36
Our hydrodynamic framework: In this work, we use relativistic hydrodynamic equations for polarized spin 1 / 2 particles to determine the space-time evolution of the spin polarization in the system using forms of the energy-momentum and spin tensors proposed by de Groot, van Leeuwen, and van Weert (GLW). S. R. De Groot,Relativistic Kinetic Theory. Principles and Applications (1980). Rajeev Singh (IFJ PAN) Spin Hydrodynamics 13 / 36
Our hydrodynamic framework: In this work, we use relativistic hydrodynamic equations for polarized spin 1 / 2 particles to determine the space-time evolution of the spin polarization in the system using forms of the energy-momentum and spin tensors proposed by de Groot, van Leeuwen, and van Weert (GLW). S. R. De Groot,Relativistic Kinetic Theory. Principles and Applications (1980). The calculations are done in a boost-invariant and transversely homogeneous setup. We show how the formalism of hydrodynamics with spin can be used to determine physical observables related to the spin polarization required for the modelling of the experimental data. Wojciech Florkowski et.al. (Phys. Rev. C 99, 044910), Wojciech Florkowski et.al. (Phys. Rev. C 97, 041901), Wojciech Florkowski et.al. (Phys. Rev. D 97, 116017). Rajeev Singh (IFJ PAN) Spin Hydrodynamics 14 / 36
Our hydrodynamic framework: In this work, we use relativistic hydrodynamic equations for polarized spin 1 / 2 particles to determine the space-time evolution of the spin polarization in the system using forms of the energy-momentum and spin tensors proposed by de Groot, van Leeuwen, and van Weert (GLW). S. R. De Groot,Relativistic Kinetic Theory. Principles and Applications (1980). The calculations are done in a boost-invariant and transversely homogeneous setup. We show how the formalism of hydrodynamics with spin can be used to determine physical observables related to the spin polarization required for the modelling of the experimental data. Wojciech Florkowski et.al. (Phys. Rev. C 99, 044910), Wojciech Florkowski et.al. (Phys. Rev. C 97, 041901), Wojciech Florkowski et.al. (Phys. Rev. D 97, 116017). Our hydrodynamic formulation does not allow for arbitrary large values of the spin polarization tensor, hence we have restricted ourselves to the leading order terms in the ω µν . Rajeev Singh (IFJ PAN) Spin Hydrodynamics 15 / 36
Spin polarization tensor: The spin polarization tensor ω µν is anti-symmetric and can be defined by the four-vectors κ µ and ω µ , ω µν = κ µ U ν − κ ν U µ + ǫ µναβ U α ω β , Note that, any part of the 4-vectors κ µ and ω µ which is parallel to U µ does not contribute, therefore κ µ and ω µ satisfy the following orthogonality conditions: κ · U = 0 , ω · U = 0 We can express κ µ and ω µ in terms of ω µν , namely ω µ = 1 κ µ = ω µα U α , 2 ǫ µαβγ ω αβ U γ Rajeev Singh (IFJ PAN) Spin Hydrodynamics 16 / 36
Conservation of charge: ∂ α N α ( x ) = 0, N α = nU α , where, n = 4 sinh( ξ ) n (0) ( T ). The quantity n (0) ( T ) defines the number density of spinless and neutral massive Boltzmann particles, 2 π 2 T 3 ˆ 1 m 2 K 2 ( ˆ n (0) ( T ) = � p · U � 0 = m ) dP ( · · · ) e − β · p denotes the thermal average, � where, � · · · � 0 ≡ m ≡ m / T denotes the ratio of the particle mass ( m ) and the temperature ˆ ( T ), and K 2 ( ˆ m ) denotes the modified Bessel function. e ξ − e − ξ � � The factor, 4 sinh( ξ ) = 2 accounts for spin degeneracy and presence of both particles and antiparticles in the system and the variable ξ denotes the ratio of the baryon chemical potential µ and the temperature T , ξ = µ/ T . Rajeev Singh (IFJ PAN) Spin Hydrodynamics 17 / 36
Conservation of energy and linear momentum: ∂ α T αβ GLW ( x ) = 0 where the energy-momentum tensor T αβ GLW has the perfect-fluid form: GLW ( x ) = ( ε + P ) U α U β − Pg αβ T αβ with energy density ε = 4 cosh( ξ ) ε (0) ( T ) and pressure P = 4 cosh( ξ ) P (0) ( T ) The auxiliary quantities are: ε (0) ( T ) = � ( p · U ) 2 � 0 and P (0) ( T ) = − (1 / 3) � p · p − ( p · U ) 2 � 0 are the energy density and pressure of the spin-less ideal gas respectively. In case of ideal relativistic gas of classical massive particles, m 2 � � 2 π 2 T 4 ˆ 1 ε (0) ( T ) = 3 K 2 ( ˆ m ) + ˆ mK 1 ( ˆ m ) , P (0) ( T ) = Tn (0) ( T ) where, K 1 and K 2 are the modified Bessel functions of 1st and 2nd kind respectively. Rajeev Singh (IFJ PAN) Spin Hydrodynamics 18 / 36
Conservation of energy and linear momentum: Above conservation laws (charge and energy-linear momentum) provide closed system of five equations for five unknown functions: ξ , T , and three independent components of U µ (hydrodynamic flow vector) which needs to be solved to get the hydrodynamic background. Rajeev Singh (IFJ PAN) Spin Hydrodynamics 19 / 36
Conservation of total angular momentum: ∂ µ J µ,αβ ( x ) = 0, J µ,αβ ( x ) = − J µ,βα ( x ) Total angular momentum consists of orbital and spin parts: J µ,αβ ( x ) = L µ,αβ ( x ) + S µ,αβ ( x ), L µ,αβ ( x ) = x α T µβ ( x ) − x β T µα ( x ) Since the energy-momentum tensor is symmetric, the conservation of the angular momentum implies the conservation of its spin part. ∂ λ J λ,µν ( x ) = 0, ∂ µ T µν ( x ) = 0 ⇒ ∂ λ S λ,µν ( x ) = T νµ ( x ) − T µν ( x ) = Hence, the spin tensor S µ,αβ ( x ) is separately conserved in GLW formulation. Rajeev Singh (IFJ PAN) Spin Hydrodynamics 20 / 36
Conservation of spin angular momentum: ∂ α S α,βγ GLW ( x ) = 0 GLW spin tensor in the leading order of ω µν is: � � n (0) ( T ) U α ω βγ + S α,βγ S α,βγ GLW = cosh( ξ ) ∆ GLW Here, ω βγ is known as spin polarization tensor, whereas the auxiliary tensor S α,βγ ∆ GLW is: S α,βγ ∆ GLW = A (0) U α U δ U [ β ω γ ] δ � � U [ β ∆ αδ ω γ ] δ + U α ∆ δ [ β ω γ ] δ + U δ ∆ α [ β ω γ ] + B (0) , δ with, B (0) = − 2 m 2 s (0) ( T ) ˆ A (0) = − 3 B (0) + 2 n (0) ( T ) Rajeev Singh (IFJ PAN) Spin Hydrodynamics 21 / 36
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